    1 Introduction
        1.1 ALGLIB license
        1.2 Documentation license
        1.3 Reference Manual and User Guide
        1.4 Acknowledgements
    2 Getting started with ALGLIB
        2.1 ALGLIB structure
            2.1.1 Packages
            2.1.2 Subpackages
        2.2 Compatibility
        2.3 Compiling ALGLIB
            2.3.1 Adding to your project
        2.4 Using ALGLIB
            2.4.1 ALGLIB classes
            2.4.2 Datatypes
            2.4.3 Functions
            2.4.4 Using functions: 'expert' and 'friendly' interfaces
        2.5 Advanced topics
            2.5.1 Testing ALGLIB
    3 ALGLIB reference manual

# 1 Introduction

## 1.1 ALGLIB license

ALGLIB is a free software which uses dual licensing. You can either use it under GPL license (version 2 or at your option any later version) or buy commercial license without copyleft requirement. A copy of the GNU General Public License is available at http://www.fsf.org/licensing/licenses A copy of the commercial license can be found at http://www.alglib.net/commercial.php

## 1.2 Documentation license

This reference manual is licensed under BSD-like documentation license:

Copyright 1994-2009 Sergey Bochkanov, ALGLIB Project. All rights reserved.

Redistribution and use of this document (ALGLIB Reference Manual) with or without modification, are permitted provided that such redistributions will retain the above copyright notice, this condition and the following disclaimer as the first (or last) lines of this file.

THIS DOCUMENTATION IS PROVIDED BY THE ALGLIB PROJECT "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE ALGLIB PROJECT BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS DOCUMENTATION, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

## 1.3 Reference Manual and User Guide

ALGLIB Project provides two sources of information: ALGLIB Reference Manual (this document) and ALGLIB User Guide.

ALGLIB Reference Manual contains full description of all publicly accessible ALGLIB units accompanied with examples. Reference Manual is focused on the source code: it documents units, functions, structures and so on. If you want to know what unit `YYY` can do or what subroutines unit `ZZZ` contains Reference Manual is a place to go. Free software needs free documentation - that's why ALGLIB Reference Manual is licensed under BSD-like documentation license.

Additionally to the Reference Manual we provide you User Guide. User Guide is focused on more general questions: how fast ALGLIB is? how reliable it is? what are the strong and weak sides of the algorithms used? We aim to make ALGLIB User Guide an important source of information both about ALGLIB and numerical analysis algorithms in general. We want it to be a book about algorithms, not just software documentation. And we want it to be unique - that's why ALGLIB User Guide is distributed under less-permissive personal-use-only license.

## 1.4 Acknowledgements

ALGLIB was not possible without contribution of the next open source projects:

- LAPACK
- Cephes
- GNU MP
- MPFR

# 2 Getting started with ALGLIB

## 2.1 ALGLIB structure

### 2.1.1 Packages

ALGLIB for C# is a pure C# 2.0 library automatically generated by code generation tools developed within ALGLIB project. Pre-3.0 versions of ALGLIB included more than 100 units, but it was difficult to work with such large number of files. Since ALGLIB 3.0 all units are merged into 11 packages and two support units:

- alglibmisc.cs - contains different algorithms which are hard to classify
- dataanalysis.cs - contains data mining algorithms
- diffequations.cs - contains differential equation solvers
- fasttransforms.cs - contains FFT and other related algorithms
- integration.cs - contains numerical integration algorithms
- interpolation.cs - contains interpolation algorithms
- linalg.cs - contains linear algebra algorithms
- optimization.cs - contains optimization algorithms
- solvers.cs - contains linear and nonlinear solvers
- specialfunctions.cs - contains special functions
- statistics.cs - statistics
- alglibinternal.cs - contains internal functions which are used by other packages, but not exposed to the external world
- ap.cs - contains publicly accessible vector/matrix classes, most important and general functions and other "basic" functionality

One package may rely on other ones, but we have tried to reduce number of dependencies. Every package relies on `ap.cs` and many packages rely on `alglibinternal.cs`. But many packages require only these two to work, and many other packages need significantly less than 13 packages. For example, `statistics.cs` requires two packages mentioned above and only one additional package - `specialfunctions.cs`.

### 2.1.2 Subpackages

There is one more concept to learn - subpackages. Every package was created from several source files. For example (as of ALGLIB 3.0.0), `linalg.cs` was created by merging together 14 .cs files. These files provide different functionality: one of them calculates triangular factorizations, another generates random matrices, and so on. We've merged source code, but what to do with their documentation?

Of course, we can merge their documentation (as we've merged units) in one big list of functions and data structures, but such list will be hard to read. Instead, we have decided to merge source code, but leave documentation separate.

If you look at the list of ALGLIB packages, you will see that each package includes several subpackages. For example, `linalg.cs` includes `trfac`, `svd`, `evd` and other subpackages. These subpackages do no exist as separate files, namespaces or other entities. They are just subsets of one large file, one large class, which provide significantly different functionality. They have separate documentation sections, but if you want to use `svd` subpackage, you have to compile `linalg.cs`, not `svd.cs`.

## 2.2 Compatibility

ALGLIB for C# is compatible with:

- any kind of optimization settings
- any CPU which supports .NET
- both Microsoft-provided implementation of .NET framework (2.0 or higher) and open source implementation of .NET (Mono).
- any C# compiler which support C# 2.0 or later (MS C# compiler or Mono C# compiler)
- any OS where .NET framework exists

## 2.3 Compiling ALGLIB

### 2.3.1 Adding to your project

Adding ALGLIB to your project is easy - just pick packages you need and... add them to your project! It will work without any additional settings.

As you see, ALGLIB has no project files. Why? There are several reasons:

- first, because many ALGLIB users don't need separate shared library - they prefer to integrate source code in their projects. We have provided script-based build system before, but majority of our users prefer to build ALGLIB themselves.
- second, because we want ALGLIB to be usable in any programming environment, whether it is Visual Studio or Mono. The best solution is to write package which doesn't depend on any particular programming environment.

In any case, compiling ALGLIB is so simple that even without project file you can do it in several minutes.

## 2.4 Using ALGLIB

### 2.4.1 ALGLIB classes

All ALGLIB functionality is provided by one large class - `alglib`.

### 2.4.2 Datatypes

ALGLIB defines several "basic" datatypes (types which are used across many packages) and many package-specific datatypes. "Basic" datatypes are defined in `ap.cs`. Here is short list:

- `alglib.complex` - double precision complex datatype, which supports arithmetic operations
- `alglib.alglibexception` - exception which is thrown by library
- `alglib.ndimensional_func` - delegate used by optimizers/solvers
- `alglib.ndimensional_grad` - delegate used by optimizers/solvers
- `alglib.ndimensional_hess` - delegate used by optimizers/solvers
- `alglib.ndimensional_jac` - delegate used by optimizers/solvers
- `alglib.ndimensional_pfunc` - delegate used by optimizers/solvers
- `alglib.ndimensional_pgrad` - delegate used by optimizers/solvers
- `alglib.ndimensional_phess` - delegate used by optimizers/solvers
- `alglib.ndimensional_rep` - delegate used by optimizers/solvers
- `alglib.ndimensional_ode_rp` - delegate used by ODE solver
- `alglib.integrator1_func` - delegate used by 1-dimensional integrator

Package-specific datatypes are classes which can be divided into two distinct groups:

- "struct-like" classes with public fields
- "object-like" classes which have no public fields. You should use ALGLIB functions to work with them.

### 2.4.3 Functions

The most important "basic" functions from `alglib` class are:

- `alglib.randomreal()` - returns random real number from [0,1)
- `alglib.randominteger(mx)` - returns random integer number from [0,nx)
- `alglib.sqr(x)` - returns square of real x
- `alglib.csqr(z)` - returns square of complex z
- `alglib.conj(z)` - returns conjugate of complex z
- `alglib.abscomplex(z)` - returns magnitude of complex z

### 2.4.4 Using functions: 'expert' and 'friendly' interfaces

Most ALGLIB functions provide two interfaces: 'expert' and 'friendly'. What is the difference between two? When you use 'friendly' interface, ALGLIB:

- tries to automatically determine size of input arguments
- throws an exception when arguments have inconsistent size (for example, square matrix is expected, but non-square is passed; another example - two parameters must have same size, but have different size)
- if semantics of input parameter assumes that it is symmetric/Hermitian matrix, checks that lower triangle is equal to upper triangle (conjugate of upper triangle) and throws an exception otherwise
- if semantics of output parameter assumes that it is symmetric/Hermitian matrix, returns full matrix (both upper and lower triangles)

When you use 'expert' interface, ALGLIB:

- requires caller to explicitly specify size of input arguments. If vector/matrix is larger than size being specified (say, N), only N leading elements are used
- if semantics of input parameter assumes that it is symmetric/Hermitian matrix, uses only upper or lower triangle of input matrix and requires caller to specify what triangle to use
- if semantics of output parameter assumes that it is symmetric/Hermitian matrix, returns only upper or lower triangle (when you look at specific function, it is clear what triangle is returned)

Here are several examples of 'friendly' and 'expert' interfaces:

    double[] x = new double[]{0,1,2,3};
    double[] y = new double[]{1,5,3,9};
    double[] y2 = new double[]{1,5,3,9,0};
    alglib.spline1dinterpolant s;

    alglib.spline1dbuildlinear(x, y, 4, out s);  // 'expert' interface is used
    alglib.spline1dbuildlinear(x, y, out s);     // 'friendly' interface - input size is
                                                 // automatically determined

    alglib.spline1dbuildlinear(x, y2, 4, out s); // y2.length() is 5, but it will work

    alglib.spline1dbuildlinear(x, y2, out s);    // it won't work because sizes of x and y2
                                                 // are inconsistent

'Friendly' interface - matrix semantics:

    double[,] a;
    alglib.matinvreport  rep;
    alglib.ae_int_t      info;

    //
    // 'Friendly' interface: spdmatrixinverse() accepts and returns symmetric matrix
    //

    // symmetric positive definite matrix
    a = new double[,]{{2,1},{1,2}};

    // after this line A will contain [[0.66,-0.33],[-0.33,0.66]]
    // which is symmetric too
    alglib.spdmatrixinverse(ref a, out info, out rep);

    // you may try to pass nonsymmetric matrix
    a = new double[,]{{2,1},{0,2}};

    // but exception will be thrown in such case
    alglib.spdmatrixinverse(ref a, out info, out rep);

Same function but with 'expert' interface:

    double[,] a;
    alglib.matinvreport  rep;
    alglib.ae_int_t      info;

    //
    // 'Expert' interface, spdmatrixinverse()
    //

    // only upper triangle is used; a[1][0] is initialized by NAN,
    // but it can be arbitrary number
    a = new double[,]{{2,1},{Double.NaN,2}};

    // after this line A will contain [[0.66,-0.33],[NAN,0.66]]
    // only upper triangle is modified
    alglib.spdmatrixinverse(ref a, 2 /* N */, true /* upper triangle is used */, out info, out rep);

## 2.5 Advanced topics

### 2.5.1 Testing ALGLIB

There are two test suites in ALGLIB: computational tests and interface tests. Computational tests are located in `/tests/test_c.cs`. They are focused on numerical properties of algorithms, stress testing and "deep" tests (large automatically generated problems). They require significant amount of time to finish (tens of minutes).

Interface tests are located in `/tests/test_i.cs`. These tests are focused on ability to correctly pass data between computational core and caller, ability to detect simple problems in inputs, and on ability to at least compile ALGLIB with your compiler. They are very fast (about a minute to finish including compilation time).

Running test suite is easy - just

1. compile one of these files (`test_c.cs` or `test_i.cs`) along with the rest of the library
2. launch executable you will get. It may take from several seconds (interface tests) to several minutes (computational tests) to get final results

However, there is no strong reasons to launch tests before using ALGLIB. It was tested with many different compiler settings, and the very nature of .NET framework and its inherent portability allows us to say that ALGLIB for .NET will work on any system.

# 3 ALGLIB reference manual

## Packages and subpackages

`AlglibMisc` package
hqrnd |  | High quality random numbers generator
nearestneighbor |  | Nearest neighbor search: approximate and exact
`DataAnalysis` package
bdss |  | Basic dataset functions
dforest |  | Decision forest classifier (regression model)
kmeans |  | K-means++ clustering
lda |  | Linear discriminant analysis
linreg |  | Linear models
logit |  | Logit models
mcpd |  |
mlpbase |  | Basic neural network operations
mlpe |  | Neural network ensemble models
mlptrain |  | Neural network training
pca |  | Principal component analysis
`DiffEquations` package
odesolver |  | Ordinary differential equation solver
`FastTransforms` package
conv |  | Fast real/complex convolution
corr |  | Fast real/complex cross-correlation
fft |  | Real/complex FFT
fht |  | Real Fast Hartley Transform
`Integration` package
autogk |  | Adaptive 1-dimensional integration
gkq |  | Gauss-Kronrod quadrature generator
gq |  | Gaussian quadrature generator
`Interpolation` package
idwint |  | Inverse distance weighting: interpolation/fitting
lsfit |  | Linear and nonlinear least-squares solvers
polint |  | Polynomial interpolation/fitting
pspline |  | Parametric spline interpolation
ratint |  | Rational interpolation/fitting
spline1d |  | 1D spline interpolation/fitting
spline2d |  | 2D spline interpolation
`LinAlg` package
ablas |  | Level 2 and Level 3 BLAS operations
bdsvd |  | Bidiagonal SVD
evd |  | Eigensolvers
inverseupdate |  | Sherman-Morrison update of the inverse matrix
matdet |  | Determinant calculation
matgen |  | Random matrix generation
matinv |  | Matrix inverse
ortfac |  | Real/complex QR/LQ, bi(tri)diagonal, Hessenberg decompositions
rcond |  | Condition number estimate
schur |  | Schur decomposition
spdgevd |  | Generalized symmetric eigensolver
svd |  | Singular value decomposition
trfac |  | LU and Cholesky decompositions
`Optimization` package
minbleic |  | Bound constrained optimizer with additional linear equality/inequality constraints
mincg |  | Conjugate gradient optimizer
mincomp |  | Backward compatibility functions
minlbfgs |  | Limited memory BFGS optimizer
minlm |  | Improved Levenberg-Marquardt optimizer
minqp |  | Quadratic programming with bound and linear equality/inequality constraints
`Solvers` package
densesolver |  | Dense linear system solver
nleq |  | Solvers for nonlinear equations
`SpecialFunctions` package
airyf |  | Airy functions
bessel |  | Bessel functions
betaf |  | Beta function
binomialdistr |  | Binomial distribution
chebyshev |  | Chebyshev polynomials
chisquaredistr |  | Chi-Square distribution
dawson |  | Dawson integral
elliptic |  | Elliptic integrals
expintegrals |  | Exponential integrals
fdistr |  | F-distribution
fresnel |  | Fresnel integrals
gammafunc |  | Gamma function
hermite |  | Hermite polynomials
ibetaf |  | Incomplete beta function
igammaf |  | Incomplete gamma function
jacobianelliptic |  | Jacobian elliptic functions
laguerre |  | Laguerre polynomials
legendre |  | Legendre polynomials
normaldistr |  | Normal distribution
poissondistr |  | Poisson distribution
psif |  | Psi function
studenttdistr |  | Student's t-distribution
trigintegrals |  | Trigonometric integrals
`Statistics` package
basestat |  | Mean, variance, covariance, correlation, etc.
correlationtests |  | Hypothesis testing: correlation tests
jarquebera |  | Hypothesis testing: Jarque-Bera test
mannwhitneyu |  | Hypothesis testing: Mann-Whitney-U test
stest |  | Hypothesis testing: sign test
studentttests |  | Hypothesis testing: Student's t-test
variancetests |  | Hypothesis testing: F-test and one-sample variance test
wsr |  | Hypothesis testing: Wilcoxon signed rank test

## `ablas` subpackage

### Functions

cmatrixcopy
cmatrixgemm
cmatrixlefttrsm
cmatrixmv
cmatrixrank1
cmatrixrighttrsm
cmatrixsyrk
cmatrixtranspose
rmatrixcopy
rmatrixgemm
rmatrixlefttrsm
rmatrixmv
rmatrixrank1
rmatrixrighttrsm
rmatrixsyrk
rmatrixtranspose

### Examples

### `cmatrixcopy` function

    /*************************************************************************
    Copy

    Input parameters:
        M   -   number of rows
        N   -   number of columns
        A   -   source matrix, MxN submatrix is copied and transposed
        IA  -   submatrix offset (row index)
        JA  -   submatrix offset (column index)
        B   -   destination matrix
        IB  -   submatrix offset (row index)
        JB  -   submatrix offset (column index)
    *************************************************************************/
    public static void cmatrixcopy(
        int m,
        int n,
        complex[,] a,
        int ia,
        int ja,
        ref complex[,] b,
        int ib,
        int jb)

### `cmatrixgemm` function

    /*************************************************************************
    This subroutine calculates C = alpha*op1(A)*op2(B) +beta*C where:
    * C is MxN general matrix
    * op1(A) is MxK matrix
    * op2(B) is KxN matrix
    * "op" may be identity transformation, transposition, conjugate transposition

    Additional info:
    * cache-oblivious algorithm is used.
    * multiplication result replaces C. If Beta=0, C elements are not used in
      calculations (not multiplied by zero - just not referenced)
    * if Alpha=0, A is not used (not multiplied by zero - just not referenced)
    * if both Beta and Alpha are zero, C is filled by zeros.

    INPUT PARAMETERS
        N       -   matrix size, N>0
        M       -   matrix size, N>0
        K       -   matrix size, K>0
        Alpha   -   coefficient
        A       -   matrix
        IA      -   submatrix offset
        JA      -   submatrix offset
        OpTypeA -   transformation type:
                    * 0 - no transformation
                    * 1 - transposition
                    * 2 - conjugate transposition
        B       -   matrix
        IB      -   submatrix offset
        JB      -   submatrix offset
        OpTypeB -   transformation type:
                    * 0 - no transformation
                    * 1 - transposition
                    * 2 - conjugate transposition
        Beta    -   coefficient
        C       -   matrix
        IC      -   submatrix offset
        JC      -   submatrix offset

      -- ALGLIB routine --
         16.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixgemm(
        int m,
        int n,
        int k,
        complex alpha,
        complex[,] a,
        int ia,
        int ja,
        int optypea,
        complex[,] b,
        int ib,
        int jb,
        int optypeb,
        complex beta,
        ref complex[,] c,
        int ic,
        int jc)

### `cmatrixlefttrsm` function

    /*************************************************************************
    This subroutine calculates op(A^-1)*X where:
    * X is MxN general matrix
    * A is MxM upper/lower triangular/unitriangular matrix
    * "op" may be identity transformation, transposition, conjugate transposition

    Multiplication result replaces X.
    Cache-oblivious algorithm is used.

    INPUT PARAMETERS
        N   -   matrix size, N>=0
        M   -   matrix size, N>=0
        A       -   matrix, actial matrix is stored in A[I1:I1+M-1,J1:J1+M-1]
        I1      -   submatrix offset
        J1      -   submatrix offset
        IsUpper -   whether matrix is upper triangular
        IsUnit  -   whether matrix is unitriangular
        OpType  -   transformation type:
                    * 0 - no transformation
                    * 1 - transposition
                    * 2 - conjugate transposition
        C   -   matrix, actial matrix is stored in C[I2:I2+M-1,J2:J2+N-1]
        I2  -   submatrix offset
        J2  -   submatrix offset

      -- ALGLIB routine --
         15.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixlefttrsm(
        int m,
        int n,
        complex[,] a,
        int i1,
        int j1,
        bool isupper,
        bool isunit,
        int optype,
        ref complex[,] x,
        int i2,
        int j2)

### `cmatrixmv` function

    /*************************************************************************
    Matrix-vector product: y := op(A)*x

    INPUT PARAMETERS:
        M   -   number of rows of op(A)
                M>=0
        N   -   number of columns of op(A)
                N>=0
        A   -   target matrix
        IA  -   submatrix offset (row index)
        JA  -   submatrix offset (column index)
        OpA -   operation type:
                * OpA=0     =>  op(A) = A
                * OpA=1     =>  op(A) = A^T
                * OpA=2     =>  op(A) = A^H
        X   -   input vector
        IX  -   subvector offset
        IY  -   subvector offset

    OUTPUT PARAMETERS:
        Y   -   vector which stores result

    if M=0, then subroutine does nothing.
    if N=0, Y is filled by zeros.

      -- ALGLIB routine --

         28.01.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixmv(
        int m,
        int n,
        complex[,] a,
        int ia,
        int ja,
        int opa,
        complex[] x,
        int ix,
        ref complex[] y,
        int iy)

### `cmatrixrank1` function

    /*************************************************************************
    Rank-1 correction: A := A + u*v'

    INPUT PARAMETERS:
        M   -   number of rows
        N   -   number of columns
        A   -   target matrix, MxN submatrix is updated
        IA  -   submatrix offset (row index)
        JA  -   submatrix offset (column index)
        U   -   vector #1
        IU  -   subvector offset
        V   -   vector #2
        IV  -   subvector offset
    *************************************************************************/
    public static void cmatrixrank1(
        int m,
        int n,
        ref complex[,] a,
        int ia,
        int ja,
        ref complex[] u,
        int iu,
        ref complex[] v,
        int iv)

### `cmatrixrighttrsm` function

    /*************************************************************************
    This subroutine calculates X*op(A^-1) where:
    * X is MxN general matrix
    * A is NxN upper/lower triangular/unitriangular matrix
    * "op" may be identity transformation, transposition, conjugate transposition

    Multiplication result replaces X.
    Cache-oblivious algorithm is used.

    INPUT PARAMETERS
        N   -   matrix size, N>=0
        M   -   matrix size, N>=0
        A       -   matrix, actial matrix is stored in A[I1:I1+N-1,J1:J1+N-1]
        I1      -   submatrix offset
        J1      -   submatrix offset
        IsUpper -   whether matrix is upper triangular
        IsUnit  -   whether matrix is unitriangular
        OpType  -   transformation type:
                    * 0 - no transformation
                    * 1 - transposition
                    * 2 - conjugate transposition
        C   -   matrix, actial matrix is stored in C[I2:I2+M-1,J2:J2+N-1]
        I2  -   submatrix offset
        J2  -   submatrix offset

      -- ALGLIB routine --
         15.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixrighttrsm(
        int m,
        int n,
        complex[,] a,
        int i1,
        int j1,
        bool isupper,
        bool isunit,
        int optype,
        ref complex[,] x,
        int i2,
        int j2)

### `cmatrixsyrk` function

    /*************************************************************************
    This subroutine calculates  C=alpha*A*A^H+beta*C  or  C=alpha*A^H*A+beta*C
    where:
    * C is NxN Hermitian matrix given by its upper/lower triangle
    * A is NxK matrix when A*A^H is calculated, KxN matrix otherwise

    Additional info:
    * cache-oblivious algorithm is used.
    * multiplication result replaces C. If Beta=0, C elements are not used in
      calculations (not multiplied by zero - just not referenced)
    * if Alpha=0, A is not used (not multiplied by zero - just not referenced)
    * if both Beta and Alpha are zero, C is filled by zeros.

    INPUT PARAMETERS
        N       -   matrix size, N>=0
        K       -   matrix size, K>=0
        Alpha   -   coefficient
        A       -   matrix
        IA      -   submatrix offset
        JA      -   submatrix offset
        OpTypeA -   multiplication type:
                    * 0 - A*A^H is calculated
                    * 2 - A^H*A is calculated
        Beta    -   coefficient
        C       -   matrix
        IC      -   submatrix offset
        JC      -   submatrix offset
        IsUpper -   whether C is upper triangular or lower triangular

      -- ALGLIB routine --
         16.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixsyrk(
        int n,
        int k,
        double alpha,
        complex[,] a,
        int ia,
        int ja,
        int optypea,
        double beta,
        ref complex[,] c,
        int ic,
        int jc,
        bool isupper)

### `cmatrixtranspose` function

    /*************************************************************************
    Cache-oblivous complex "copy-and-transpose"

    Input parameters:
        M   -   number of rows
        N   -   number of columns
        A   -   source matrix, MxN submatrix is copied and transposed
        IA  -   submatrix offset (row index)
        JA  -   submatrix offset (column index)
        A   -   destination matrix
        IB  -   submatrix offset (row index)
        JB  -   submatrix offset (column index)
    *************************************************************************/
    public static void cmatrixtranspose(
        int m,
        int n,
        complex[,] a,
        int ia,
        int ja,
        ref complex[,] b,
        int ib,
        int jb)

### `rmatrixcopy` function

    /*************************************************************************
    Copy

    Input parameters:
        M   -   number of rows
        N   -   number of columns
        A   -   source matrix, MxN submatrix is copied and transposed
        IA  -   submatrix offset (row index)
        JA  -   submatrix offset (column index)
        B   -   destination matrix
        IB  -   submatrix offset (row index)
        JB  -   submatrix offset (column index)
    *************************************************************************/
    public static void rmatrixcopy(
        int m,
        int n,
        double[,] a,
        int ia,
        int ja,
        ref double[,] b,
        int ib,
        int jb)

### `rmatrixgemm` function

    /*************************************************************************
    Same as CMatrixGEMM, but for real numbers.
    OpType may be only 0 or 1.

      -- ALGLIB routine --
         16.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixgemm(
        int m,
        int n,
        int k,
        double alpha,
        double[,] a,
        int ia,
        int ja,
        int optypea,
        double[,] b,
        int ib,
        int jb,
        int optypeb,
        double beta,
        ref double[,] c,
        int ic,
        int jc)

### `rmatrixlefttrsm` function

    /*************************************************************************
    Same as CMatrixLeftTRSM, but for real matrices

    OpType may be only 0 or 1.

      -- ALGLIB routine --
         15.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixlefttrsm(
        int m,
        int n,
        double[,] a,
        int i1,
        int j1,
        bool isupper,
        bool isunit,
        int optype,
        ref double[,] x,
        int i2,
        int j2)

### `rmatrixmv` function

    /*************************************************************************
    Matrix-vector product: y := op(A)*x

    INPUT PARAMETERS:
        M   -   number of rows of op(A)
        N   -   number of columns of op(A)
        A   -   target matrix
        IA  -   submatrix offset (row index)
        JA  -   submatrix offset (column index)
        OpA -   operation type:
                * OpA=0     =>  op(A) = A
                * OpA=1     =>  op(A) = A^T
        X   -   input vector
        IX  -   subvector offset
        IY  -   subvector offset

    OUTPUT PARAMETERS:
        Y   -   vector which stores result

    if M=0, then subroutine does nothing.
    if N=0, Y is filled by zeros.

      -- ALGLIB routine --

         28.01.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixmv(
        int m,
        int n,
        double[,] a,
        int ia,
        int ja,
        int opa,
        double[] x,
        int ix,
        ref double[] y,
        int iy)

### `rmatrixrank1` function

    /*************************************************************************
    Rank-1 correction: A := A + u*v'

    INPUT PARAMETERS:
        M   -   number of rows
        N   -   number of columns
        A   -   target matrix, MxN submatrix is updated
        IA  -   submatrix offset (row index)
        JA  -   submatrix offset (column index)
        U   -   vector #1
        IU  -   subvector offset
        V   -   vector #2
        IV  -   subvector offset
    *************************************************************************/
    public static void rmatrixrank1(
        int m,
        int n,
        ref double[,] a,
        int ia,
        int ja,
        ref double[] u,
        int iu,
        ref double[] v,
        int iv)

### `rmatrixrighttrsm` function

    /*************************************************************************
    Same as CMatrixRightTRSM, but for real matrices

    OpType may be only 0 or 1.

      -- ALGLIB routine --
         15.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixrighttrsm(
        int m,
        int n,
        double[,] a,
        int i1,
        int j1,
        bool isupper,
        bool isunit,
        int optype,
        ref double[,] x,
        int i2,
        int j2)

### `rmatrixsyrk` function

    /*************************************************************************
    Same as CMatrixSYRK, but for real matrices

    OpType may be only 0 or 1.

      -- ALGLIB routine --
         16.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixsyrk(
        int n,
        int k,
        double alpha,
        double[,] a,
        int ia,
        int ja,
        int optypea,
        double beta,
        ref double[,] c,
        int ic,
        int jc,
        bool isupper)

### `rmatrixtranspose` function

    /*************************************************************************
    Cache-oblivous real "copy-and-transpose"

    Input parameters:
        M   -   number of rows
        N   -   number of columns
        A   -   source matrix, MxN submatrix is copied and transposed
        IA  -   submatrix offset (row index)
        JA  -   submatrix offset (column index)
        A   -   destination matrix
        IB  -   submatrix offset (row index)
        JB  -   submatrix offset (column index)
    *************************************************************************/
    public static void rmatrixtranspose(
        int m,
        int n,
        double[,] a,
        int ia,
        int ja,
        ref double[,] b,
        int ib,
        int jb)

## `airyf` subpackage

### Functions

airy

### Examples

### `airy` function

    /*************************************************************************
    Airy function

    Solution of the differential equation

    y"(x) = xy.

    The function returns the two independent solutions Ai, Bi
    and their first derivatives Ai'(x), Bi'(x).

    Evaluation is by power series summation for small x,
    by rational minimax approximations for large x.

    ACCURACY:
    Error criterion is absolute when function <= 1, relative
    when function > 1, except * denotes relative error criterion.
    For large negative x, the absolute error increases as x^1.5.
    For large positive x, the relative error increases as x^1.5.

    Arithmetic  domain   function  # trials      peak         rms
    IEEE        -10, 0     Ai        10000       1.6e-15     2.7e-16
    IEEE          0, 10    Ai        10000       2.3e-14*    1.8e-15*
    IEEE        -10, 0     Ai'       10000       4.6e-15     7.6e-16
    IEEE          0, 10    Ai'       10000       1.8e-14*    1.5e-15*
    IEEE        -10, 10    Bi        30000       4.2e-15     5.3e-16
    IEEE        -10, 10    Bi'       30000       4.9e-15     7.3e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static void airy(
        double x,
        out double ai,
        out double aip,
        out double bi,
        out double bip)

## `autogk` subpackage

### Classes

autogkreport
autogkstate

### Functions

autogkintegrate
autogkresults
autogksingular
autogksmooth
autogksmoothw

### Examples

autogk_d1 |  | Integrating f=exp(x) by adaptive integrator

### `autogkreport` class

    /*************************************************************************
    Integration report:
    * TerminationType = completetion code:
        * -5    non-convergence of Gauss-Kronrod nodes
                calculation subroutine.
        * -1    incorrect parameters were specified
        *  1    OK
    * Rep.NFEV countains number of function calculations
    * Rep.NIntervals contains number of intervals [a,b]
      was partitioned into.
    *************************************************************************/
    public class autogkreport
    {
        public int                  terminationtype;
        public int                  nfev;
        public int                  nintervals;
    }

### `autogkstate` class

    /*************************************************************************
    This structure stores state of the integration algorithm.

    Although this class has public fields,  they are not intended for external
    use. You should use ALGLIB functions to work with this class:
    * autogksmooth()/AutoGKSmoothW()/... to create objects
    * autogkintegrate() to begin integration
    * autogkresults() to get results
    *************************************************************************/
    public class autogkstate
    {
    }

### `autogkintegrate` function

    /*************************************************************************
    This function is used to launcn iterations of ODE solver

    It accepts following parameters:
        diff    -   callback which calculates dy/dx for given y and x
        obj     -   optional object which is passed to diff; can be NULL

      -- ALGLIB --
         Copyright 07.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void autogkintegrate(autogkstate state, integrator1_func func, object obj)

Examples: [1]

### `autogkresults` function

    /*************************************************************************
    Adaptive integration results

    Called after AutoGKIteration returned False.

    Input parameters:
        State   -   algorithm state (used by AutoGKIteration).

    Output parameters:
        V       -   integral(f(x)dx,a,b)
        Rep     -   optimization report (see AutoGKReport description)

      -- ALGLIB --
         Copyright 14.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void autogkresults(
        autogkstate state,
        out double v,
        out autogkreport rep)

Examples: [1]

### `autogksingular` function

    /*************************************************************************
    Integration on a finite interval [A,B].
    Integrand have integrable singularities at A/B.

    F(X) must diverge as "(x-A)^alpha" at A, as "(B-x)^beta" at B,  with known
    alpha/beta (alpha>-1, beta>-1).  If alpha/beta  are  not known,  estimates
    from below can be used (but these estimates should be greater than -1 too).

    One  of  alpha/beta variables (or even both alpha/beta) may be equal to 0,
    which means than function F(x) is non-singular at A/B. Anyway (singular at
    bounds or not), function F(x) is supposed to be continuous on (A,B).

    Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
    is calculated with accuracy close to the machine precision.

    INPUT PARAMETERS:
        A, B    -   interval boundaries (A<B, A=B or A>B)
        Alpha   -   power-law coefficient of the F(x) at A,
                    Alpha>-1
        Beta    -   power-law coefficient of the F(x) at B,
                    Beta>-1

    OUTPUT PARAMETERS
        State   -   structure which stores algorithm state

    SEE ALSO
        AutoGKSmooth, AutoGKSmoothW, AutoGKResults.

      -- ALGLIB --
         Copyright 06.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void autogksingular(
        double a,
        double b,
        double alpha,
        double beta,
        out autogkstate state)

### `autogksmooth` function

    /*************************************************************************
    Integration of a smooth function F(x) on a finite interval [a,b].

    Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
    is calculated with accuracy close to the machine precision.

    Algorithm works well only with smooth integrands.  It  may  be  used  with
    continuous non-smooth integrands, but with  less  performance.

    It should never be used with integrands which have integrable singularities
    at lower or upper limits - algorithm may crash. Use AutoGKSingular in such
    cases.

    INPUT PARAMETERS:
        A, B    -   interval boundaries (A<B, A=B or A>B)

    OUTPUT PARAMETERS
        State   -   structure which stores algorithm state

    SEE ALSO
        AutoGKSmoothW, AutoGKSingular, AutoGKResults.

      -- ALGLIB --
         Copyright 06.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void autogksmooth(double a, double b, out autogkstate state)

Examples: [1]

### `autogksmoothw` function

    /*************************************************************************
    Integration of a smooth function F(x) on a finite interval [a,b].

    This subroutine is same as AutoGKSmooth(), but it guarantees that interval
    [a,b] is partitioned into subintervals which have width at most XWidth.

    Subroutine  can  be  used  when  integrating nearly-constant function with
    narrow "bumps" (about XWidth wide). If "bumps" are too narrow, AutoGKSmooth
    subroutine can overlook them.

    INPUT PARAMETERS:
        A, B    -   interval boundaries (A<B, A=B or A>B)

    OUTPUT PARAMETERS
        State   -   structure which stores algorithm state

    SEE ALSO
        AutoGKSmooth, AutoGKSingular, AutoGKResults.

      -- ALGLIB --
         Copyright 06.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void autogksmoothw(
        double a,
        double b,
        double xwidth,
        out autogkstate state)

### autogk_d1 example

    public static void int_function_1_func(double x, double xminusa, double bminusx, ref double y, object obj)
    {
        // this callback calculates f(x)=exp(x)
        y = Math.Exp(x);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates integration of f=exp(x) on [0,1]:
        // * first, autogkstate is initialized
        // * then we call integration function
        // * and finally we obtain results with autogkresults() call
        //
        double a = 0;
        double b = 1;
        alglib.autogkstate s;
        double v;
        alglib.autogkreport rep;

        alglib.autogksmooth(a, b, out s);
        alglib.autogkintegrate(s, int_function_1_func, null);
        alglib.autogkresults(s, out v, out rep);

        System.Console.WriteLine("{0:F2", v); // EXPECTED: 1.7182
        System.Console.ReadLine();
        return 0;
    }

## `basestat` subpackage

### Functions

cov2
covm
covm2
pearsoncorr2
pearsoncorrelation
pearsoncorrm
pearsoncorrm2
sampleadev
samplemedian
samplemoments
samplepercentile
spearmancorr2
spearmancorrm
spearmancorrm2
spearmanrankcorrelation

### Examples

basestat_d_base |  | Basic functionality (moments, adev, median, percentile)
basestat_d_c2 |  | Correlation (covariance) between two random variables
basestat_d_cm |  | Correlation (covariance) between components of random vector
basestat_d_cm2 |  | Correlation (covariance) between two random vectors

### `cov2` function

    /*************************************************************************
    2-sample covariance

    Input parameters:
        X       -   sample 1 (array indexes: [0..N-1])
        Y       -   sample 2 (array indexes: [0..N-1])
        N       -   N>=0, sample size:
                    * if given, only N leading elements of X/Y are processed
                    * if not given, automatically determined from input sizes

    Result:
        covariance (zero for N=0 or N=1)

      -- ALGLIB --
         Copyright 28.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static double cov2(double[] x, double[] y)
    public static double cov2(double[] x, double[] y, int n)

Examples: [1]

### `covm` function

    /*************************************************************************
    Covariance matrix

    INPUT PARAMETERS:
        X   -   array[N,M], sample matrix:
                * J-th column corresponds to J-th variable
                * I-th row corresponds to I-th observation
        N   -   N>=0, number of observations:
                * if given, only leading N rows of X are used
                * if not given, automatically determined from input size
        M   -   M>0, number of variables:
                * if given, only leading M columns of X are used
                * if not given, automatically determined from input size

    OUTPUT PARAMETERS:
        C   -   array[M,M], covariance matrix (zero if N=0 or N=1)

      -- ALGLIB --
         Copyright 28.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void covm(double[,] x, out double[,] c)
    public static void covm(double[,] x, int n, int m, out double[,] c)

Examples: [1]

### `covm2` function

    /*************************************************************************
    Cross-covariance matrix

    INPUT PARAMETERS:
        X   -   array[N,M1], sample matrix:
                * J-th column corresponds to J-th variable
                * I-th row corresponds to I-th observation
        Y   -   array[N,M2], sample matrix:
                * J-th column corresponds to J-th variable
                * I-th row corresponds to I-th observation
        N   -   N>=0, number of observations:
                * if given, only leading N rows of X/Y are used
                * if not given, automatically determined from input sizes
        M1  -   M1>0, number of variables in X:
                * if given, only leading M1 columns of X are used
                * if not given, automatically determined from input size
        M2  -   M2>0, number of variables in Y:
                * if given, only leading M1 columns of X are used
                * if not given, automatically determined from input size

    OUTPUT PARAMETERS:
        C   -   array[M1,M2], cross-covariance matrix (zero if N=0 or N=1)

      -- ALGLIB --
         Copyright 28.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void covm2(double[,] x, double[,] y, out double[,] c)
    public static void covm2(
        double[,] x,
        double[,] y,
        int n,
        int m1,
        int m2,
        out double[,] c)

Examples: [1]

### `pearsoncorr2` function

    /*************************************************************************
    Pearson product-moment correlation coefficient

    Input parameters:
        X       -   sample 1 (array indexes: [0..N-1])
        Y       -   sample 2 (array indexes: [0..N-1])
        N       -   N>=0, sample size:
                    * if given, only N leading elements of X/Y are processed
                    * if not given, automatically determined from input sizes

    Result:
        Pearson product-moment correlation coefficient
        (zero for N=0 or N=1)

      -- ALGLIB --
         Copyright 28.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static double pearsoncorr2(double[] x, double[] y)
    public static double pearsoncorr2(double[] x, double[] y, int n)

Examples: [1]

### `pearsoncorrelation` function

    /*************************************************************************
    Obsolete function, we recommend to use PearsonCorr2().

      -- ALGLIB --
         Copyright 09.04.2007 by Bochkanov Sergey
    *************************************************************************/
    public static double pearsoncorrelation(double[] x, double[] y, int n)

### `pearsoncorrm` function

    /*************************************************************************
    Pearson product-moment correlation matrix

    INPUT PARAMETERS:
        X   -   array[N,M], sample matrix:
                * J-th column corresponds to J-th variable
                * I-th row corresponds to I-th observation
        N   -   N>=0, number of observations:
                * if given, only leading N rows of X are used
                * if not given, automatically determined from input size
        M   -   M>0, number of variables:
                * if given, only leading M columns of X are used
                * if not given, automatically determined from input size

    OUTPUT PARAMETERS:
        C   -   array[M,M], correlation matrix (zero if N=0 or N=1)

      -- ALGLIB --
         Copyright 28.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pearsoncorrm(double[,] x, out double[,] c)
    public static void pearsoncorrm(
        double[,] x,
        int n,
        int m,
        out double[,] c)

Examples: [1]

### `pearsoncorrm2` function

    /*************************************************************************
    Pearson product-moment cross-correlation matrix

    INPUT PARAMETERS:
        X   -   array[N,M1], sample matrix:
                * J-th column corresponds to J-th variable
                * I-th row corresponds to I-th observation
        Y   -   array[N,M2], sample matrix:
                * J-th column corresponds to J-th variable
                * I-th row corresponds to I-th observation
        N   -   N>=0, number of observations:
                * if given, only leading N rows of X/Y are used
                * if not given, automatically determined from input sizes
        M1  -   M1>0, number of variables in X:
                * if given, only leading M1 columns of X are used
                * if not given, automatically determined from input size
        M2  -   M2>0, number of variables in Y:
                * if given, only leading M1 columns of X are used
                * if not given, automatically determined from input size

    OUTPUT PARAMETERS:
        C   -   array[M1,M2], cross-correlation matrix (zero if N=0 or N=1)

      -- ALGLIB --
         Copyright 28.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pearsoncorrm2(
        double[,] x,
        double[,] y,
        out double[,] c)
    public static void pearsoncorrm2(
        double[,] x,
        double[,] y,
        int n,
        int m1,
        int m2,
        out double[,] c)

Examples: [1]

### `sampleadev` function

    /*************************************************************************
    ADev

    Input parameters:
        X   -   sample
        N   -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

    Output parameters:
        ADev-   ADev

      -- ALGLIB --
         Copyright 06.09.2006 by Bochkanov Sergey
    *************************************************************************/
    public static void sampleadev(double[] x, out double adev)
    public static void sampleadev(double[] x, int n, out double adev)

Examples: [1]

### `samplemedian` function

    /*************************************************************************
    Median calculation.

    Input parameters:
        X   -   sample (array indexes: [0..N-1])
        N   -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

    Output parameters:
        Median

      -- ALGLIB --
         Copyright 06.09.2006 by Bochkanov Sergey
    *************************************************************************/
    public static void samplemedian(double[] x, out double median)
    public static void samplemedian(double[] x, int n, out double median)

Examples: [1]

### `samplemoments` function

    /*************************************************************************
    Calculation of the distribution moments: mean, variance, skewness, kurtosis.

    INPUT PARAMETERS:
        X       -   sample
        N       -   N>=0, sample size:
                    * if given, only leading N elements of X are processed
                    * if not given, automatically determined from size of X

    OUTPUT PARAMETERS
        Mean    -   mean.
        Variance-   variance.
        Skewness-   skewness (if variance<>0; zero otherwise).
        Kurtosis-   kurtosis (if variance<>0; zero otherwise).

      -- ALGLIB --
         Copyright 06.09.2006 by Bochkanov Sergey
    *************************************************************************/
    public static void samplemoments(
        double[] x,
        out double mean,
        out double variance,
        out double skewness,
        out double kurtosis)
    public static void samplemoments(
        double[] x,
        int n,
        out double mean,
        out double variance,
        out double skewness,
        out double kurtosis)

Examples: [1]

### `samplepercentile` function

    /*************************************************************************
    Percentile calculation.

    Input parameters:
        X   -   sample (array indexes: [0..N-1])
        N   -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X
        P   -   percentile (0<=P<=1)

    Output parameters:
        V   -   percentile

      -- ALGLIB --
         Copyright 01.03.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void samplepercentile(double[] x, double p, out double v)
    public static void samplepercentile(
        double[] x,
        int n,
        double p,
        out double v)

Examples: [1]

### `spearmancorr2` function

    /*************************************************************************
    Spearman's rank correlation coefficient

    Input parameters:
        X       -   sample 1 (array indexes: [0..N-1])
        Y       -   sample 2 (array indexes: [0..N-1])
        N       -   N>=0, sample size:
                    * if given, only N leading elements of X/Y are processed
                    * if not given, automatically determined from input sizes

    Result:
        Spearman's rank correlation coefficient
        (zero for N=0 or N=1)

      -- ALGLIB --
         Copyright 09.04.2007 by Bochkanov Sergey
    *************************************************************************/
    public static double spearmancorr2(double[] x, double[] y)
    public static double spearmancorr2(double[] x, double[] y, int n)

Examples: [1]

### `spearmancorrm` function

    /*************************************************************************
    Spearman's rank correlation matrix

    INPUT PARAMETERS:
        X   -   array[N,M], sample matrix:
                * J-th column corresponds to J-th variable
                * I-th row corresponds to I-th observation
        N   -   N>=0, number of observations:
                * if given, only leading N rows of X are used
                * if not given, automatically determined from input size
        M   -   M>0, number of variables:
                * if given, only leading M columns of X are used
                * if not given, automatically determined from input size

    OUTPUT PARAMETERS:
        C   -   array[M,M], correlation matrix (zero if N=0 or N=1)

      -- ALGLIB --
         Copyright 28.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spearmancorrm(double[,] x, out double[,] c)
    public static void spearmancorrm(
        double[,] x,
        int n,
        int m,
        out double[,] c)

Examples: [1]

### `spearmancorrm2` function

    /*************************************************************************
    Spearman's rank cross-correlation matrix

    INPUT PARAMETERS:
        X   -   array[N,M1], sample matrix:
                * J-th column corresponds to J-th variable
                * I-th row corresponds to I-th observation
        Y   -   array[N,M2], sample matrix:
                * J-th column corresponds to J-th variable
                * I-th row corresponds to I-th observation
        N   -   N>=0, number of observations:
                * if given, only leading N rows of X/Y are used
                * if not given, automatically determined from input sizes
        M1  -   M1>0, number of variables in X:
                * if given, only leading M1 columns of X are used
                * if not given, automatically determined from input size
        M2  -   M2>0, number of variables in Y:
                * if given, only leading M1 columns of X are used
                * if not given, automatically determined from input size

    OUTPUT PARAMETERS:
        C   -   array[M1,M2], cross-correlation matrix (zero if N=0 or N=1)

      -- ALGLIB --
         Copyright 28.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spearmancorrm2(
        double[,] x,
        double[,] y,
        out double[,] c)
    public static void spearmancorrm2(
        double[,] x,
        double[,] y,
        int n,
        int m1,
        int m2,
        out double[,] c)

Examples: [1]

### `spearmanrankcorrelation` function

    /*************************************************************************
    Obsolete function, we recommend to use SpearmanCorr2().

        -- ALGLIB --
        Copyright 09.04.2007 by Bochkanov Sergey
    *************************************************************************/
    public static double spearmanrankcorrelation(
        double[] x,
        double[] y,
        int n)

### basestat_d_base example

    public static int Main(string[] args)
    {
        double[] x = new double[]{0,1,4,9,16,25,36,49,64,81};
        double mean;
        double variance;
        double skewness;
        double kurtosis;
        double adev;
        double p;
        double v;

        //
        // Here we demonstrate calculation of sample moments
        // (mean, variance, skewness, kurtosis)
        //
        alglib.samplemoments(x, out mean, out variance, out skewness, out kurtosis);
        System.Console.WriteLine("{0:F1", mean); // EXPECTED: 28.5
        System.Console.WriteLine("{0:F1", variance); // EXPECTED: 801.1667
        System.Console.WriteLine("{0:F1", skewness); // EXPECTED: 0.5751
        System.Console.WriteLine("{0:F1", kurtosis); // EXPECTED: -1.2666

        //
        // Average deviation
        //
        alglib.sampleadev(x, out adev);
        System.Console.WriteLine("{0:F1", adev); // EXPECTED: 23.2

        //
        // Median and percentile
        //
        alglib.samplemedian(x, out v);
        System.Console.WriteLine("{0:F1", v); // EXPECTED: 20.5
        p = 0.5;
        alglib.samplepercentile(x, p, out v);
        System.Console.WriteLine("{0:F1", v); // EXPECTED: 20.5
        System.Console.ReadLine();
        return 0;
    }

### basestat_d_c2 example

    public static int Main(string[] args)
    {
        //
        // We have two samples - x and y, and want to measure dependency between them
        //
        double[] x = new double[]{0,1,4,9,16,25,36,49,64,81};
        double[] y = new double[]{0,1,2,3,4,5,6,7,8,9};
        double v;

        //
        // Three dependency measures are calculated:
        // * covariation
        // * Pearson correlation
        // * Spearman rank correlation
        //
        v = alglib.cov2(x, y);
        System.Console.WriteLine("{0:F2", v); // EXPECTED: 82.5
        v = alglib.pearsoncorr2(x, y);
        System.Console.WriteLine("{0:F2", v); // EXPECTED: 0.9627
        v = alglib.spearmancorr2(x, y);
        System.Console.WriteLine("{0:F2", v); // EXPECTED: 1.000
        System.Console.ReadLine();
        return 0;
    }

### basestat_d_cm example

    public static int Main(string[] args)
    {
        //
        // X is a sample matrix:
        // * I-th row corresponds to I-th observation
        // * J-th column corresponds to J-th variable
        //
        double[,] x = new double[,]{{1,0,1},{1,1,0},{-1,1,0},{-2,-1,1},{-1,0,9}};
        double[,] c;

        //
        // Three dependency measures are calculated:
        // * covariation
        // * Pearson correlation
        // * Spearman rank correlation
        //
        // Result is stored into C, with C[i,j] equal to correlation
        // (covariance) between I-th and J-th variables of X.
        //
        alglib.covm(x, out c);
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [[1.80,0.60,-1.40],[0.60,0.70,-0.80],[-1.40,-0.80,14.70]]
        alglib.pearsoncorrm(x, out c);
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [[1.000,0.535,-0.272],[0.535,1.000,-0.249],[-0.272,-0.249,1.000]]
        alglib.spearmancorrm(x, out c);
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [[1.000,0.556,-0.306],[0.556,1.000,-0.750],[-0.306,-0.750,1.000]]
        System.Console.ReadLine();
        return 0;
    }

### basestat_d_cm2 example

    public static int Main(string[] args)
    {
        //
        // X and Y are sample matrices:
        // * I-th row corresponds to I-th observation
        // * J-th column corresponds to J-th variable
        //
        double[,] x = new double[,]{{1,0,1},{1,1,0},{-1,1,0},{-2,-1,1},{-1,0,9}};
        double[,] y = new double[,]{{2,3},{2,1},{-1,6},{-9,9},{7,1}};
        double[,] c;

        //
        // Three dependency measures are calculated:
        // * covariation
        // * Pearson correlation
        // * Spearman rank correlation
        //
        // Result is stored into C, with C[i,j] equal to correlation
        // (covariance) between I-th variable of X and J-th variable of Y.
        //
        alglib.covm2(x, y, out c);
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [[4.100,-3.250],[2.450,-1.500],[13.450,-5.750]]
        alglib.pearsoncorrm2(x, y, out c);
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [[0.519,-0.699],[0.497,-0.518],[0.596,-0.433]]
        alglib.spearmancorrm2(x, y, out c);
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [[0.541,-0.649],[0.216,-0.433],[0.433,-0.135]]
        System.Console.ReadLine();
        return 0;
    }

## `bdss` subpackage

### Functions

dsoptimalsplit2
dsoptimalsplit2fast

### Examples

### `dsoptimalsplit2` function

    /*************************************************************************
    Optimal binary classification

    Algorithms finds optimal (=with minimal cross-entropy) binary partition.
    Internal subroutine.

    INPUT PARAMETERS:
        A       -   array[0..N-1], variable
        C       -   array[0..N-1], class numbers (0 or 1).
        N       -   array size

    OUTPUT PARAMETERS:
        Info    -   completetion code:
                    * -3, all values of A[] are same (partition is impossible)
                    * -2, one of C[] is incorrect (<0, >1)
                    * -1, incorrect pararemets were passed (N<=0).
                    *  1, OK
        Threshold-  partiton boundary. Left part contains values which are
                    strictly less than Threshold. Right part contains values
                    which are greater than or equal to Threshold.
        PAL, PBL-   probabilities P(0|v<Threshold) and P(1|v<Threshold)
        PAR, PBR-   probabilities P(0|v>=Threshold) and P(1|v>=Threshold)
        CVE     -   cross-validation estimate of cross-entropy

      -- ALGLIB --
         Copyright 22.05.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void dsoptimalsplit2(
        double[] a,
        int[] c,
        int n,
        out int info,
        out double threshold,
        out double pal,
        out double pbl,
        out double par,
        out double pbr,
        out double cve)

### `dsoptimalsplit2fast` function

    /*************************************************************************
    Optimal partition, internal subroutine. Fast version.

    Accepts:
        A       array[0..N-1]       array of attributes     array[0..N-1]
        C       array[0..N-1]       array of class labels
        TiesBuf array[0..N]         temporaries (ties)
        CntBuf  array[0..2*NC-1]    temporaries (counts)
        Alpha                       centering factor (0<=alpha<=1, recommended value - 0.05)
        BufR    array[0..N-1]       temporaries
        BufI    array[0..N-1]       temporaries

    Output:
        Info    error code (">0"=OK, "<0"=bad)
        RMS     training set RMS error
        CVRMS   leave-one-out RMS error

    Note:
        content of all arrays is changed by subroutine;
        it doesn't allocate temporaries.

      -- ALGLIB --
         Copyright 11.12.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void dsoptimalsplit2fast(
        ref double[] a,
        ref int[] c,
        ref int[] tiesbuf,
        ref int[] cntbuf,
        ref double[] bufr,
        ref int[] bufi,
        int n,
        int nc,
        double alpha,
        out int info,
        out double threshold,
        out double rms,
        out double cvrms)

## `bdsvd` subpackage

### Functions

rmatrixbdsvd

### Examples

### `rmatrixbdsvd` function

    /*************************************************************************
    Singular value decomposition of a bidiagonal matrix (extended algorithm)

    The algorithm performs the singular value decomposition  of  a  bidiagonal
    matrix B (upper or lower) representing it as B = Q*S*P^T, where Q and  P -
    orthogonal matrices, S - diagonal matrix with non-negative elements on the
    main diagonal, in descending order.

    The  algorithm  finds  singular  values.  In  addition,  the algorithm can
    calculate  matrices  Q  and P (more precisely, not the matrices, but their
    product  with  given  matrices U and VT - U*Q and (P^T)*VT)).  Of  course,
    matrices U and VT can be of any type, including identity. Furthermore, the
    algorithm can calculate Q'*C (this product is calculated more  effectively
    than U*Q,  because  this calculation operates with rows instead  of matrix
    columns).

    The feature of the algorithm is its ability to find  all  singular  values
    including those which are arbitrarily close to 0  with  relative  accuracy
    close to  machine precision. If the parameter IsFractionalAccuracyRequired
    is set to True, all singular values will have high relative accuracy close
    to machine precision. If the parameter is set to False, only  the  biggest
    singular value will have relative accuracy  close  to  machine  precision.
    The absolute error of other singular values is equal to the absolute error
    of the biggest singular value.

    Input parameters:
        D       -   main diagonal of matrix B.
                    Array whose index ranges within [0..N-1].
        E       -   superdiagonal (or subdiagonal) of matrix B.
                    Array whose index ranges within [0..N-2].
        N       -   size of matrix B.
        IsUpper -   True, if the matrix is upper bidiagonal.
        IsFractionalAccuracyRequired -
                    accuracy to search singular values with.
        U       -   matrix to be multiplied by Q.
                    Array whose indexes range within [0..NRU-1, 0..N-1].
                    The matrix can be bigger, in that case only the  submatrix
                    [0..NRU-1, 0..N-1] will be multiplied by Q.
        NRU     -   number of rows in matrix U.
        C       -   matrix to be multiplied by Q'.
                    Array whose indexes range within [0..N-1, 0..NCC-1].
                    The matrix can be bigger, in that case only the  submatrix
                    [0..N-1, 0..NCC-1] will be multiplied by Q'.
        NCC     -   number of columns in matrix C.
        VT      -   matrix to be multiplied by P^T.
                    Array whose indexes range within [0..N-1, 0..NCVT-1].
                    The matrix can be bigger, in that case only the  submatrix
                    [0..N-1, 0..NCVT-1] will be multiplied by P^T.
        NCVT    -   number of columns in matrix VT.

    Output parameters:
        D       -   singular values of matrix B in descending order.
        U       -   if NRU>0, contains matrix U*Q.
        VT      -   if NCVT>0, contains matrix (P^T)*VT.
        C       -   if NCC>0, contains matrix Q'*C.

    Result:
        True, if the algorithm has converged.
        False, if the algorithm hasn't converged (rare case).

    Additional information:
        The type of convergence is controlled by the internal  parameter  TOL.
        If the parameter is greater than 0, the singular values will have
        relative accuracy TOL. If TOL<0, the singular values will have
        absolute accuracy ABS(TOL)*norm(B).
        By default, |TOL| falls within the range of 10*Epsilon and 100*Epsilon,
        where Epsilon is the machine precision. It is not  recommended  to  use
        TOL less than 10*Epsilon since this will  considerably  slow  down  the
        algorithm and may not lead to error decreasing.
    History:
        * 31 March, 2007.
            changed MAXITR from 6 to 12.

      -- LAPACK routine (version 3.0) --
         Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
         Courant Institute, Argonne National Lab, and Rice University
         October 31, 1999.
    *************************************************************************/
    public static bool rmatrixbdsvd(
        ref double[] d,
        double[] e,
        int n,
        bool isupper,
        bool isfractionalaccuracyrequired,
        ref double[,] u,
        int nru,
        ref double[,] c,
        int ncc,
        ref double[,] vt,
        int ncvt)

## `bessel` subpackage

### Functions

besseli0
besseli1
besselj0
besselj1
besseljn
besselk0
besselk1
besselkn
bessely0
bessely1
besselyn

### Examples

### `besseli0` function

    /*************************************************************************
    Modified Bessel function of order zero

    Returns modified Bessel function of order zero of the
    argument.

    The function is defined as i0(x) = j0( ix ).

    The range is partitioned into the two intervals [0,8] and
    (8, infinity).  Chebyshev polynomial expansions are employed
    in each interval.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE      0,30        30000       5.8e-16     1.4e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double besseli0(double x)

### `besseli1` function

    /*************************************************************************
    Modified Bessel function of order one

    Returns modified Bessel function of order one of the
    argument.

    The function is defined as i1(x) = -i j1( ix ).

    The range is partitioned into the two intervals [0,8] and
    (8, infinity).  Chebyshev polynomial expansions are employed
    in each interval.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE      0, 30       30000       1.9e-15     2.1e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1985, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double besseli1(double x)

### `besselj0` function

    /*************************************************************************
    Bessel function of order zero

    Returns Bessel function of order zero of the argument.

    The domain is divided into the intervals [0, 5] and
    (5, infinity). In the first interval the following rational
    approximation is used:

           2         2
    (w - r  ) (w - r  ) P (w) / Q (w)
          1         2    3       8

               2
    where w = x  and the two r's are zeros of the function.

    In the second interval, the Hankel asymptotic expansion
    is employed with two rational functions of degree 6/6
    and 7/7.

    ACCURACY:

                         Absolute error:
    arithmetic   domain     # trials      peak         rms
       IEEE      0, 30       60000       4.2e-16     1.1e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double besselj0(double x)

### `besselj1` function

    /*************************************************************************
    Bessel function of order one

    Returns Bessel function of order one of the argument.

    The domain is divided into the intervals [0, 8] and
    (8, infinity). In the first interval a 24 term Chebyshev
    expansion is used. In the second, the asymptotic
    trigonometric representation is employed using two
    rational functions of degree 5/5.

    ACCURACY:

                         Absolute error:
    arithmetic   domain      # trials      peak         rms
       IEEE      0, 30       30000       2.6e-16     1.1e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double besselj1(double x)

### `besseljn` function

    /*************************************************************************
    Bessel function of integer order

    Returns Bessel function of order n, where n is a
    (possibly negative) integer.

    The ratio of jn(x) to j0(x) is computed by backward
    recurrence.  First the ratio jn/jn-1 is found by a
    continued fraction expansion.  Then the recurrence
    relating successive orders is applied until j0 or j1 is
    reached.

    If n = 0 or 1 the routine for j0 or j1 is called
    directly.

    ACCURACY:

                         Absolute error:
    arithmetic   range      # trials      peak         rms
       IEEE      0, 30        5000       4.4e-16     7.9e-17

    Not suitable for large n or x. Use jv() (fractional order) instead.

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double besseljn(int n, double x)

### `besselk0` function

    /*************************************************************************
    Modified Bessel function, second kind, order zero

    Returns modified Bessel function of the second kind
    of order zero of the argument.

    The range is partitioned into the two intervals [0,8] and
    (8, infinity).  Chebyshev polynomial expansions are employed
    in each interval.

    ACCURACY:

    Tested at 2000 random points between 0 and 8.  Peak absolute
    error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE      0, 30       30000       1.2e-15     1.6e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double besselk0(double x)

### `besselk1` function

    /*************************************************************************
    Modified Bessel function, second kind, order one

    Computes the modified Bessel function of the second kind
    of order one of the argument.

    The range is partitioned into the two intervals [0,2] and
    (2, infinity).  Chebyshev polynomial expansions are employed
    in each interval.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE      0, 30       30000       1.2e-15     1.6e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double besselk1(double x)

### `besselkn` function

    /*************************************************************************
    Modified Bessel function, second kind, integer order

    Returns modified Bessel function of the second kind
    of order n of the argument.

    The range is partitioned into the two intervals [0,9.55] and
    (9.55, infinity).  An ascending power series is used in the
    low range, and an asymptotic expansion in the high range.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE      0,30        90000       1.8e-8      3.0e-10

    Error is high only near the crossover point x = 9.55
    between the two expansions used.

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double besselkn(int nn, double x)

### `bessely0` function

    /*************************************************************************
    Bessel function of the second kind, order zero

    Returns Bessel function of the second kind, of order
    zero, of the argument.

    The domain is divided into the intervals [0, 5] and
    (5, infinity). In the first interval a rational approximation
    R(x) is employed to compute
      y0(x)  = R(x)  +   2 * log(x) * j0(x) / PI.
    Thus a call to j0() is required.

    In the second interval, the Hankel asymptotic expansion
    is employed with two rational functions of degree 6/6
    and 7/7.

    ACCURACY:

     Absolute error, when y0(x) < 1; else relative error:

    arithmetic   domain     # trials      peak         rms
       IEEE      0, 30       30000       1.3e-15     1.6e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double bessely0(double x)

### `bessely1` function

    /*************************************************************************
    Bessel function of second kind of order one

    Returns Bessel function of the second kind of order one
    of the argument.

    The domain is divided into the intervals [0, 8] and
    (8, infinity). In the first interval a 25 term Chebyshev
    expansion is used, and a call to j1() is required.
    In the second, the asymptotic trigonometric representation
    is employed using two rational functions of degree 5/5.

    ACCURACY:

                         Absolute error:
    arithmetic   domain      # trials      peak         rms
       IEEE      0, 30       30000       1.0e-15     1.3e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double bessely1(double x)

### `besselyn` function

    /*************************************************************************
    Bessel function of second kind of integer order

    Returns Bessel function of order n, where n is a
    (possibly negative) integer.

    The function is evaluated by forward recurrence on
    n, starting with values computed by the routines
    y0() and y1().

    If n = 0 or 1 the routine for y0 or y1 is called
    directly.

    ACCURACY:
                         Absolute error, except relative
                         when y > 1:
    arithmetic   domain     # trials      peak         rms
       IEEE      0, 30       30000       3.4e-15     4.3e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double besselyn(int n, double x)

## `betaf` subpackage

### Functions

beta

### Examples

### `beta` function

    /*************************************************************************
    Beta function

                      -     -
                     | (a) | (b)
    beta( a, b )  =  -----------.
                        -
                       | (a+b)

    For large arguments the logarithm of the function is
    evaluated using lgam(), then exponentiated.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE       0,30       30000       8.1e-14     1.1e-14

    Cephes Math Library Release 2.0:  April, 1987
    Copyright 1984, 1987 by Stephen L. Moshier
    *************************************************************************/
    public static double beta(double a, double b)

## `binomialdistr` subpackage

### Functions

binomialcdistribution
binomialdistribution
invbinomialdistribution

### Examples

### `binomialcdistribution` function

    /*************************************************************************
    Complemented binomial distribution

    Returns the sum of the terms k+1 through n of the Binomial
    probability density:

      n
      --  ( n )   j      n-j
      >   (   )  p  (1-p)
      --  ( j )
     j=k+1

    The terms are not summed directly; instead the incomplete
    beta integral is employed, according to the formula

    y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).

    The arguments must be positive, with p ranging from 0 to 1.

    ACCURACY:

    Tested at random points (a,b,p).

                  a,b                     Relative error:
    arithmetic  domain     # trials      peak         rms
     For p between 0.001 and 1:
       IEEE     0,100       100000      6.7e-15     8.2e-16
     For p between 0 and .001:
       IEEE     0,100       100000      1.5e-13     2.7e-15

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double binomialcdistribution(int k, int n, double p)

### `binomialdistribution` function

    /*************************************************************************
    Binomial distribution

    Returns the sum of the terms 0 through k of the Binomial
    probability density:

      k
      --  ( n )   j      n-j
      >   (   )  p  (1-p)
      --  ( j )
     j=0

    The terms are not summed directly; instead the incomplete
    beta integral is employed, according to the formula

    y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).

    The arguments must be positive, with p ranging from 0 to 1.

    ACCURACY:

    Tested at random points (a,b,p), with p between 0 and 1.

                  a,b                     Relative error:
    arithmetic  domain     # trials      peak         rms
     For p between 0.001 and 1:
       IEEE     0,100       100000      4.3e-15     2.6e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double binomialdistribution(int k, int n, double p)

### `invbinomialdistribution` function

    /*************************************************************************
    Inverse binomial distribution

    Finds the event probability p such that the sum of the
    terms 0 through k of the Binomial probability density
    is equal to the given cumulative probability y.

    This is accomplished using the inverse beta integral
    function and the relation

    1 - p = incbi( n-k, k+1, y ).

    ACCURACY:

    Tested at random points (a,b,p).

                  a,b                     Relative error:
    arithmetic  domain     # trials      peak         rms
     For p between 0.001 and 1:
       IEEE     0,100       100000      2.3e-14     6.4e-16
       IEEE     0,10000     100000      6.6e-12     1.2e-13
     For p between 10^-6 and 0.001:
       IEEE     0,100       100000      2.0e-12     1.3e-14
       IEEE     0,10000     100000      1.5e-12     3.2e-14

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double invbinomialdistribution(int k, int n, double y)

## `chebyshev` subpackage

### Functions

chebyshevcalculate
chebyshevcoefficients
chebyshevsum
fromchebyshev

### Examples

### `chebyshevcalculate` function

    /*************************************************************************
    Calculation of the value of the Chebyshev polynomials of the
    first and second kinds.

    Parameters:
        r   -   polynomial kind, either 1 or 2.
        n   -   degree, n>=0
        x   -   argument, -1 <= x <= 1

    Result:
        the value of the Chebyshev polynomial at x
    *************************************************************************/
    public static double chebyshevcalculate(int r, int n, double x)

### `chebyshevcoefficients` function

    /*************************************************************************
    Representation of Tn as C[0] + C[1]*X + ... + C[N]*X^N

    Input parameters:
        N   -   polynomial degree, n>=0

    Output parameters:
        C   -   coefficients
    *************************************************************************/
    public static void chebyshevcoefficients(int n, out double[] c)

### `chebyshevsum` function

    /*************************************************************************
    Summation of Chebyshev polynomials using Clenshaw’s recurrence formula.

    This routine calculates
        c[0]*T0(x) + c[1]*T1(x) + ... + c[N]*TN(x)
    or
        c[0]*U0(x) + c[1]*U1(x) + ... + c[N]*UN(x)
    depending on the R.

    Parameters:
        r   -   polynomial kind, either 1 or 2.
        n   -   degree, n>=0
        x   -   argument

    Result:
        the value of the Chebyshev polynomial at x
    *************************************************************************/
    public static double chebyshevsum(double[] c, int r, int n, double x)

### `fromchebyshev` function

    /*************************************************************************
    Conversion of a series of Chebyshev polynomials to a power series.

    Represents A[0]*T0(x) + A[1]*T1(x) + ... + A[N]*Tn(x) as
    B[0] + B[1]*X + ... + B[N]*X^N.

    Input parameters:
        A   -   Chebyshev series coefficients
        N   -   degree, N>=0

    Output parameters
        B   -   power series coefficients
    *************************************************************************/
    public static void fromchebyshev(double[] a, int n, out double[] b)

## `chisquaredistr` subpackage

### Functions

chisquarecdistribution
chisquaredistribution
invchisquaredistribution

### Examples

### `chisquarecdistribution` function

    /*************************************************************************
    Complemented Chi-square distribution

    Returns the area under the right hand tail (from x to
    infinity) of the Chi square probability density function
    with v degrees of freedom:

                                     inf.
                                       -
                           1          | |  v/2-1  -t/2
     P( x | v )   =   -----------     |   t      e     dt
                       v/2  -       | |
                      2    | (v/2)   -
                                      x

    where x is the Chi-square variable.

    The incomplete gamma integral is used, according to the
    formula

    y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ).

    The arguments must both be positive.

    ACCURACY:

    See incomplete gamma function

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double chisquarecdistribution(double v, double x)

### `chisquaredistribution` function

    /*************************************************************************
    Chi-square distribution

    Returns the area under the left hand tail (from 0 to x)
    of the Chi square probability density function with
    v degrees of freedom.

                                      x
                                       -
                           1          | |  v/2-1  -t/2
     P( x | v )   =   -----------     |   t      e     dt
                       v/2  -       | |
                      2    | (v/2)   -
                                      0

    where x is the Chi-square variable.

    The incomplete gamma integral is used, according to the
    formula

    y = chdtr( v, x ) = igam( v/2.0, x/2.0 ).

    The arguments must both be positive.

    ACCURACY:

    See incomplete gamma function

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double chisquaredistribution(double v, double x)

### `invchisquaredistribution` function

    /*************************************************************************
    Inverse of complemented Chi-square distribution

    Finds the Chi-square argument x such that the integral
    from x to infinity of the Chi-square density is equal
    to the given cumulative probability y.

    This is accomplished using the inverse gamma integral
    function and the relation

       x/2 = igami( df/2, y );

    ACCURACY:

    See inverse incomplete gamma function

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double invchisquaredistribution(double v, double y)

## `conv` subpackage

### Functions

convc1d
convc1dcircular
convc1dcircularinv
convc1dinv
convr1d
convr1dcircular
convr1dcircularinv
convr1dinv

### Examples

### `convc1d` function

    /*************************************************************************
    1-dimensional complex convolution.

    For given A/B returns conv(A,B) (non-circular). Subroutine can automatically
    choose between three implementations: straightforward O(M*N)  formula  for
    very small N (or M), overlap-add algorithm for  cases  where  max(M,N)  is
    significantly larger than min(M,N), but O(M*N) algorithm is too slow,  and
    general FFT-based formula for cases where two previois algorithms are  too
    slow.

    Algorithm has max(M,N)*log(max(M,N)) complexity for any M/N.

    INPUT PARAMETERS
        A   -   array[0..M-1] - complex function to be transformed
        M   -   problem size
        B   -   array[0..N-1] - complex function to be transformed
        N   -   problem size

    OUTPUT PARAMETERS
        R   -   convolution: A*B. array[0..N+M-2].

    NOTE:
        It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
    functions have non-zero values at negative T's, you  can  still  use  this
    subroutine - just shift its result correspondingly.

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void convc1d(
        complex[] a,
        int m,
        complex[] b,
        int n,
        out complex[] r)

### `convc1dcircular` function

    /*************************************************************************
    1-dimensional circular complex convolution.

    For given S/R returns conv(S,R) (circular). Algorithm has linearithmic
    complexity for any M/N.

    IMPORTANT:  normal convolution is commutative,  i.e.   it  is symmetric  -
    conv(A,B)=conv(B,A).  Cyclic convolution IS NOT.  One function - S - is  a
    signal,  periodic function, and another - R - is a response,  non-periodic
    function with limited length.

    INPUT PARAMETERS
        S   -   array[0..M-1] - complex periodic signal
        M   -   problem size
        B   -   array[0..N-1] - complex non-periodic response
        N   -   problem size

    OUTPUT PARAMETERS
        R   -   convolution: A*B. array[0..M-1].

    NOTE:
        It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
    negative T's, you can still use this subroutine - just  shift  its  result
    correspondingly.

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void convc1dcircular(
        complex[] s,
        int m,
        complex[] r,
        int n,
        out complex[] c)

### `convc1dcircularinv` function

    /*************************************************************************
    1-dimensional circular complex deconvolution (inverse of ConvC1DCircular()).

    Algorithm has M*log(M)) complexity for any M (composite or prime).

    INPUT PARAMETERS
        A   -   array[0..M-1] - convolved periodic signal, A = conv(R, B)
        M   -   convolved signal length
        B   -   array[0..N-1] - non-periodic response
        N   -   response length

    OUTPUT PARAMETERS
        R   -   deconvolved signal. array[0..M-1].

    NOTE:
        deconvolution is unstable process and may result in division  by  zero
    (if your response function is degenerate, i.e. has zero Fourier coefficient).

    NOTE:
        It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
    negative T's, you can still use this subroutine - just  shift  its  result
    correspondingly.

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void convc1dcircularinv(
        complex[] a,
        int m,
        complex[] b,
        int n,
        out complex[] r)

### `convc1dinv` function

    /*************************************************************************
    1-dimensional complex non-circular deconvolution (inverse of ConvC1D()).

    Algorithm has M*log(M)) complexity for any M (composite or prime).

    INPUT PARAMETERS
        A   -   array[0..M-1] - convolved signal, A = conv(R, B)
        M   -   convolved signal length
        B   -   array[0..N-1] - response
        N   -   response length, N<=M

    OUTPUT PARAMETERS
        R   -   deconvolved signal. array[0..M-N].

    NOTE:
        deconvolution is unstable process and may result in division  by  zero
    (if your response function is degenerate, i.e. has zero Fourier coefficient).

    NOTE:
        It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
    functions have non-zero values at negative T's, you  can  still  use  this
    subroutine - just shift its result correspondingly.

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void convc1dinv(
        complex[] a,
        int m,
        complex[] b,
        int n,
        out complex[] r)

### `convr1d` function

    /*************************************************************************
    1-dimensional real convolution.

    Analogous to ConvC1D(), see ConvC1D() comments for more details.

    INPUT PARAMETERS
        A   -   array[0..M-1] - real function to be transformed
        M   -   problem size
        B   -   array[0..N-1] - real function to be transformed
        N   -   problem size

    OUTPUT PARAMETERS
        R   -   convolution: A*B. array[0..N+M-2].

    NOTE:
        It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
    functions have non-zero values at negative T's, you  can  still  use  this
    subroutine - just shift its result correspondingly.

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void convr1d(
        double[] a,
        int m,
        double[] b,
        int n,
        out double[] r)

### `convr1dcircular` function

    /*************************************************************************
    1-dimensional circular real convolution.

    Analogous to ConvC1DCircular(), see ConvC1DCircular() comments for more details.

    INPUT PARAMETERS
        S   -   array[0..M-1] - real signal
        M   -   problem size
        B   -   array[0..N-1] - real response
        N   -   problem size

    OUTPUT PARAMETERS
        R   -   convolution: A*B. array[0..M-1].

    NOTE:
        It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
    negative T's, you can still use this subroutine - just  shift  its  result
    correspondingly.

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void convr1dcircular(
        double[] s,
        int m,
        double[] r,
        int n,
        out double[] c)

### `convr1dcircularinv` function

    /*************************************************************************
    1-dimensional complex deconvolution (inverse of ConvC1D()).

    Algorithm has M*log(M)) complexity for any M (composite or prime).

    INPUT PARAMETERS
        A   -   array[0..M-1] - convolved signal, A = conv(R, B)
        M   -   convolved signal length
        B   -   array[0..N-1] - response
        N   -   response length

    OUTPUT PARAMETERS
        R   -   deconvolved signal. array[0..M-N].

    NOTE:
        deconvolution is unstable process and may result in division  by  zero
    (if your response function is degenerate, i.e. has zero Fourier coefficient).

    NOTE:
        It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
    negative T's, you can still use this subroutine - just  shift  its  result
    correspondingly.

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void convr1dcircularinv(
        double[] a,
        int m,
        double[] b,
        int n,
        out double[] r)

### `convr1dinv` function

    /*************************************************************************
    1-dimensional real deconvolution (inverse of ConvC1D()).

    Algorithm has M*log(M)) complexity for any M (composite or prime).

    INPUT PARAMETERS
        A   -   array[0..M-1] - convolved signal, A = conv(R, B)
        M   -   convolved signal length
        B   -   array[0..N-1] - response
        N   -   response length, N<=M

    OUTPUT PARAMETERS
        R   -   deconvolved signal. array[0..M-N].

    NOTE:
        deconvolution is unstable process and may result in division  by  zero
    (if your response function is degenerate, i.e. has zero Fourier coefficient).

    NOTE:
        It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
    functions have non-zero values at negative T's, you  can  still  use  this
    subroutine - just shift its result correspondingly.

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void convr1dinv(
        double[] a,
        int m,
        double[] b,
        int n,
        out double[] r)

## `corr` subpackage

### Functions

corrc1d
corrc1dcircular
corrr1d
corrr1dcircular

### Examples

### `corrc1d` function

    /*************************************************************************
    1-dimensional complex cross-correlation.

    For given Pattern/Signal returns corr(Pattern,Signal) (non-circular).

    Correlation is calculated using reduction to  convolution.  Algorithm with
    max(N,N)*log(max(N,N)) complexity is used (see  ConvC1D()  for  more  info
    about performance).

    IMPORTANT:
        for  historical reasons subroutine accepts its parameters in  reversed
        order: CorrC1D(Signal, Pattern) = Pattern x Signal (using  traditional
        definition of cross-correlation, denoting cross-correlation as "x").

    INPUT PARAMETERS
        Signal  -   array[0..N-1] - complex function to be transformed,
                    signal containing pattern
        N       -   problem size
        Pattern -   array[0..M-1] - complex function to be transformed,
                    pattern to search withing signal
        M       -   problem size

    OUTPUT PARAMETERS
        R       -   cross-correlation, array[0..N+M-2]:
                    * positive lags are stored in R[0..N-1],
                      R[i] = sum(conj(pattern[j])*signal[i+j]
                    * negative lags are stored in R[N..N+M-2],
                      R[N+M-1-i] = sum(conj(pattern[j])*signal[-i+j]

    NOTE:
        It is assumed that pattern domain is [0..M-1].  If Pattern is non-zero
    on [-K..M-1],  you can still use this subroutine, just shift result by K.

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void corrc1d(
        complex[] signal,
        int n,
        complex[] pattern,
        int m,
        out complex[] r)

### `corrc1dcircular` function

    /*************************************************************************
    1-dimensional circular complex cross-correlation.

    For given Pattern/Signal returns corr(Pattern,Signal) (circular).
    Algorithm has linearithmic complexity for any M/N.

    IMPORTANT:
        for  historical reasons subroutine accepts its parameters in  reversed
        order:   CorrC1DCircular(Signal, Pattern) = Pattern x Signal    (using
        traditional definition of cross-correlation, denoting cross-correlation
        as "x").

    INPUT PARAMETERS
        Signal  -   array[0..N-1] - complex function to be transformed,
                    periodic signal containing pattern
        N       -   problem size
        Pattern -   array[0..M-1] - complex function to be transformed,
                    non-periodic pattern to search withing signal
        M       -   problem size

    OUTPUT PARAMETERS
        R   -   convolution: A*B. array[0..M-1].

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void corrc1dcircular(
        complex[] signal,
        int m,
        complex[] pattern,
        int n,
        out complex[] c)

### `corrr1d` function

    /*************************************************************************
    1-dimensional real cross-correlation.

    For given Pattern/Signal returns corr(Pattern,Signal) (non-circular).

    Correlation is calculated using reduction to  convolution.  Algorithm with
    max(N,N)*log(max(N,N)) complexity is used (see  ConvC1D()  for  more  info
    about performance).

    IMPORTANT:
        for  historical reasons subroutine accepts its parameters in  reversed
        order: CorrR1D(Signal, Pattern) = Pattern x Signal (using  traditional
        definition of cross-correlation, denoting cross-correlation as "x").

    INPUT PARAMETERS
        Signal  -   array[0..N-1] - real function to be transformed,
                    signal containing pattern
        N       -   problem size
        Pattern -   array[0..M-1] - real function to be transformed,
                    pattern to search withing signal
        M       -   problem size

    OUTPUT PARAMETERS
        R       -   cross-correlation, array[0..N+M-2]:
                    * positive lags are stored in R[0..N-1],
                      R[i] = sum(pattern[j]*signal[i+j]
                    * negative lags are stored in R[N..N+M-2],
                      R[N+M-1-i] = sum(pattern[j]*signal[-i+j]

    NOTE:
        It is assumed that pattern domain is [0..M-1].  If Pattern is non-zero
    on [-K..M-1],  you can still use this subroutine, just shift result by K.

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void corrr1d(
        double[] signal,
        int n,
        double[] pattern,
        int m,
        out double[] r)

### `corrr1dcircular` function

    /*************************************************************************
    1-dimensional circular real cross-correlation.

    For given Pattern/Signal returns corr(Pattern,Signal) (circular).
    Algorithm has linearithmic complexity for any M/N.

    IMPORTANT:
        for  historical reasons subroutine accepts its parameters in  reversed
        order:   CorrR1DCircular(Signal, Pattern) = Pattern x Signal    (using
        traditional definition of cross-correlation, denoting cross-correlation
        as "x").

    INPUT PARAMETERS
        Signal  -   array[0..N-1] - real function to be transformed,
                    periodic signal containing pattern
        N       -   problem size
        Pattern -   array[0..M-1] - real function to be transformed,
                    non-periodic pattern to search withing signal
        M       -   problem size

    OUTPUT PARAMETERS
        R   -   convolution: A*B. array[0..M-1].

      -- ALGLIB --
         Copyright 21.07.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void corrr1dcircular(
        double[] signal,
        int m,
        double[] pattern,
        int n,
        out double[] c)

## `correlationtests` subpackage

### Functions

pearsoncorrelationsignificance
spearmanrankcorrelationsignificance

### Examples

### `pearsoncorrelationsignificance` function

    /*************************************************************************
    Pearson's correlation coefficient significance test

    This test checks hypotheses about whether X  and  Y  are  samples  of  two
    continuous  distributions  having  zero  correlation  or   whether   their
    correlation is non-zero.

    The following tests are performed:
        * two-tailed test (null hypothesis - X and Y have zero correlation)
        * left-tailed test (null hypothesis - the correlation  coefficient  is
          greater than or equal to 0)
        * right-tailed test (null hypothesis - the correlation coefficient  is
          less than or equal to 0).

    Requirements:
        * the number of elements in each sample is not less than 5
        * normality of distributions of X and Y.

    Input parameters:
        R   -   Pearson's correlation coefficient for X and Y
        N   -   number of elements in samples, N>=5.

    Output parameters:
        BothTails   -   p-value for two-tailed test.
                        If BothTails is less than the given significance level
                        the null hypothesis is rejected.
        LeftTail    -   p-value for left-tailed test.
                        If LeftTail is less than the given significance level,
                        the null hypothesis is rejected.
        RightTail   -   p-value for right-tailed test.
                        If RightTail is less than the given significance level
                        the null hypothesis is rejected.

      -- ALGLIB --
         Copyright 09.04.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void pearsoncorrelationsignificance(
        double r,
        int n,
        out double bothtails,
        out double lefttail,
        out double righttail)

### `spearmanrankcorrelationsignificance` function

    /*************************************************************************
    Spearman's rank correlation coefficient significance test

    This test checks hypotheses about whether X  and  Y  are  samples  of  two
    continuous  distributions  having  zero  correlation  or   whether   their
    correlation is non-zero.

    The following tests are performed:
        * two-tailed test (null hypothesis - X and Y have zero correlation)
        * left-tailed test (null hypothesis - the correlation  coefficient  is
          greater than or equal to 0)
        * right-tailed test (null hypothesis - the correlation coefficient  is
          less than or equal to 0).

    Requirements:
        * the number of elements in each sample is not less than 5.

    The test is non-parametric and doesn't require distributions X and Y to be
    normal.

    Input parameters:
        R   -   Spearman's rank correlation coefficient for X and Y
        N   -   number of elements in samples, N>=5.

    Output parameters:
        BothTails   -   p-value for two-tailed test.
                        If BothTails is less than the given significance level
                        the null hypothesis is rejected.
        LeftTail    -   p-value for left-tailed test.
                        If LeftTail is less than the given significance level,
                        the null hypothesis is rejected.
        RightTail   -   p-value for right-tailed test.
                        If RightTail is less than the given significance level
                        the null hypothesis is rejected.

      -- ALGLIB --
         Copyright 09.04.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spearmanrankcorrelationsignificance(
        double r,
        int n,
        out double bothtails,
        out double lefttail,
        out double righttail)

## `dawson` subpackage

### Functions

dawsonintegral

### Examples

### `dawsonintegral` function

    /*************************************************************************
    Dawson's Integral

    Approximates the integral

                                x
                                -
                         2     | |        2
     dawsn(x)  =  exp( -x  )   |    exp( t  ) dt
                             | |
                              -
                              0

    Three different rational approximations are employed, for
    the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE      0,10        10000       6.9e-16     1.0e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double dawsonintegral(double x)

## `densesolver` subpackage

### Classes

densesolverlsreport
densesolverreport

### Functions

cmatrixlusolve
cmatrixlusolvem
cmatrixmixedsolve
cmatrixmixedsolvem
cmatrixsolve
cmatrixsolvem
hpdmatrixcholeskysolve
hpdmatrixcholeskysolvem
hpdmatrixsolve
hpdmatrixsolvem
rmatrixlusolve
rmatrixlusolvem
rmatrixmixedsolve
rmatrixmixedsolvem
rmatrixsolve
rmatrixsolvels
rmatrixsolvem
spdmatrixcholeskysolve
spdmatrixcholeskysolvem
spdmatrixsolve
spdmatrixsolvem

### Examples

### `densesolverlsreport` class

    /*************************************************************************

    *************************************************************************/
    public class densesolverlsreport
    {
        public double               r2;
        public double[,]            cx;
        public int                  n;
        public int                  k;
    }

### `densesolverreport` class

    /*************************************************************************

    *************************************************************************/
    public class densesolverreport
    {
        public double               r1;
        public double               rinf;
    }

### `cmatrixlusolve` function

    /*************************************************************************
    Dense solver. Same as RMatrixLUSolve(), but for complex matrices.

    Algorithm features:
    * automatic detection of degenerate cases
    * O(N^2) complexity
    * condition number estimation

    No iterative refinement is provided because exact form of original matrix
    is not known to subroutine. Use CMatrixSolve or CMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        LUA     -   array[0..N-1,0..N-1], LU decomposition, CMatrixLU result
        P       -   array[0..N-1], pivots array, CMatrixLU result
        N       -   size of A
        B       -   array[0..N-1], right part

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixlusolve(
        complex[,] lua,
        int[] p,
        int n,
        complex[] b,
        out int info,
        out densesolverreport rep,
        out complex[] x)

### `cmatrixlusolvem` function

    /*************************************************************************
    Dense solver. Same as RMatrixLUSolveM(), but for complex matrices.

    Algorithm features:
    * automatic detection of degenerate cases
    * O(M*N^2) complexity
    * condition number estimation

    No iterative refinement  is provided because exact form of original matrix
    is not known to subroutine. Use CMatrixSolve or CMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
        P       -   array[0..N-1], pivots array, RMatrixLU result
        N       -   size of A
        B       -   array[0..N-1,0..M-1], right part
        M       -   right part size

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixlusolvem(
        complex[,] lua,
        int[] p,
        int n,
        complex[,] b,
        int m,
        out int info,
        out densesolverreport rep,
        out complex[,] x)

### `cmatrixmixedsolve` function

    /*************************************************************************
    Dense solver. Same as RMatrixMixedSolve(), but for complex matrices.

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * iterative refinement
    * O(N^2) complexity

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        LUA     -   array[0..N-1,0..N-1], LU decomposition, CMatrixLU result
        P       -   array[0..N-1], pivots array, CMatrixLU result
        N       -   size of A
        B       -   array[0..N-1], right part

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolveM
        Rep     -   same as in RMatrixSolveM
        X       -   same as in RMatrixSolveM

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixmixedsolve(
        complex[,] a,
        complex[,] lua,
        int[] p,
        int n,
        complex[] b,
        out int info,
        out densesolverreport rep,
        out complex[] x)

### `cmatrixmixedsolvem` function

    /*************************************************************************
    Dense solver. Same as RMatrixMixedSolveM(), but for complex matrices.

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * iterative refinement
    * O(M*N^2) complexity

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        LUA     -   array[0..N-1,0..N-1], LU decomposition, CMatrixLU result
        P       -   array[0..N-1], pivots array, CMatrixLU result
        N       -   size of A
        B       -   array[0..N-1,0..M-1], right part
        M       -   right part size

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolveM
        Rep     -   same as in RMatrixSolveM
        X       -   same as in RMatrixSolveM

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixmixedsolvem(
        complex[,] a,
        complex[,] lua,
        int[] p,
        int n,
        complex[,] b,
        int m,
        out int info,
        out densesolverreport rep,
        out complex[,] x)

### `cmatrixsolve` function

    /*************************************************************************
    Dense solver. Same as RMatrixSolve(), but for complex matrices.

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * iterative refinement
    * O(N^3) complexity

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        N       -   size of A
        B       -   array[0..N-1], right part

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixsolve(
        complex[,] a,
        int n,
        complex[] b,
        out int info,
        out densesolverreport rep,
        out complex[] x)

### `cmatrixsolvem` function

    /*************************************************************************
    Dense solver. Same as RMatrixSolveM(), but for complex matrices.

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * iterative refinement
    * O(N^3+M*N^2) complexity

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        N       -   size of A
        B       -   array[0..N-1,0..M-1], right part
        M       -   right part size
        RFS     -   iterative refinement switch:
                    * True - refinement is used.
                      Less performance, more precision.
                    * False - refinement is not used.
                      More performance, less precision.

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixsolvem(
        complex[,] a,
        int n,
        complex[,] b,
        int m,
        bool rfs,
        out int info,
        out densesolverreport rep,
        out complex[,] x)

### `hpdmatrixcholeskysolve` function

    /*************************************************************************
    Dense solver. Same as RMatrixLUSolve(), but for  HPD matrices  represented
    by their Cholesky decomposition.

    Algorithm features:
    * automatic detection of degenerate cases
    * O(N^2) complexity
    * condition number estimation
    * matrix is represented by its upper or lower triangle

    No iterative refinement is provided because such partial representation of
    matrix does not allow efficient calculation of extra-precise  matrix-vector
    products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        CHA     -   array[0..N-1,0..N-1], Cholesky decomposition,
                    SPDMatrixCholesky result
        N       -   size of A
        IsUpper -   what half of CHA is provided
        B       -   array[0..N-1], right part

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void hpdmatrixcholeskysolve(
        complex[,] cha,
        int n,
        bool isupper,
        complex[] b,
        out int info,
        out densesolverreport rep,
        out complex[] x)

### `hpdmatrixcholeskysolvem` function

    /*************************************************************************
    Dense solver. Same as RMatrixLUSolveM(), but for HPD matrices  represented
    by their Cholesky decomposition.

    Algorithm features:
    * automatic detection of degenerate cases
    * O(M*N^2) complexity
    * condition number estimation
    * matrix is represented by its upper or lower triangle

    No iterative refinement is provided because such partial representation of
    matrix does not allow efficient calculation of extra-precise  matrix-vector
    products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        CHA     -   array[0..N-1,0..N-1], Cholesky decomposition,
                    HPDMatrixCholesky result
        N       -   size of CHA
        IsUpper -   what half of CHA is provided
        B       -   array[0..N-1,0..M-1], right part
        M       -   right part size

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void hpdmatrixcholeskysolvem(
        complex[,] cha,
        int n,
        bool isupper,
        complex[,] b,
        int m,
        out int info,
        out densesolverreport rep,
        out complex[,] x)

### `hpdmatrixsolve` function

    /*************************************************************************
    Dense solver. Same as RMatrixSolve(),  but for Hermitian positive definite
    matrices.

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * O(N^3) complexity
    * matrix is represented by its upper or lower triangle

    No iterative refinement is provided because such partial representation of
    matrix does not allow efficient calculation of extra-precise  matrix-vector
    products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        N       -   size of A
        IsUpper -   what half of A is provided
        B       -   array[0..N-1], right part

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
                    Returns -3 for non-HPD matrices.
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void hpdmatrixsolve(
        complex[,] a,
        int n,
        bool isupper,
        complex[] b,
        out int info,
        out densesolverreport rep,
        out complex[] x)

### `hpdmatrixsolvem` function

    /*************************************************************************
    Dense solver. Same as RMatrixSolveM(), but for Hermitian positive definite
    matrices.

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * O(N^3+M*N^2) complexity
    * matrix is represented by its upper or lower triangle

    No iterative refinement is provided because such partial representation of
    matrix does not allow efficient calculation of extra-precise  matrix-vector
    products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        N       -   size of A
        IsUpper -   what half of A is provided
        B       -   array[0..N-1,0..M-1], right part
        M       -   right part size

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve.
                    Returns -3 for non-HPD matrices.
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void hpdmatrixsolvem(
        complex[,] a,
        int n,
        bool isupper,
        complex[,] b,
        int m,
        out int info,
        out densesolverreport rep,
        out complex[,] x)

### `rmatrixlusolve` function

    /*************************************************************************
    Dense solver.

    This  subroutine  solves  a  system  A*X=B,  where A is NxN non-denegerate
    real matrix given by its LU decomposition, X and B are NxM real matrices.

    Algorithm features:
    * automatic detection of degenerate cases
    * O(N^2) complexity
    * condition number estimation

    No iterative refinement  is provided because exact form of original matrix
    is not known to subroutine. Use RMatrixSolve or RMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
        P       -   array[0..N-1], pivots array, RMatrixLU result
        N       -   size of A
        B       -   array[0..N-1], right part

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixlusolve(
        double[,] lua,
        int[] p,
        int n,
        double[] b,
        out int info,
        out densesolverreport rep,
        out double[] x)

### `rmatrixlusolvem` function

    /*************************************************************************
    Dense solver.

    Similar to RMatrixLUSolve() but solves task with multiple right parts
    (where b and x are NxM matrices).

    Algorithm features:
    * automatic detection of degenerate cases
    * O(M*N^2) complexity
    * condition number estimation

    No iterative refinement  is provided because exact form of original matrix
    is not known to subroutine. Use RMatrixSolve or RMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
        P       -   array[0..N-1], pivots array, RMatrixLU result
        N       -   size of A
        B       -   array[0..N-1,0..M-1], right part
        M       -   right part size

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixlusolvem(
        double[,] lua,
        int[] p,
        int n,
        double[,] b,
        int m,
        out int info,
        out densesolverreport rep,
        out double[,] x)

### `rmatrixmixedsolve` function

    /*************************************************************************
    Dense solver.

    This  subroutine  solves  a  system  A*x=b,  where BOTH ORIGINAL A AND ITS
    LU DECOMPOSITION ARE KNOWN. You can use it if for some  reasons  you  have
    both A and its LU decomposition.

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * iterative refinement
    * O(N^2) complexity

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
        P       -   array[0..N-1], pivots array, RMatrixLU result
        N       -   size of A
        B       -   array[0..N-1], right part

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolveM
        Rep     -   same as in RMatrixSolveM
        X       -   same as in RMatrixSolveM

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixmixedsolve(
        double[,] a,
        double[,] lua,
        int[] p,
        int n,
        double[] b,
        out int info,
        out densesolverreport rep,
        out double[] x)

### `rmatrixmixedsolvem` function

    /*************************************************************************
    Dense solver.

    Similar to RMatrixMixedSolve() but  solves task with multiple right  parts
    (where b and x are NxM matrices).

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * iterative refinement
    * O(M*N^2) complexity

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
        P       -   array[0..N-1], pivots array, RMatrixLU result
        N       -   size of A
        B       -   array[0..N-1,0..M-1], right part
        M       -   right part size

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolveM
        Rep     -   same as in RMatrixSolveM
        X       -   same as in RMatrixSolveM

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixmixedsolvem(
        double[,] a,
        double[,] lua,
        int[] p,
        int n,
        double[,] b,
        int m,
        out int info,
        out densesolverreport rep,
        out double[,] x)

### `rmatrixsolve` function

    /*************************************************************************
    Dense solver.

    This  subroutine  solves  a  system  A*x=b,  where A is NxN non-denegerate
    real matrix, x and b are vectors.

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * iterative refinement
    * O(N^3) complexity

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        N       -   size of A
        B       -   array[0..N-1], right part

    OUTPUT PARAMETERS
        Info    -   return code:
                    * -3    A is singular, or VERY close to singular.
                            X is filled by zeros in such cases.
                    * -1    N<=0 was passed
                    *  1    task is solved (but matrix A may be ill-conditioned,
                            check R1/RInf parameters for condition numbers).
        Rep     -   solver report, see below for more info
        X       -   array[0..N-1], it contains:
                    * solution of A*x=b if A is non-singular (well-conditioned
                      or ill-conditioned, but not very close to singular)
                    * zeros,  if  A  is  singular  or  VERY  close to singular
                      (in this case Info=-3).

    SOLVER REPORT

    Subroutine sets following fields of the Rep structure:
    * R1        reciprocal of condition number: 1/cond(A), 1-norm.
    * RInf      reciprocal of condition number: 1/cond(A), inf-norm.

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixsolve(
        double[,] a,
        int n,
        double[] b,
        out int info,
        out densesolverreport rep,
        out double[] x)

### `rmatrixsolvels` function

    /*************************************************************************
    Dense solver.

    This subroutine finds solution of the linear system A*X=B with non-square,
    possibly degenerate A.  System  is  solved in the least squares sense, and
    general least squares solution  X = X0 + CX*y  which  minimizes |A*X-B| is
    returned. If A is non-degenerate, solution in the  usual sense is returned

    Algorithm features:
    * automatic detection of degenerate cases
    * iterative refinement
    * O(N^3) complexity

    INPUT PARAMETERS
        A       -   array[0..NRows-1,0..NCols-1], system matrix
        NRows   -   vertical size of A
        NCols   -   horizontal size of A
        B       -   array[0..NCols-1], right part
        Threshold-  a number in [0,1]. Singular values  beyond  Threshold  are
                    considered  zero.  Set  it to 0.0, if you don't understand
                    what it means, so the solver will choose good value on its
                    own.

    OUTPUT PARAMETERS
        Info    -   return code:
                    * -4    SVD subroutine failed
                    * -1    if NRows<=0 or NCols<=0 or Threshold<0 was passed
                    *  1    if task is solved
        Rep     -   solver report, see below for more info
        X       -   array[0..N-1,0..M-1], it contains:
                    * solution of A*X=B if A is non-singular (well-conditioned
                      or ill-conditioned, but not very close to singular)
                    * zeros,  if  A  is  singular  or  VERY  close to singular
                      (in this case Info=-3).

    SOLVER REPORT

    Subroutine sets following fields of the Rep structure:
    * R2        reciprocal of condition number: 1/cond(A), 2-norm.
    * N         = NCols
    * K         dim(Null(A))
    * CX        array[0..N-1,0..K-1], kernel of A.
                Columns of CX store such vectors that A*CX[i]=0.

      -- ALGLIB --
         Copyright 24.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixsolvels(
        double[,] a,
        int nrows,
        int ncols,
        double[] b,
        double threshold,
        out int info,
        out densesolverlsreport rep,
        out double[] x)

### `rmatrixsolvem` function

    /*************************************************************************
    Dense solver.

    Similar to RMatrixSolve() but solves task with multiple right parts (where
    b and x are NxM matrices).

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * optional iterative refinement
    * O(N^3+M*N^2) complexity

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        N       -   size of A
        B       -   array[0..N-1,0..M-1], right part
        M       -   right part size
        RFS     -   iterative refinement switch:
                    * True - refinement is used.
                      Less performance, more precision.
                    * False - refinement is not used.
                      More performance, less precision.

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixsolvem(
        double[,] a,
        int n,
        double[,] b,
        int m,
        bool rfs,
        out int info,
        out densesolverreport rep,
        out double[,] x)

### `spdmatrixcholeskysolve` function

    /*************************************************************************
    Dense solver. Same as RMatrixLUSolve(), but for  SPD matrices  represented
    by their Cholesky decomposition.

    Algorithm features:
    * automatic detection of degenerate cases
    * O(N^2) complexity
    * condition number estimation
    * matrix is represented by its upper or lower triangle

    No iterative refinement is provided because such partial representation of
    matrix does not allow efficient calculation of extra-precise  matrix-vector
    products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        CHA     -   array[0..N-1,0..N-1], Cholesky decomposition,
                    SPDMatrixCholesky result
        N       -   size of A
        IsUpper -   what half of CHA is provided
        B       -   array[0..N-1], right part

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spdmatrixcholeskysolve(
        double[,] cha,
        int n,
        bool isupper,
        double[] b,
        out int info,
        out densesolverreport rep,
        out double[] x)

### `spdmatrixcholeskysolvem` function

    /*************************************************************************
    Dense solver. Same as RMatrixLUSolveM(), but for SPD matrices  represented
    by their Cholesky decomposition.

    Algorithm features:
    * automatic detection of degenerate cases
    * O(M*N^2) complexity
    * condition number estimation
    * matrix is represented by its upper or lower triangle

    No iterative refinement is provided because such partial representation of
    matrix does not allow efficient calculation of extra-precise  matrix-vector
    products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        CHA     -   array[0..N-1,0..N-1], Cholesky decomposition,
                    SPDMatrixCholesky result
        N       -   size of CHA
        IsUpper -   what half of CHA is provided
        B       -   array[0..N-1,0..M-1], right part
        M       -   right part size

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spdmatrixcholeskysolvem(
        double[,] cha,
        int n,
        bool isupper,
        double[,] b,
        int m,
        out int info,
        out densesolverreport rep,
        out double[,] x)

### `spdmatrixsolve` function

    /*************************************************************************
    Dense solver. Same as RMatrixSolve(), but for SPD matrices.

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * O(N^3) complexity
    * matrix is represented by its upper or lower triangle

    No iterative refinement is provided because such partial representation of
    matrix does not allow efficient calculation of extra-precise  matrix-vector
    products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        N       -   size of A
        IsUpper -   what half of A is provided
        B       -   array[0..N-1], right part

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve
                    Returns -3 for non-SPD matrices.
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spdmatrixsolve(
        double[,] a,
        int n,
        bool isupper,
        double[] b,
        out int info,
        out densesolverreport rep,
        out double[] x)

### `spdmatrixsolvem` function

    /*************************************************************************
    Dense solver. Same as RMatrixSolveM(), but for symmetric positive definite
    matrices.

    Algorithm features:
    * automatic detection of degenerate cases
    * condition number estimation
    * O(N^3+M*N^2) complexity
    * matrix is represented by its upper or lower triangle

    No iterative refinement is provided because such partial representation of
    matrix does not allow efficient calculation of extra-precise  matrix-vector
    products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
    need iterative refinement.

    INPUT PARAMETERS
        A       -   array[0..N-1,0..N-1], system matrix
        N       -   size of A
        IsUpper -   what half of A is provided
        B       -   array[0..N-1,0..M-1], right part
        M       -   right part size

    OUTPUT PARAMETERS
        Info    -   same as in RMatrixSolve.
                    Returns -3 for non-SPD matrices.
        Rep     -   same as in RMatrixSolve
        X       -   same as in RMatrixSolve

      -- ALGLIB --
         Copyright 27.01.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spdmatrixsolvem(
        double[,] a,
        int n,
        bool isupper,
        double[,] b,
        int m,
        out int info,
        out densesolverreport rep,
        out double[,] x)

## `dforest` subpackage

### Classes

decisionforest
dfreport

### Functions

dfavgce
dfavgerror
dfavgrelerror
dfbuildrandomdecisionforest
dfbuildrandomdecisionforestx1
dfprocess
dfprocessi
dfrelclserror
dfrmserror
dfserialize
dfunserialize

### Examples

### `decisionforest` class

    /*************************************************************************

    *************************************************************************/
    public class decisionforest
    {
    }

### `dfreport` class

    /*************************************************************************

    *************************************************************************/
    public class dfreport
    {
        public double               relclserror;
        public double               avgce;
        public double               rmserror;
        public double               avgerror;
        public double               avgrelerror;
        public double               oobrelclserror;
        public double               oobavgce;
        public double               oobrmserror;
        public double               oobavgerror;
        public double               oobavgrelerror;
    }

### `dfavgce` function

    /*************************************************************************
    Average cross-entropy (in bits per element) on the test set

    INPUT PARAMETERS:
        DF      -   decision forest model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        CrossEntropy/(NPoints*LN(2)).
        Zero if model solves regression task.

      -- ALGLIB --
         Copyright 16.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double dfavgce(decisionforest df, double[,] xy, int npoints)

### `dfavgerror` function

    /*************************************************************************
    Average error on the test set

    INPUT PARAMETERS:
        DF      -   decision forest model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        Its meaning for regression task is obvious. As for
        classification task, it means average error when estimating posterior
        probabilities.

      -- ALGLIB --
         Copyright 16.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double dfavgerror(
        decisionforest df,
        double[,] xy,
        int npoints)

### `dfavgrelerror` function

    /*************************************************************************
    Average relative error on the test set

    INPUT PARAMETERS:
        DF      -   decision forest model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        Its meaning for regression task is obvious. As for
        classification task, it means average relative error when estimating
        posterior probability of belonging to the correct class.

      -- ALGLIB --
         Copyright 16.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double dfavgrelerror(
        decisionforest df,
        double[,] xy,
        int npoints)

### `dfbuildrandomdecisionforest` function

    /*************************************************************************
    This subroutine builds random decision forest.

    INPUT PARAMETERS:
        XY          -   training set
        NPoints     -   training set size, NPoints>=1
        NVars       -   number of independent variables, NVars>=1
        NClasses    -   task type:
                        * NClasses=1 - regression task with one
                                       dependent variable
                        * NClasses>1 - classification task with
                                       NClasses classes.
        NTrees      -   number of trees in a forest, NTrees>=1.
                        recommended values: 50-100.
        R           -   percent of a training set used to build
                        individual trees. 0<R<=1.
                        recommended values: 0.1 <= R <= 0.66.

    OUTPUT PARAMETERS:
        Info        -   return code:
                        * -2, if there is a point with class number
                              outside of [0..NClasses-1].
                        * -1, if incorrect parameters was passed
                              (NPoints<1, NVars<1, NClasses<1, NTrees<1, R<=0
                              or R>1).
                        *  1, if task has been solved
        DF          -   model built
        Rep         -   training report, contains error on a training set
                        and out-of-bag estimates of generalization error.

      -- ALGLIB --
         Copyright 19.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void dfbuildrandomdecisionforest(
        double[,] xy,
        int npoints,
        int nvars,
        int nclasses,
        int ntrees,
        double r,
        out int info,
        out decisionforest df,
        out dfreport rep)

### `dfbuildrandomdecisionforestx1` function

    /*************************************************************************
    This subroutine builds random decision forest.
    This function gives ability to tune number of variables used when choosing
    best split.

    INPUT PARAMETERS:
        XY          -   training set
        NPoints     -   training set size, NPoints>=1
        NVars       -   number of independent variables, NVars>=1
        NClasses    -   task type:
                        * NClasses=1 - regression task with one
                                       dependent variable
                        * NClasses>1 - classification task with
                                       NClasses classes.
        NTrees      -   number of trees in a forest, NTrees>=1.
                        recommended values: 50-100.
        NRndVars    -   number of variables used when choosing best split
        R           -   percent of a training set used to build
                        individual trees. 0<R<=1.
                        recommended values: 0.1 <= R <= 0.66.

    OUTPUT PARAMETERS:
        Info        -   return code:
                        * -2, if there is a point with class number
                              outside of [0..NClasses-1].
                        * -1, if incorrect parameters was passed
                              (NPoints<1, NVars<1, NClasses<1, NTrees<1, R<=0
                              or R>1).
                        *  1, if task has been solved
        DF          -   model built
        Rep         -   training report, contains error on a training set
                        and out-of-bag estimates of generalization error.

      -- ALGLIB --
         Copyright 19.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void dfbuildrandomdecisionforestx1(
        double[,] xy,
        int npoints,
        int nvars,
        int nclasses,
        int ntrees,
        int nrndvars,
        double r,
        out int info,
        out decisionforest df,
        out dfreport rep)

### `dfprocess` function

    /*************************************************************************
    Procesing

    INPUT PARAMETERS:
        DF      -   decision forest model
        X       -   input vector,  array[0..NVars-1].

    OUTPUT PARAMETERS:
        Y       -   result. Regression estimate when solving regression  task,
                    vector of posterior probabilities for classification task.

    See also DFProcessI.

      -- ALGLIB --
         Copyright 16.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void dfprocess(
        decisionforest df,
        double[] x,
        ref double[] y)

### `dfprocessi` function

    /*************************************************************************
    'interactive' variant of DFProcess for languages like Python which support
    constructs like "Y = DFProcessI(DF,X)" and interactive mode of interpreter

    This function allocates new array on each call,  so  it  is  significantly
    slower than its 'non-interactive' counterpart, but it is  more  convenient
    when you call it from command line.

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void dfprocessi(
        decisionforest df,
        double[] x,
        out double[] y)

### `dfrelclserror` function

    /*************************************************************************
    Relative classification error on the test set

    INPUT PARAMETERS:
        DF      -   decision forest model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        percent of incorrectly classified cases.
        Zero if model solves regression task.

      -- ALGLIB --
         Copyright 16.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double dfrelclserror(
        decisionforest df,
        double[,] xy,
        int npoints)

### `dfrmserror` function

    /*************************************************************************
    RMS error on the test set

    INPUT PARAMETERS:
        DF      -   decision forest model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        root mean square error.
        Its meaning for regression task is obvious. As for
        classification task, RMS error means error when estimating posterior
        probabilities.

      -- ALGLIB --
         Copyright 16.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double dfrmserror(
        decisionforest df,
        double[,] xy,
        int npoints)

### `dfserialize` function

    /*************************************************************************
    This function serializes data structure to string.

    Important properties of s_out:
    * it contains alphanumeric characters, dots, underscores, minus signs
    * these symbols are grouped into words, which are separated by spaces
      and Windows-style (CR+LF) newlines
    * although  serializer  uses  spaces and CR+LF as separators, you can
      replace any separator character by arbitrary combination of spaces,
      tabs, Windows or Unix newlines. It allows flexible reformatting  of
      the  string  in  case you want to include it into text or XML file.
      But you should not insert separators into the middle of the "words"
      nor you should change case of letters.
    * s_out can be freely moved between 32-bit and 64-bit systems, little
      and big endian machines, and so on. You can serialize structure  on
      32-bit machine and unserialize it on 64-bit one (or vice versa), or
      serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also
      serialize  it  in  C++ version of ALGLIB and unserialize in C# one,
      and vice versa.
    *************************************************************************/
    public static void dfserialize(decisionforest obj, out string s_out)

### `dfunserialize` function

    /*************************************************************************
    This function unserializes data structure from string.
    *************************************************************************/
    public static void dfunserialize(string s_in, out decisionforest obj)

## `elliptic` subpackage

### Functions

ellipticintegrale
ellipticintegralk
ellipticintegralkhighprecision
incompleteellipticintegrale
incompleteellipticintegralk

### Examples

### `ellipticintegrale` function

    /*************************************************************************
    Complete elliptic integral of the second kind

    Approximates the integral

               pi/2
                -
               | |                 2
    E(m)  =    |    sqrt( 1 - m sin t ) dt
             | |
              -
               0

    using the approximation

         P(x)  -  x log x Q(x).

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE       0, 1       10000       2.1e-16     7.3e-17

    Cephes Math Library, Release 2.8: June, 2000
    Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double ellipticintegrale(double m)

### `ellipticintegralk` function

    /*************************************************************************
    Complete elliptic integral of the first kind

    Approximates the integral

               pi/2
                -
               | |
               |           dt
    K(m)  =    |    ------------------
               |                   2
             | |    sqrt( 1 - m sin t )
              -
               0

    using the approximation

        P(x)  -  log x Q(x).

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE       0,1        30000       2.5e-16     6.8e-17

    Cephes Math Library, Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double ellipticintegralk(double m)

### `ellipticintegralkhighprecision` function

    /*************************************************************************
    Complete elliptic integral of the first kind

    Approximates the integral

               pi/2
                -
               | |
               |           dt
    K(m)  =    |    ------------------
               |                   2
             | |    sqrt( 1 - m sin t )
              -
               0

    where m = 1 - m1, using the approximation

        P(x)  -  log x Q(x).

    The argument m1 is used rather than m so that the logarithmic
    singularity at m = 1 will be shifted to the origin; this
    preserves maximum accuracy.

    K(0) = pi/2.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE       0,1        30000       2.5e-16     6.8e-17

    Àëãîðèòì âçÿò èç áèáëèîòåêè Cephes
    *************************************************************************/
    public static double ellipticintegralkhighprecision(double m1)

### `incompleteellipticintegrale` function

    /*************************************************************************
    Incomplete elliptic integral of the second kind

    Approximates the integral

                   phi
                    -
                   | |
                   |                   2
    E(phi_\m)  =    |    sqrt( 1 - m sin t ) dt
                   |
                 | |
                  -
                   0

    of amplitude phi and modulus m, using the arithmetic -
    geometric mean algorithm.

    ACCURACY:

    Tested at random arguments with phi in [-10, 10] and m in
    [0, 1].
                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE     -10,10      150000       3.3e-15     1.4e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1993, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double incompleteellipticintegrale(double phi, double m)

### `incompleteellipticintegralk` function

    /*************************************************************************
    Incomplete elliptic integral of the first kind F(phi|m)

    Approximates the integral

                   phi
                    -
                   | |
                   |           dt
    F(phi_\m)  =    |    ------------------
                   |                   2
                 | |    sqrt( 1 - m sin t )
                  -
                   0

    of amplitude phi and modulus m, using the arithmetic -
    geometric mean algorithm.

    ACCURACY:

    Tested at random points with m in [0, 1] and phi as indicated.

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE     -10,10       200000      7.4e-16     1.0e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double incompleteellipticintegralk(double phi, double m)

## `evd` subpackage

### Functions

hmatrixevd
hmatrixevdi
hmatrixevdr
rmatrixevd
smatrixevd
smatrixevdi
smatrixevdr
smatrixtdevd
smatrixtdevdi
smatrixtdevdr

### Examples

### `hmatrixevd` function

    /*************************************************************************
    Finding the eigenvalues and eigenvectors of a Hermitian matrix

    The algorithm finds eigen pairs of a Hermitian matrix by  reducing  it  to
    real tridiagonal form and using the QL/QR algorithm.

    Input parameters:
        A       -   Hermitian matrix which is given  by  its  upper  or  lower
                    triangular part.
                    Array whose indexes range within [0..N-1, 0..N-1].
        N       -   size of matrix A.
        IsUpper -   storage format.
        ZNeeded -   flag controlling whether the eigenvectors  are  needed  or
                    not. If ZNeeded is equal to:
                     * 0, the eigenvectors are not returned;
                     * 1, the eigenvectors are returned.

    Output parameters:
        D       -   eigenvalues in ascending order.
                    Array whose index ranges within [0..N-1].
        Z       -   if ZNeeded is equal to:
                     * 0, Z hasn’t changed;
                     * 1, Z contains the eigenvectors.
                    Array whose indexes range within [0..N-1, 0..N-1].
                    The eigenvectors are stored in the matrix columns.

    Result:
        True, if the algorithm has converged.
        False, if the algorithm hasn't converged (rare case).

    Note:
        eigenvectors of Hermitian matrix are defined up to  multiplication  by
        a complex number L, such that |L|=1.

      -- ALGLIB --
         Copyright 2005, 23 March 2007 by Bochkanov Sergey
    *************************************************************************/
    public static bool hmatrixevd(
        complex[,] a,
        int n,
        int zneeded,
        bool isupper,
        out double[] d,
        out complex[,] z)

### `hmatrixevdi` function

    /*************************************************************************
    Subroutine for finding the eigenvalues and  eigenvectors  of  a  Hermitian
    matrix with given indexes by using bisection and inverse iteration methods

    Input parameters:
        A       -   Hermitian matrix which is given  by  its  upper  or  lower
                    triangular part.
                    Array whose indexes range within [0..N-1, 0..N-1].
        N       -   size of matrix A.
        ZNeeded -   flag controlling whether the eigenvectors  are  needed  or
                    not. If ZNeeded is equal to:
                     * 0, the eigenvectors are not returned;
                     * 1, the eigenvectors are returned.
        IsUpperA -  storage format of matrix A.
        I1, I2 -    index interval for searching (from I1 to I2).
                    0 <= I1 <= I2 <= N-1.

    Output parameters:
        W       -   array of the eigenvalues found.
                    Array whose index ranges within [0..I2-I1].
        Z       -   if ZNeeded is equal to:
                     * 0, Z hasn’t changed;
                     * 1, Z contains eigenvectors.
                    Array whose indexes range within [0..N-1, 0..I2-I1].
                    In  that  case,  the eigenvectors are stored in the matrix
                    columns.

    Result:
        True, if successful. W contains the eigenvalues, Z contains the
        eigenvectors (if needed).

        False, if the bisection method subroutine  wasn't  able  to  find  the
        eigenvalues  in  the  given  interval  or  if  the  inverse  iteration
        subroutine wasn't able to find  all  the  corresponding  eigenvectors.
        In that case, the eigenvalues and eigenvectors are not returned.

    Note:
        eigen vectors of Hermitian matrix are defined up to multiplication  by
        a complex number L, such as |L|=1.

      -- ALGLIB --
         Copyright 07.01.2006, 24.03.2007 by Bochkanov Sergey.
    *************************************************************************/
    public static bool hmatrixevdi(
        complex[,] a,
        int n,
        int zneeded,
        bool isupper,
        int i1,
        int i2,
        out double[] w,
        out complex[,] z)

### `hmatrixevdr` function

    /*************************************************************************
    Subroutine for finding the eigenvalues (and eigenvectors) of  a  Hermitian
    matrix  in  a  given half-interval (A, B] by using a bisection and inverse
    iteration

    Input parameters:
        A       -   Hermitian matrix which is given  by  its  upper  or  lower
                    triangular  part.  Array  whose   indexes   range   within
                    [0..N-1, 0..N-1].
        N       -   size of matrix A.
        ZNeeded -   flag controlling whether the eigenvectors  are  needed  or
                    not. If ZNeeded is equal to:
                     * 0, the eigenvectors are not returned;
                     * 1, the eigenvectors are returned.
        IsUpperA -  storage format of matrix A.
        B1, B2 -    half-interval (B1, B2] to search eigenvalues in.

    Output parameters:
        M       -   number of eigenvalues found in a given half-interval, M>=0
        W       -   array of the eigenvalues found.
                    Array whose index ranges within [0..M-1].
        Z       -   if ZNeeded is equal to:
                     * 0, Z hasn’t changed;
                     * 1, Z contains eigenvectors.
                    Array whose indexes range within [0..N-1, 0..M-1].
                    The eigenvectors are stored in the matrix columns.

    Result:
        True, if successful. M contains the number of eigenvalues in the given
        half-interval (could be equal to 0), W contains the eigenvalues,
        Z contains the eigenvectors (if needed).

        False, if the bisection method subroutine  wasn't  able  to  find  the
        eigenvalues  in  the  given  interval  or  if  the  inverse  iteration
        subroutine  wasn't  able  to  find all the corresponding eigenvectors.
        In that case, the eigenvalues and eigenvectors are not returned, M  is
        equal to 0.

    Note:
        eigen vectors of Hermitian matrix are defined up to multiplication  by
        a complex number L, such as |L|=1.

      -- ALGLIB --
         Copyright 07.01.2006, 24.03.2007 by Bochkanov Sergey.
    *************************************************************************/
    public static bool hmatrixevdr(
        complex[,] a,
        int n,
        int zneeded,
        bool isupper,
        double b1,
        double b2,
        out int m,
        out double[] w,
        out complex[,] z)

### `rmatrixevd` function

    /*************************************************************************
    Finding eigenvalues and eigenvectors of a general matrix

    The algorithm finds eigenvalues and eigenvectors of a general matrix by
    using the QR algorithm with multiple shifts. The algorithm can find
    eigenvalues and both left and right eigenvectors.

    The right eigenvector is a vector x such that A*x = w*x, and the left
    eigenvector is a vector y such that y'*A = w*y' (here y' implies a complex
    conjugate transposition of vector y).

    Input parameters:
        A       -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
        N       -   size of matrix A.
        VNeeded -   flag controlling whether eigenvectors are needed or not.
                    If VNeeded is equal to:
                     * 0, eigenvectors are not returned;
                     * 1, right eigenvectors are returned;
                     * 2, left eigenvectors are returned;
                     * 3, both left and right eigenvectors are returned.

    Output parameters:
        WR      -   real parts of eigenvalues.
                    Array whose index ranges within [0..N-1].
        WR      -   imaginary parts of eigenvalues.
                    Array whose index ranges within [0..N-1].
        VL, VR  -   arrays of left and right eigenvectors (if they are needed).
                    If WI[i]=0, the respective eigenvalue is a real number,
                    and it corresponds to the column number I of matrices VL/VR.
                    If WI[i]>0, we have a pair of complex conjugate numbers with
                    positive and negative imaginary parts:
                        the first eigenvalue WR[i] + sqrt(-1)*WI[i];
                        the second eigenvalue WR[i+1] + sqrt(-1)*WI[i+1];
                        WI[i]>0
                        WI[i+1] = -WI[i] < 0
                    In that case, the eigenvector  corresponding to the first
                    eigenvalue is located in i and i+1 columns of matrices
                    VL/VR (the column number i contains the real part, and the
                    column number i+1 contains the imaginary part), and the vector
                    corresponding to the second eigenvalue is a complex conjugate to
                    the first vector.
                    Arrays whose indexes range within [0..N-1, 0..N-1].

    Result:
        True, if the algorithm has converged.
        False, if the algorithm has not converged.

    Note 1:
        Some users may ask the following question: what if WI[N-1]>0?
        WI[N] must contain an eigenvalue which is complex conjugate to the
        N-th eigenvalue, but the array has only size N?
        The answer is as follows: such a situation cannot occur because the
        algorithm finds a pairs of eigenvalues, therefore, if WI[i]>0, I is
        strictly less than N-1.

    Note 2:
        The algorithm performance depends on the value of the internal parameter
        NS of the InternalSchurDecomposition subroutine which defines the number
        of shifts in the QR algorithm (similarly to the block width in block-matrix
        algorithms of linear algebra). If you require maximum performance
        on your machine, it is recommended to adjust this parameter manually.

    See also the InternalTREVC subroutine.

    The algorithm is based on the LAPACK 3.0 library.
    *************************************************************************/
    public static bool rmatrixevd(
        double[,] a,
        int n,
        int vneeded,
        out double[] wr,
        out double[] wi,
        out double[,] vl,
        out double[,] vr)

### `smatrixevd` function

    /*************************************************************************
    Finding the eigenvalues and eigenvectors of a symmetric matrix

    The algorithm finds eigen pairs of a symmetric matrix by reducing it to
    tridiagonal form and using the QL/QR algorithm.

    Input parameters:
        A       -   symmetric matrix which is given by its upper or lower
                    triangular part.
                    Array whose indexes range within [0..N-1, 0..N-1].
        N       -   size of matrix A.
        ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                    If ZNeeded is equal to:
                     * 0, the eigenvectors are not returned;
                     * 1, the eigenvectors are returned.
        IsUpper -   storage format.

    Output parameters:
        D       -   eigenvalues in ascending order.
                    Array whose index ranges within [0..N-1].
        Z       -   if ZNeeded is equal to:
                     * 0, Z hasn’t changed;
                     * 1, Z contains the eigenvectors.
                    Array whose indexes range within [0..N-1, 0..N-1].
                    The eigenvectors are stored in the matrix columns.

    Result:
        True, if the algorithm has converged.
        False, if the algorithm hasn't converged (rare case).

      -- ALGLIB --
         Copyright 2005-2008 by Bochkanov Sergey
    *************************************************************************/
    public static bool smatrixevd(
        double[,] a,
        int n,
        int zneeded,
        bool isupper,
        out double[] d,
        out double[,] z)

### `smatrixevdi` function

    /*************************************************************************
    Subroutine for finding the eigenvalues and  eigenvectors  of  a  symmetric
    matrix with given indexes by using bisection and inverse iteration methods.

    Input parameters:
        A       -   symmetric matrix which is given by its upper or lower
                    triangular part. Array whose indexes range within [0..N-1, 0..N-1].
        N       -   size of matrix A.
        ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                    If ZNeeded is equal to:
                     * 0, the eigenvectors are not returned;
                     * 1, the eigenvectors are returned.
        IsUpperA -  storage format of matrix A.
        I1, I2 -    index interval for searching (from I1 to I2).
                    0 <= I1 <= I2 <= N-1.

    Output parameters:
        W       -   array of the eigenvalues found.
                    Array whose index ranges within [0..I2-I1].
        Z       -   if ZNeeded is equal to:
                     * 0, Z hasn’t changed;
                     * 1, Z contains eigenvectors.
                    Array whose indexes range within [0..N-1, 0..I2-I1].
                    In that case, the eigenvectors are stored in the matrix columns.

    Result:
        True, if successful. W contains the eigenvalues, Z contains the
        eigenvectors (if needed).

        False, if the bisection method subroutine wasn't able to find the
        eigenvalues in the given interval or if the inverse iteration subroutine
        wasn't able to find all the corresponding eigenvectors.
        In that case, the eigenvalues and eigenvectors are not returned.

      -- ALGLIB --
         Copyright 07.01.2006 by Bochkanov Sergey
    *************************************************************************/
    public static bool smatrixevdi(
        double[,] a,
        int n,
        int zneeded,
        bool isupper,
        int i1,
        int i2,
        out double[] w,
        out double[,] z)

### `smatrixevdr` function

    /*************************************************************************
    Subroutine for finding the eigenvalues (and eigenvectors) of  a  symmetric
    matrix  in  a  given half open interval (A, B] by using  a  bisection  and
    inverse iteration

    Input parameters:
        A       -   symmetric matrix which is given by its upper or lower
                    triangular part. Array [0..N-1, 0..N-1].
        N       -   size of matrix A.
        ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                    If ZNeeded is equal to:
                     * 0, the eigenvectors are not returned;
                     * 1, the eigenvectors are returned.
        IsUpperA -  storage format of matrix A.
        B1, B2 -    half open interval (B1, B2] to search eigenvalues in.

    Output parameters:
        M       -   number of eigenvalues found in a given half-interval (M>=0).
        W       -   array of the eigenvalues found.
                    Array whose index ranges within [0..M-1].
        Z       -   if ZNeeded is equal to:
                     * 0, Z hasn’t changed;
                     * 1, Z contains eigenvectors.
                    Array whose indexes range within [0..N-1, 0..M-1].
                    The eigenvectors are stored in the matrix columns.

    Result:
        True, if successful. M contains the number of eigenvalues in the given
        half-interval (could be equal to 0), W contains the eigenvalues,
        Z contains the eigenvectors (if needed).

        False, if the bisection method subroutine wasn't able to find the
        eigenvalues in the given interval or if the inverse iteration subroutine
        wasn't able to find all the corresponding eigenvectors.
        In that case, the eigenvalues and eigenvectors are not returned,
        M is equal to 0.

      -- ALGLIB --
         Copyright 07.01.2006 by Bochkanov Sergey
    *************************************************************************/
    public static bool smatrixevdr(
        double[,] a,
        int n,
        int zneeded,
        bool isupper,
        double b1,
        double b2,
        out int m,
        out double[] w,
        out double[,] z)

### `smatrixtdevd` function

    /*************************************************************************
    Finding the eigenvalues and eigenvectors of a tridiagonal symmetric matrix

    The algorithm finds the eigen pairs of a tridiagonal symmetric matrix by
    using an QL/QR algorithm with implicit shifts.

    Input parameters:
        D       -   the main diagonal of a tridiagonal matrix.
                    Array whose index ranges within [0..N-1].
        E       -   the secondary diagonal of a tridiagonal matrix.
                    Array whose index ranges within [0..N-2].
        N       -   size of matrix A.
        ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                    If ZNeeded is equal to:
                     * 0, the eigenvectors are not needed;
                     * 1, the eigenvectors of a tridiagonal matrix
                       are multiplied by the square matrix Z. It is used if the
                       tridiagonal matrix is obtained by the similarity
                       transformation of a symmetric matrix;
                     * 2, the eigenvectors of a tridiagonal matrix replace the
                       square matrix Z;
                     * 3, matrix Z contains the first row of the eigenvectors
                       matrix.
        Z       -   if ZNeeded=1, Z contains the square matrix by which the
                    eigenvectors are multiplied.
                    Array whose indexes range within [0..N-1, 0..N-1].

    Output parameters:
        D       -   eigenvalues in ascending order.
                    Array whose index ranges within [0..N-1].
        Z       -   if ZNeeded is equal to:
                     * 0, Z hasn’t changed;
                     * 1, Z contains the product of a given matrix (from the left)
                       and the eigenvectors matrix (from the right);
                     * 2, Z contains the eigenvectors.
                     * 3, Z contains the first row of the eigenvectors matrix.
                    If ZNeeded<3, Z is the array whose indexes range within [0..N-1, 0..N-1].
                    In that case, the eigenvectors are stored in the matrix columns.
                    If ZNeeded=3, Z is the array whose indexes range within [0..0, 0..N-1].

    Result:
        True, if the algorithm has converged.
        False, if the algorithm hasn't converged.

      -- LAPACK routine (version 3.0) --
         Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
         Courant Institute, Argonne National Lab, and Rice University
         September 30, 1994
    *************************************************************************/
    public static bool smatrixtdevd(
        ref double[] d,
        double[] e,
        int n,
        int zneeded,
        ref double[,] z)

### `smatrixtdevdi` function

    /*************************************************************************
    Subroutine for finding tridiagonal matrix eigenvalues/vectors with given
    indexes (in ascending order) by using the bisection and inverse iteraion.

    Input parameters:
        D       -   the main diagonal of a tridiagonal matrix.
                    Array whose index ranges within [0..N-1].
        E       -   the secondary diagonal of a tridiagonal matrix.
                    Array whose index ranges within [0..N-2].
        N       -   size of matrix. N>=0.
        ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                    If ZNeeded is equal to:
                     * 0, the eigenvectors are not needed;
                     * 1, the eigenvectors of a tridiagonal matrix are multiplied
                       by the square matrix Z. It is used if the
                       tridiagonal matrix is obtained by the similarity transformation
                       of a symmetric matrix.
                     * 2, the eigenvectors of a tridiagonal matrix replace
                       matrix Z.
        I1, I2  -   index interval for searching (from I1 to I2).
                    0 <= I1 <= I2 <= N-1.
        Z       -   if ZNeeded is equal to:
                     * 0, Z isn't used and remains unchanged;
                     * 1, Z contains the square matrix (array whose indexes range within [0..N-1, 0..N-1])
                       which reduces the given symmetric matrix to  tridiagonal form;
                     * 2, Z isn't used (but changed on the exit).

    Output parameters:
        D       -   array of the eigenvalues found.
                    Array whose index ranges within [0..I2-I1].
        Z       -   if ZNeeded is equal to:
                     * 0, doesn't contain any information;
                     * 1, contains the product of a given NxN matrix Z (from the left) and
                       Nx(I2-I1) matrix of the eigenvectors found (from the right).
                       Array whose indexes range within [0..N-1, 0..I2-I1].
                     * 2, contains the matrix of the eigenvalues found.
                       Array whose indexes range within [0..N-1, 0..I2-I1].

    Result:

        True, if successful. In that case, D contains the eigenvalues,
        Z contains the eigenvectors (if needed).
        It should be noted that the subroutine changes the size of arrays D and Z.

        False, if the bisection method subroutine wasn't able to find the eigenvalues
        in the given interval or if the inverse iteration subroutine wasn't able
        to find all the corresponding eigenvectors. In that case, the eigenvalues
        and eigenvectors are not returned.

      -- ALGLIB --
         Copyright 25.12.2005 by Bochkanov Sergey
    *************************************************************************/
    public static bool smatrixtdevdi(
        ref double[] d,
        double[] e,
        int n,
        int zneeded,
        int i1,
        int i2,
        ref double[,] z)

### `smatrixtdevdr` function

    /*************************************************************************
    Subroutine for finding the tridiagonal matrix eigenvalues/vectors in a
    given half-interval (A, B] by using bisection and inverse iteration.

    Input parameters:
        D       -   the main diagonal of a tridiagonal matrix.
                    Array whose index ranges within [0..N-1].
        E       -   the secondary diagonal of a tridiagonal matrix.
                    Array whose index ranges within [0..N-2].
        N       -   size of matrix, N>=0.
        ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                    If ZNeeded is equal to:
                     * 0, the eigenvectors are not needed;
                     * 1, the eigenvectors of a tridiagonal matrix are multiplied
                       by the square matrix Z. It is used if the tridiagonal
                       matrix is obtained by the similarity transformation
                       of a symmetric matrix.
                     * 2, the eigenvectors of a tridiagonal matrix replace matrix Z.
        A, B    -   half-interval (A, B] to search eigenvalues in.
        Z       -   if ZNeeded is equal to:
                     * 0, Z isn't used and remains unchanged;
                     * 1, Z contains the square matrix (array whose indexes range
                       within [0..N-1, 0..N-1]) which reduces the given symmetric
                       matrix to tridiagonal form;
                     * 2, Z isn't used (but changed on the exit).

    Output parameters:
        D       -   array of the eigenvalues found.
                    Array whose index ranges within [0..M-1].
        M       -   number of eigenvalues found in the given half-interval (M>=0).
        Z       -   if ZNeeded is equal to:
                     * 0, doesn't contain any information;
                     * 1, contains the product of a given NxN matrix Z (from the
                       left) and NxM matrix of the eigenvectors found (from the
                       right). Array whose indexes range within [0..N-1, 0..M-1].
                     * 2, contains the matrix of the eigenvectors found.
                       Array whose indexes range within [0..N-1, 0..M-1].

    Result:

        True, if successful. In that case, M contains the number of eigenvalues
        in the given half-interval (could be equal to 0), D contains the eigenvalues,
        Z contains the eigenvectors (if needed).
        It should be noted that the subroutine changes the size of arrays D and Z.

        False, if the bisection method subroutine wasn't able to find the
        eigenvalues in the given interval or if the inverse iteration subroutine
        wasn't able to find all the corresponding eigenvectors. In that case,
        the eigenvalues and eigenvectors are not returned, M is equal to 0.

      -- ALGLIB --
         Copyright 31.03.2008 by Bochkanov Sergey
    *************************************************************************/
    public static bool smatrixtdevdr(
        ref double[] d,
        double[] e,
        int n,
        int zneeded,
        double a,
        double b,
        out int m,
        ref double[,] z)

## `expintegrals` subpackage

### Functions

exponentialintegralei
exponentialintegralen

### Examples

### `exponentialintegralei` function

    /*************************************************************************
    Exponential integral Ei(x)

                  x
                   -     t
                  | |   e
       Ei(x) =   -|-   ---  dt .
                | |     t
                 -
                -inf

    Not defined for x <= 0.
    See also expn.c.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE       0,100       50000      8.6e-16     1.3e-16

    Cephes Math Library Release 2.8:  May, 1999
    Copyright 1999 by Stephen L. Moshier
    *************************************************************************/
    public static double exponentialintegralei(double x)

### `exponentialintegralen` function

    /*************************************************************************
    Exponential integral En(x)

    Evaluates the exponential integral

                    inf.
                      -
                     | |   -xt
                     |    e
         E (x)  =    |    ----  dt.
          n          |      n
                   | |     t
                    -
                     1

    Both n and x must be nonnegative.

    The routine employs either a power series, a continued
    fraction, or an asymptotic formula depending on the
    relative values of n and x.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE      0, 30       10000       1.7e-15     3.6e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1985, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double exponentialintegralen(double x, int n)

## `fdistr` subpackage

### Functions

fcdistribution
fdistribution
invfdistribution

### Examples

### `fcdistribution` function

    /*************************************************************************
    Complemented F distribution

    Returns the area from x to infinity under the F density
    function (also known as Snedcor's density or the
    variance ratio density).

                         inf.
                          -
                 1       | |  a-1      b-1
    1-P(x)  =  ------    |   t    (1-t)    dt
               B(a,b)  | |
                        -
                         x

    The incomplete beta integral is used, according to the
    formula

    P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).

    ACCURACY:

    Tested at random points (a,b,x) in the indicated intervals.
                   x     a,b                     Relative error:
    arithmetic  domain  domain     # trials      peak         rms
       IEEE      0,1    1,100       100000      3.7e-14     5.9e-16
       IEEE      1,5    1,100       100000      8.0e-15     1.6e-15
       IEEE      0,1    1,10000     100000      1.8e-11     3.5e-13
       IEEE      1,5    1,10000     100000      2.0e-11     3.0e-12

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double fcdistribution(int a, int b, double x)

### `fdistribution` function

    /*************************************************************************
    F distribution

    Returns the area from zero to x under the F density
    function (also known as Snedcor's density or the
    variance ratio density).  This is the density
    of x = (u1/df1)/(u2/df2), where u1 and u2 are random
    variables having Chi square distributions with df1
    and df2 degrees of freedom, respectively.
    The incomplete beta integral is used, according to the
    formula

    P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).

    The arguments a and b are greater than zero, and x is
    nonnegative.

    ACCURACY:

    Tested at random points (a,b,x).

                   x     a,b                     Relative error:
    arithmetic  domain  domain     # trials      peak         rms
       IEEE      0,1    0,100       100000      9.8e-15     1.7e-15
       IEEE      1,5    0,100       100000      6.5e-15     3.5e-16
       IEEE      0,1    1,10000     100000      2.2e-11     3.3e-12
       IEEE      1,5    1,10000     100000      1.1e-11     1.7e-13

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double fdistribution(int a, int b, double x)

### `invfdistribution` function

    /*************************************************************************
    Inverse of complemented F distribution

    Finds the F density argument x such that the integral
    from x to infinity of the F density is equal to the
    given probability p.

    This is accomplished using the inverse beta integral
    function and the relations

         z = incbi( df2/2, df1/2, p )
         x = df2 (1-z) / (df1 z).

    Note: the following relations hold for the inverse of
    the uncomplemented F distribution:

         z = incbi( df1/2, df2/2, p )
         x = df2 z / (df1 (1-z)).

    ACCURACY:

    Tested at random points (a,b,p).

                 a,b                     Relative error:
    arithmetic  domain     # trials      peak         rms
     For p between .001 and 1:
       IEEE     1,100       100000      8.3e-15     4.7e-16
       IEEE     1,10000     100000      2.1e-11     1.4e-13
     For p between 10^-6 and 10^-3:
       IEEE     1,100        50000      1.3e-12     8.4e-15
       IEEE     1,10000      50000      3.0e-12     4.8e-14

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double invfdistribution(int a, int b, double y)

## `fft` subpackage

### Functions

fftc1d
fftc1dinv
fftr1d
fftr1dinv

### Examples

fft_complex_d1 |  | Complex FFT: simple example
fft_complex_d2 |  | Complex FFT: advanced example
fft_real_d1 |  | Real FFT: simple example
fft_real_d2 |  | Real FFT: advanced example

### `fftc1d` function

    /*************************************************************************
    1-dimensional complex FFT.

    Array size N may be arbitrary number (composite or prime).  Composite  N's
    are handled with cache-oblivious variation of  a  Cooley-Tukey  algorithm.
    Small prime-factors are transformed using hard coded  codelets (similar to
    FFTW codelets, but without low-level  optimization),  large  prime-factors
    are handled with Bluestein's algorithm.

    Fastests transforms are for smooth N's (prime factors are 2, 3,  5  only),
    most fast for powers of 2. When N have prime factors  larger  than  these,
    but orders of magnitude smaller than N, computations will be about 4 times
    slower than for nearby highly composite N's. When N itself is prime, speed
    will be 6 times lower.

    Algorithm has O(N*logN) complexity for any N (composite or prime).

    INPUT PARAMETERS
        A   -   array[0..N-1] - complex function to be transformed
        N   -   problem size

    OUTPUT PARAMETERS
        A   -   DFT of a input array, array[0..N-1]
                A_out[j] = SUM(A_in[k]*exp(-2*pi*sqrt(-1)*j*k/N), k = 0..N-1)

      -- ALGLIB --
         Copyright 29.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void fftc1d(ref complex[] a)
    public static void fftc1d(ref complex[] a, int n)

Examples: [1] [2]

### `fftc1dinv` function

    /*************************************************************************
    1-dimensional complex inverse FFT.

    Array size N may be arbitrary number (composite or prime).  Algorithm  has
    O(N*logN) complexity for any N (composite or prime).

    See FFTC1D() description for more information about algorithm performance.

    INPUT PARAMETERS
        A   -   array[0..N-1] - complex array to be transformed
        N   -   problem size

    OUTPUT PARAMETERS
        A   -   inverse DFT of a input array, array[0..N-1]
                A_out[j] = SUM(A_in[k]/N*exp(+2*pi*sqrt(-1)*j*k/N), k = 0..N-1)

      -- ALGLIB --
         Copyright 29.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void fftc1dinv(ref complex[] a)
    public static void fftc1dinv(ref complex[] a, int n)

Examples: [1] [2]

### `fftr1d` function

    /*************************************************************************
    1-dimensional real FFT.

    Algorithm has O(N*logN) complexity for any N (composite or prime).

    INPUT PARAMETERS
        A   -   array[0..N-1] - real function to be transformed
        N   -   problem size

    OUTPUT PARAMETERS
        F   -   DFT of a input array, array[0..N-1]
                F[j] = SUM(A[k]*exp(-2*pi*sqrt(-1)*j*k/N), k = 0..N-1)

    NOTE:
        F[] satisfies symmetry property F[k] = conj(F[N-k]),  so just one half
    of  array  is  usually needed. But for convinience subroutine returns full
    complex array (with frequencies above N/2), so its result may be  used  by
    other FFT-related subroutines.

      -- ALGLIB --
         Copyright 01.06.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void fftr1d(double[] a, out complex[] f)
    public static void fftr1d(double[] a, int n, out complex[] f)

Examples: [1] [2]

### `fftr1dinv` function

    /*************************************************************************
    1-dimensional real inverse FFT.

    Algorithm has O(N*logN) complexity for any N (composite or prime).

    INPUT PARAMETERS
        F   -   array[0..floor(N/2)] - frequencies from forward real FFT
        N   -   problem size

    OUTPUT PARAMETERS
        A   -   inverse DFT of a input array, array[0..N-1]

    NOTE:
        F[] should satisfy symmetry property F[k] = conj(F[N-k]), so just  one
    half of frequencies array is needed - elements from 0 to floor(N/2).  F[0]
    is ALWAYS real. If N is even F[floor(N/2)] is real too. If N is odd,  then
    F[floor(N/2)] has no special properties.

    Relying on properties noted above, FFTR1DInv subroutine uses only elements
    from 0th to floor(N/2)-th. It ignores imaginary part of F[0],  and in case
    N is even it ignores imaginary part of F[floor(N/2)] too.

    When you call this function using full arguments list - "FFTR1DInv(F,N,A)"
    - you can pass either either frequencies array with N elements or  reduced
    array with roughly N/2 elements - subroutine will  successfully  transform
    both.

    If you call this function using reduced arguments list -  "FFTR1DInv(F,A)"
    - you must pass FULL array with N elements (although higher  N/2 are still
    not used) because array size is used to automatically determine FFT length

      -- ALGLIB --
         Copyright 01.06.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void fftr1dinv(complex[] f, out double[] a)
    public static void fftr1dinv(complex[] f, int n, out double[] a)

Examples: [1] [2]

### fft_complex_d1 example

    public static int Main(string[] args)
    {
        //
        // first we demonstrate forward FFT:
        // [1i,1i,1i,1i] is converted to [4i, 0, 0, 0]
        //
        alglib.complex[] z = new alglib.complex[]{new alglib.complex(0,1),new alglib.complex(0,1),new alglib.complex(0,1),new alglib.complex(0,1)};
        alglib.fftc1d(ref z);
        System.Console.WriteLine("{0}", alglib.ap.format(z,3)); // EXPECTED: [4i,0,0,0]

        //
        // now we convert [4i, 0, 0, 0] back to [1i,1i,1i,1i]
        // with backward FFT
        //
        alglib.fftc1dinv(ref z);
        System.Console.WriteLine("{0}", alglib.ap.format(z,3)); // EXPECTED: [1i,1i,1i,1i]
        System.Console.ReadLine();
        return 0;
    }

### fft_complex_d2 example

    public static int Main(string[] args)
    {
        //
        // first we demonstrate forward FFT:
        // [0,1,0,1i] is converted to [1+1i, -1-1i, -1-1i, 1+1i]
        //
        alglib.complex[] z = new alglib.complex[]{0,1,0,new alglib.complex(0,1)};
        alglib.fftc1d(ref z);
        System.Console.WriteLine("{0}", alglib.ap.format(z,3)); // EXPECTED: [1+1i, -1-1i, -1-1i, 1+1i]

        //
        // now we convert result back with backward FFT
        //
        alglib.fftc1dinv(ref z);
        System.Console.WriteLine("{0}", alglib.ap.format(z,3)); // EXPECTED: [0,1,0,1i]
        System.Console.ReadLine();
        return 0;
    }

### fft_real_d1 example

    public static int Main(string[] args)
    {
        //
        // first we demonstrate forward FFT:
        // [1,1,1,1] is converted to [4, 0, 0, 0]
        //
        double[] x = new double[]{1,1,1,1};
        alglib.complex[] f;
        double[] x2;
        alglib.fftr1d(x, out f);
        System.Console.WriteLine("{0}", alglib.ap.format(f,3)); // EXPECTED: [4,0,0,0]

        //
        // now we convert [4, 0, 0, 0] back to [1,1,1,1]
        // with backward FFT
        //
        alglib.fftr1dinv(f, out x2);
        System.Console.WriteLine("{0}", alglib.ap.format(x2,3)); // EXPECTED: [1,1,1,1]
        System.Console.ReadLine();
        return 0;
    }

### fft_real_d2 example

    public static int Main(string[] args)
    {
        //
        // first we demonstrate forward FFT:
        // [1,2,3,4] is converted to [10, -2+2i, -2, -2-2i]
        //
        // note that output array is self-adjoint:
        // * f[0] = conj(f[0])
        // * f[1] = conj(f[3])
        // * f[2] = conj(f[2])
        //
        double[] x = new double[]{1,2,3,4};
        alglib.complex[] f;
        double[] x2;
        alglib.fftr1d(x, out f);
        System.Console.WriteLine("{0}", alglib.ap.format(f,3)); // EXPECTED: [10, -2+2i, -2, -2-2i]

        //
        // now we convert [10, -2+2i, -2, -2-2i] back to [1,2,3,4]
        //
        alglib.fftr1dinv(f, out x2);
        System.Console.WriteLine("{0}", alglib.ap.format(x2,3)); // EXPECTED: [1,2,3,4]

        //
        // remember that F is self-adjoint? It means that we can pass just half
        // (slightly larger than half) of F to inverse real FFT and still get our result.
        //
        // I.e. instead [10, -2+2i, -2, -2-2i] we pass just [10, -2+2i, -2] and everything works!
        //
        // NOTE: in this case we should explicitly pass array length (which is 4) to ALGLIB;
        // if not, it will automatically use array length to determine FFT size and
        // will erroneously make half-length FFT.
        //
        f = new alglib.complex[]{10,new alglib.complex(-2,+2),-2};
        alglib.fftr1dinv(f, 4, out x2);
        System.Console.WriteLine("{0}", alglib.ap.format(x2,3)); // EXPECTED: [1,2,3,4]
        System.Console.ReadLine();
        return 0;
    }

## `fht` subpackage

### Functions

fhtr1d
fhtr1dinv

### Examples

### `fhtr1d` function

    /*************************************************************************
    1-dimensional Fast Hartley Transform.

    Algorithm has O(N*logN) complexity for any N (composite or prime).

    INPUT PARAMETERS
        A   -   array[0..N-1] - real function to be transformed
        N   -   problem size

    OUTPUT PARAMETERS
        A   -   FHT of a input array, array[0..N-1],
                A_out[k] = sum(A_in[j]*(cos(2*pi*j*k/N)+sin(2*pi*j*k/N)), j=0..N-1)

      -- ALGLIB --
         Copyright 04.06.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void fhtr1d(ref double[] a, int n)

### `fhtr1dinv` function

    /*************************************************************************
    1-dimensional inverse FHT.

    Algorithm has O(N*logN) complexity for any N (composite or prime).

    INPUT PARAMETERS
        A   -   array[0..N-1] - complex array to be transformed
        N   -   problem size

    OUTPUT PARAMETERS
        A   -   inverse FHT of a input array, array[0..N-1]

      -- ALGLIB --
         Copyright 29.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void fhtr1dinv(ref double[] a, int n)

## `fresnel` subpackage

### Functions

fresnelintegral

### Examples

### `fresnelintegral` function

    /*************************************************************************
    Fresnel integral

    Evaluates the Fresnel integrals

              x
              -
             | |
    C(x) =   |   cos(pi/2 t**2) dt,
           | |
            -
             0

              x
              -
             | |
    S(x) =   |   sin(pi/2 t**2) dt.
           | |
            -
             0

    The integrals are evaluated by a power series for x < 1.
    For x >= 1 auxiliary functions f(x) and g(x) are employed
    such that

    C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
    S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )

    ACCURACY:

     Relative error.

    Arithmetic  function   domain     # trials      peak         rms
      IEEE       S(x)      0, 10       10000       2.0e-15     3.2e-16
      IEEE       C(x)      0, 10       10000       1.8e-15     3.3e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static void fresnelintegral(double x, ref double c, ref double s)

## `gammafunc` subpackage

### Functions

gammafunction
lngamma

### Examples

### `gammafunction` function

    /*************************************************************************
    Gamma function

    Input parameters:
        X   -   argument

    Domain:
        0 < X < 171.6
        -170 < X < 0, X is not an integer.

    Relative error:
     arithmetic   domain     # trials      peak         rms
        IEEE    -170,-33      20000       2.3e-15     3.3e-16
        IEEE     -33,  33     20000       9.4e-16     2.2e-16
        IEEE      33, 171.6   20000       2.3e-15     3.2e-16

    Cephes Math Library Release 2.8:  June, 2000
    Original copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
    Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
    *************************************************************************/
    public static double gammafunction(double x)

### `lngamma` function

    /*************************************************************************
    Natural logarithm of gamma function

    Input parameters:
        X       -   argument

    Result:
        logarithm of the absolute value of the Gamma(X).

    Output parameters:
        SgnGam  -   sign(Gamma(X))

    Domain:
        0 < X < 2.55e305
        -2.55e305 < X < 0, X is not an integer.

    ACCURACY:
    arithmetic      domain        # trials     peak         rms
       IEEE    0, 3                 28000     5.4e-16     1.1e-16
       IEEE    2.718, 2.556e305     40000     3.5e-16     8.3e-17
    The error criterion was relative when the function magnitude
    was greater than one but absolute when it was less than one.

    The following test used the relative error criterion, though
    at certain points the relative error could be much higher than
    indicated.
       IEEE    -200, -4             10000     4.8e-16     1.3e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
    Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
    *************************************************************************/
    public static double lngamma(double x, out double sgngam)

## `gkq` subpackage

### Functions

gkqgenerategaussjacobi
gkqgenerategausslegendre
gkqgeneraterec
gkqlegendrecalc
gkqlegendretbl

### Examples

### `gkqgenerategaussjacobi` function

    /*************************************************************************
    Returns   Gauss   and   Gauss-Kronrod   nodes/weights   for   Gauss-Jacobi
    quadrature on [-1,1] with weight function

        W(x)=Power(1-x,Alpha)*Power(1+x,Beta).

    INPUT PARAMETERS:
        N           -   number of Kronrod nodes, must be odd number, >=3.
        Alpha       -   power-law coefficient, Alpha>-1
        Beta        -   power-law coefficient, Beta>-1

    OUTPUT PARAMETERS:
        Info        -   error code:
                        * -5    no real and positive Gauss-Kronrod formula can
                                be created for such a weight function  with  a
                                given number of nodes.
                        * -4    an  error  was   detected   when   calculating
                                weights/nodes. Alpha or  Beta  are  too  close
                                to -1 to obtain weights/nodes with high enough
                                accuracy, or, may be, N is too large.  Try  to
                                use multiple precision version.
                        * -3    internal eigenproblem solver hasn't converged
                        * -1    incorrect N was passed
                        * +1    OK
                        * +2    OK, but quadrature rule have exterior  nodes,
                                x[0]<-1 or x[n-1]>+1
        X           -   array[0..N-1] - array of quadrature nodes, ordered in
                        ascending order.
        WKronrod    -   array[0..N-1] - Kronrod weights
        WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                        corresponding to extended Kronrod nodes).

      -- ALGLIB --
         Copyright 12.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gkqgenerategaussjacobi(
        int n,
        double alpha,
        double beta,
        out int info,
        out double[] x,
        out double[] wkronrod,
        out double[] wgauss)

### `gkqgenerategausslegendre` function

    /*************************************************************************
    Returns   Gauss   and   Gauss-Kronrod   nodes/weights  for  Gauss-Legendre
    quadrature with N points.

    GKQLegendreCalc (calculation) or  GKQLegendreTbl  (precomputed  table)  is
    used depending on machine precision and number of nodes.

    INPUT PARAMETERS:
        N           -   number of Kronrod nodes, must be odd number, >=3.

    OUTPUT PARAMETERS:
        Info        -   error code:
                        * -4    an  error   was   detected   when  calculating
                                weights/nodes.  N  is  too  large   to  obtain
                                weights/nodes  with  high   enough   accuracy.
                                Try  to   use   multiple   precision  version.
                        * -3    internal eigenproblem solver hasn't converged
                        * -1    incorrect N was passed
                        * +1    OK
        X           -   array[0..N-1] - array of quadrature nodes, ordered in
                        ascending order.
        WKronrod    -   array[0..N-1] - Kronrod weights
        WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                        corresponding to extended Kronrod nodes).

      -- ALGLIB --
         Copyright 12.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gkqgenerategausslegendre(
        int n,
        out int info,
        out double[] x,
        out double[] wkronrod,
        out double[] wgauss)

### `gkqgeneraterec` function

    /*************************************************************************
    Computation of nodes and weights of a Gauss-Kronrod quadrature formula

    The algorithm generates the N-point Gauss-Kronrod quadrature formula  with
    weight  function  given  by  coefficients  alpha  and beta of a recurrence
    relation which generates a system of orthogonal polynomials:

        P-1(x)   =  0
        P0(x)    =  1
        Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

    and zero moment Mu0

        Mu0 = integral(W(x)dx,a,b)

    INPUT PARAMETERS:
        Alpha       –   alpha coefficients, array[0..floor(3*K/2)].
        Beta        –   beta coefficients,  array[0..ceil(3*K/2)].
                        Beta[0] is not used and may be arbitrary.
                        Beta[I]>0.
        Mu0         –   zeroth moment of the weight function.
        N           –   number of nodes of the Gauss-Kronrod quadrature formula,
                        N >= 3,
                        N =  2*K+1.

    OUTPUT PARAMETERS:
        Info        -   error code:
                        * -5    no real and positive Gauss-Kronrod formula can
                                be created for such a weight function  with  a
                                given number of nodes.
                        * -4    N is too large, task may be ill  conditioned -
                                x[i]=x[i+1] found.
                        * -3    internal eigenproblem solver hasn't converged
                        * -2    Beta[i]<=0
                        * -1    incorrect N was passed
                        * +1    OK
        X           -   array[0..N-1] - array of quadrature nodes,
                        in ascending order.
        WKronrod    -   array[0..N-1] - Kronrod weights
        WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                        corresponding to extended Kronrod nodes).

      -- ALGLIB --
         Copyright 08.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gkqgeneraterec(
        double[] alpha,
        double[] beta,
        double mu0,
        int n,
        out int info,
        out double[] x,
        out double[] wkronrod,
        out double[] wgauss)

### `gkqlegendrecalc` function

    /*************************************************************************
    Returns Gauss and Gauss-Kronrod nodes for quadrature with N points.

    Reduction to tridiagonal eigenproblem is used.

    INPUT PARAMETERS:
        N           -   number of Kronrod nodes, must be odd number, >=3.

    OUTPUT PARAMETERS:
        Info        -   error code:
                        * -4    an  error   was   detected   when  calculating
                                weights/nodes.  N  is  too  large   to  obtain
                                weights/nodes  with  high   enough   accuracy.
                                Try  to   use   multiple   precision  version.
                        * -3    internal eigenproblem solver hasn't converged
                        * -1    incorrect N was passed
                        * +1    OK
        X           -   array[0..N-1] - array of quadrature nodes, ordered in
                        ascending order.
        WKronrod    -   array[0..N-1] - Kronrod weights
        WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                        corresponding to extended Kronrod nodes).

      -- ALGLIB --
         Copyright 12.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gkqlegendrecalc(
        int n,
        out int info,
        out double[] x,
        out double[] wkronrod,
        out double[] wgauss)

### `gkqlegendretbl` function

    /*************************************************************************
    Returns Gauss and Gauss-Kronrod nodes for quadrature with N  points  using
    pre-calculated table. Nodes/weights were  computed  with  accuracy  up  to
    1.0E-32 (if MPFR version of ALGLIB is used). In standard double  precision
    accuracy reduces to something about 2.0E-16 (depending  on your compiler's
    handling of long floating point constants).

    INPUT PARAMETERS:
        N           -   number of Kronrod nodes.
                        N can be 15, 21, 31, 41, 51, 61.

    OUTPUT PARAMETERS:
        X           -   array[0..N-1] - array of quadrature nodes, ordered in
                        ascending order.
        WKronrod    -   array[0..N-1] - Kronrod weights
        WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                        corresponding to extended Kronrod nodes).

      -- ALGLIB --
         Copyright 12.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gkqlegendretbl(
        int n,
        out double[] x,
        out double[] wkronrod,
        out double[] wgauss,
        out double eps)

## `gq` subpackage

### Functions

gqgenerategausshermite
gqgenerategaussjacobi
gqgenerategausslaguerre
gqgenerategausslegendre
gqgenerategausslobattorec
gqgenerategaussradaurec
gqgeneraterec

### Examples

### `gqgenerategausshermite` function

    /*************************************************************************
    Returns  nodes/weights  for  Gauss-Hermite  quadrature on (-inf,+inf) with
    weight function W(x)=Exp(-x*x)

    INPUT PARAMETERS:
        N           -   number of nodes, >=1

    OUTPUT PARAMETERS:
        Info        -   error code:
                        * -4    an  error  was   detected   when   calculating
                                weights/nodes.  May be, N is too large. Try to
                                use multiple precision version.
                        * -3    internal eigenproblem solver hasn't converged
                        * -1    incorrect N/Alpha was passed
                        * +1    OK
        X           -   array[0..N-1] - array of quadrature nodes,
                        in ascending order.
        W           -   array[0..N-1] - array of quadrature weights.

      -- ALGLIB --
         Copyright 12.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gqgenerategausshermite(
        int n,
        out int info,
        out double[] x,
        out double[] w)

### `gqgenerategaussjacobi` function

    /*************************************************************************
    Returns  nodes/weights  for  Gauss-Jacobi quadrature on [-1,1] with weight
    function W(x)=Power(1-x,Alpha)*Power(1+x,Beta).

    INPUT PARAMETERS:
        N           -   number of nodes, >=1
        Alpha       -   power-law coefficient, Alpha>-1
        Beta        -   power-law coefficient, Beta>-1

    OUTPUT PARAMETERS:
        Info        -   error code:
                        * -4    an  error  was   detected   when   calculating
                                weights/nodes. Alpha or  Beta  are  too  close
                                to -1 to obtain weights/nodes with high enough
                                accuracy, or, may be, N is too large.  Try  to
                                use multiple precision version.
                        * -3    internal eigenproblem solver hasn't converged
                        * -1    incorrect N/Alpha/Beta was passed
                        * +1    OK
        X           -   array[0..N-1] - array of quadrature nodes,
                        in ascending order.
        W           -   array[0..N-1] - array of quadrature weights.

      -- ALGLIB --
         Copyright 12.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gqgenerategaussjacobi(
        int n,
        double alpha,
        double beta,
        out int info,
        out double[] x,
        out double[] w)

### `gqgenerategausslaguerre` function

    /*************************************************************************
    Returns  nodes/weights  for  Gauss-Laguerre  quadrature  on  [0,+inf) with
    weight function W(x)=Power(x,Alpha)*Exp(-x)

    INPUT PARAMETERS:
        N           -   number of nodes, >=1
        Alpha       -   power-law coefficient, Alpha>-1

    OUTPUT PARAMETERS:
        Info        -   error code:
                        * -4    an  error  was   detected   when   calculating
                                weights/nodes. Alpha is too  close  to  -1  to
                                obtain weights/nodes with high enough accuracy
                                or, may  be,  N  is  too  large.  Try  to  use
                                multiple precision version.
                        * -3    internal eigenproblem solver hasn't converged
                        * -1    incorrect N/Alpha was passed
                        * +1    OK
        X           -   array[0..N-1] - array of quadrature nodes,
                        in ascending order.
        W           -   array[0..N-1] - array of quadrature weights.

      -- ALGLIB --
         Copyright 12.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gqgenerategausslaguerre(
        int n,
        double alpha,
        out int info,
        out double[] x,
        out double[] w)

### `gqgenerategausslegendre` function

    /*************************************************************************
    Returns nodes/weights for Gauss-Legendre quadrature on [-1,1] with N
    nodes.

    INPUT PARAMETERS:
        N           -   number of nodes, >=1

    OUTPUT PARAMETERS:
        Info        -   error code:
                        * -4    an  error   was   detected   when  calculating
                                weights/nodes.  N  is  too  large   to  obtain
                                weights/nodes  with  high   enough   accuracy.
                                Try  to   use   multiple   precision  version.
                        * -3    internal eigenproblem solver hasn't  converged
                        * -1    incorrect N was passed
                        * +1    OK
        X           -   array[0..N-1] - array of quadrature nodes,
                        in ascending order.
        W           -   array[0..N-1] - array of quadrature weights.

      -- ALGLIB --
         Copyright 12.05.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gqgenerategausslegendre(
        int n,
        out int info,
        out double[] x,
        out double[] w)

### `gqgenerategausslobattorec` function

    /*************************************************************************
    Computation of nodes and weights for a Gauss-Lobatto quadrature formula

    The algorithm generates the N-point Gauss-Lobatto quadrature formula  with
    weight function given by coefficients alpha and beta of a recurrence which
    generates a system of orthogonal polynomials.

    P-1(x)   =  0
    P0(x)    =  1
    Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

    and zeroth moment Mu0

    Mu0 = integral(W(x)dx,a,b)

    INPUT PARAMETERS:
        Alpha   –   array[0..N-2], alpha coefficients
        Beta    –   array[0..N-2], beta coefficients.
                    Zero-indexed element is not used, may be arbitrary.
                    Beta[I]>0
        Mu0     –   zeroth moment of the weighting function.
        A       –   left boundary of the integration interval.
        B       –   right boundary of the integration interval.
        N       –   number of nodes of the quadrature formula, N>=3
                    (including the left and right boundary nodes).

    OUTPUT PARAMETERS:
        Info    -   error code:
                    * -3    internal eigenproblem solver hasn't converged
                    * -2    Beta[i]<=0
                    * -1    incorrect N was passed
                    *  1    OK
        X       -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
        W       -   array[0..N-1] - array of quadrature weights.

      -- ALGLIB --
         Copyright 2005-2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gqgenerategausslobattorec(
        double[] alpha,
        double[] beta,
        double mu0,
        double a,
        double b,
        int n,
        out int info,
        out double[] x,
        out double[] w)

### `gqgenerategaussradaurec` function

    /*************************************************************************
    Computation of nodes and weights for a Gauss-Radau quadrature formula

    The algorithm generates the N-point Gauss-Radau  quadrature  formula  with
    weight function given by the coefficients alpha and  beta  of a recurrence
    which generates a system of orthogonal polynomials.

    P-1(x)   =  0
    P0(x)    =  1
    Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

    and zeroth moment Mu0

    Mu0 = integral(W(x)dx,a,b)

    INPUT PARAMETERS:
        Alpha   –   array[0..N-2], alpha coefficients.
        Beta    –   array[0..N-1], beta coefficients
                    Zero-indexed element is not used.
                    Beta[I]>0
        Mu0     –   zeroth moment of the weighting function.
        A       –   left boundary of the integration interval.
        N       –   number of nodes of the quadrature formula, N>=2
                    (including the left boundary node).

    OUTPUT PARAMETERS:
        Info    -   error code:
                    * -3    internal eigenproblem solver hasn't converged
                    * -2    Beta[i]<=0
                    * -1    incorrect N was passed
                    *  1    OK
        X       -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
        W       -   array[0..N-1] - array of quadrature weights.

      -- ALGLIB --
         Copyright 2005-2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gqgenerategaussradaurec(
        double[] alpha,
        double[] beta,
        double mu0,
        double a,
        int n,
        out int info,
        out double[] x,
        out double[] w)

### `gqgeneraterec` function

    /*************************************************************************
    Computation of nodes and weights for a Gauss quadrature formula

    The algorithm generates the N-point Gauss quadrature formula  with  weight
    function given by coefficients alpha and beta  of  a  recurrence  relation
    which generates a system of orthogonal polynomials:

    P-1(x)   =  0
    P0(x)    =  1
    Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

    and zeroth moment Mu0

    Mu0 = integral(W(x)dx,a,b)

    INPUT PARAMETERS:
        Alpha   –   array[0..N-1], alpha coefficients
        Beta    –   array[0..N-1], beta coefficients
                    Zero-indexed element is not used and may be arbitrary.
                    Beta[I]>0.
        Mu0     –   zeroth moment of the weight function.
        N       –   number of nodes of the quadrature formula, N>=1

    OUTPUT PARAMETERS:
        Info    -   error code:
                    * -3    internal eigenproblem solver hasn't converged
                    * -2    Beta[i]<=0
                    * -1    incorrect N was passed
                    *  1    OK
        X       -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
        W       -   array[0..N-1] - array of quadrature weights.

      -- ALGLIB --
         Copyright 2005-2009 by Bochkanov Sergey
    *************************************************************************/
    public static void gqgeneraterec(
        double[] alpha,
        double[] beta,
        double mu0,
        int n,
        out int info,
        out double[] x,
        out double[] w)

## `hermite` subpackage

### Functions

hermitecalculate
hermitecoefficients
hermitesum

### Examples

### `hermitecalculate` function

    /*************************************************************************
    Calculation of the value of the Hermite polynomial.

    Parameters:
        n   -   degree, n>=0
        x   -   argument

    Result:
        the value of the Hermite polynomial Hn at x
    *************************************************************************/
    public static double hermitecalculate(int n, double x)

### `hermitecoefficients` function

    /*************************************************************************
    Representation of Hn as C[0] + C[1]*X + ... + C[N]*X^N

    Input parameters:
        N   -   polynomial degree, n>=0

    Output parameters:
        C   -   coefficients
    *************************************************************************/
    public static void hermitecoefficients(int n, out double[] c)

### `hermitesum` function

    /*************************************************************************
    Summation of Hermite polynomials using Clenshaw’s recurrence formula.

    This routine calculates
        c[0]*H0(x) + c[1]*H1(x) + ... + c[N]*HN(x)

    Parameters:
        n   -   degree, n>=0
        x   -   argument

    Result:
        the value of the Hermite polynomial at x
    *************************************************************************/
    public static double hermitesum(double[] c, int n, double x)

## `hqrnd` subpackage

### Classes

hqrndstate

### Functions

hqrndexponential
hqrndnormal
hqrndnormal2
hqrndrandomize
hqrndseed
hqrnduniformi
hqrnduniformr
hqrndunit2

### Examples

### `hqrndstate` class

    /*************************************************************************
    Portable high quality random number generator state.
    Initialized with HQRNDRandomize() or HQRNDSeed().

    Fields:
        S1, S2      -   seed values
        V           -   precomputed value
        MagicV      -   'magic' value used to determine whether State structure
                        was correctly initialized.
    *************************************************************************/
    public class hqrndstate
    {
    }

### `hqrndexponential` function

    /*************************************************************************
    Random number generator: exponential distribution

    State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

      -- ALGLIB --
         Copyright 11.08.2007 by Bochkanov Sergey
    *************************************************************************/
    public static double hqrndexponential(hqrndstate state, double lambdav)

### `hqrndnormal` function

    /*************************************************************************
    Random number generator: normal numbers

    This function generates one random number from normal distribution.
    Its performance is equal to that of HQRNDNormal2()

    State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

      -- ALGLIB --
         Copyright 02.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double hqrndnormal(hqrndstate state)

### `hqrndnormal2` function

    /*************************************************************************
    Random number generator: normal numbers

    This function generates two independent random numbers from normal
    distribution. Its performance is equal to that of HQRNDNormal()

    State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

      -- ALGLIB --
         Copyright 02.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void hqrndnormal2(
        hqrndstate state,
        out double x1,
        out double x2)

### `hqrndrandomize` function

    /*************************************************************************
    HQRNDState  initialization  with  random  values  which come from standard
    RNG.

      -- ALGLIB --
         Copyright 02.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void hqrndrandomize(out hqrndstate state)

### `hqrndseed` function

    /*************************************************************************
    HQRNDState initialization with seed values

      -- ALGLIB --
         Copyright 02.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void hqrndseed(int s1, int s2, out hqrndstate state)

### `hqrnduniformi` function

    /*************************************************************************
    This function generates random integer number in [0, N)

    1. N must be less than HQRNDMax-1.
    2. State structure must be initialized with HQRNDRandomize() or HQRNDSeed()

      -- ALGLIB --
         Copyright 02.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static int hqrnduniformi(hqrndstate state, int n)

### `hqrnduniformr` function

    /*************************************************************************
    This function generates random real number in (0,1),
    not including interval boundaries

    State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

      -- ALGLIB --
         Copyright 02.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double hqrnduniformr(hqrndstate state)

### `hqrndunit2` function

    /*************************************************************************
    Random number generator: random X and Y such that X^2+Y^2=1

    State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

      -- ALGLIB --
         Copyright 02.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void hqrndunit2(
        hqrndstate state,
        out double x,
        out double y)

## `ibetaf` subpackage

### Functions

incompletebeta
invincompletebeta

### Examples

### `incompletebeta` function

    /*************************************************************************
    Incomplete beta integral

    Returns incomplete beta integral of the arguments, evaluated
    from zero to x.  The function is defined as

                     x
        -            -
       | (a+b)      | |  a-1     b-1
     -----------    |   t   (1-t)   dt.
      -     -     | |
     | (a) | (b)   -
                    0

    The domain of definition is 0 <= x <= 1.  In this
    implementation a and b are restricted to positive values.
    The integral from x to 1 may be obtained by the symmetry
    relation

       1 - incbet( a, b, x )  =  incbet( b, a, 1-x ).

    The integral is evaluated by a continued fraction expansion
    or, when b*x is small, by a power series.

    ACCURACY:

    Tested at uniformly distributed random points (a,b,x) with a and b
    in "domain" and x between 0 and 1.
                                           Relative error
    arithmetic   domain     # trials      peak         rms
       IEEE      0,5         10000       6.9e-15     4.5e-16
       IEEE      0,85       250000       2.2e-13     1.7e-14
       IEEE      0,1000      30000       5.3e-12     6.3e-13
       IEEE      0,10000    250000       9.3e-11     7.1e-12
       IEEE      0,100000    10000       8.7e-10     4.8e-11
    Outputs smaller than the IEEE gradual underflow threshold
    were excluded from these statistics.

    Cephes Math Library, Release 2.8:  June, 2000
    Copyright 1984, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double incompletebeta(double a, double b, double x)

### `invincompletebeta` function

    /*************************************************************************
    Inverse of imcomplete beta integral

    Given y, the function finds x such that

     incbet( a, b, x ) = y .

    The routine performs interval halving or Newton iterations to find the
    root of incbet(a,b,x) - y = 0.

    ACCURACY:

                         Relative error:
                   x     a,b
    arithmetic   domain  domain  # trials    peak       rms
       IEEE      0,1    .5,10000   50000    5.8e-12   1.3e-13
       IEEE      0,1   .25,100    100000    1.8e-13   3.9e-15
       IEEE      0,1     0,5       50000    1.1e-12   5.5e-15
    With a and b constrained to half-integer or integer values:
       IEEE      0,1    .5,10000   50000    5.8e-12   1.1e-13
       IEEE      0,1    .5,100    100000    1.7e-14   7.9e-16
    With a = .5, b constrained to half-integer or integer values:
       IEEE      0,1    .5,10000   10000    8.3e-11   1.0e-11

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1996, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double invincompletebeta(double a, double b, double y)

## `idwint` subpackage

### Classes

idwinterpolant

### Functions

idwbuildmodifiedshepard
idwbuildmodifiedshepardr
idwbuildnoisy
idwcalc

### Examples

### `idwinterpolant` class

    /*************************************************************************
    IDW interpolant.
    *************************************************************************/
    public class idwinterpolant
    {
    }

### `idwbuildmodifiedshepard` function

    /*************************************************************************
    IDW interpolant using modified Shepard method for uniform point
    distributions.

    INPUT PARAMETERS:
        XY  -   X and Y values, array[0..N-1,0..NX].
                First NX columns contain X-values, last column contain
                Y-values.
        N   -   number of nodes, N>0.
        NX  -   space dimension, NX>=1.
        D   -   nodal function type, either:
                * 0     constant  model.  Just  for  demonstration only, worst
                        model ever.
                * 1     linear model, least squares fitting. Simpe  model  for
                        datasets too small for quadratic models
                * 2     quadratic  model,  least  squares  fitting. Best model
                        available (if your dataset is large enough).
                * -1    "fast"  linear  model,  use  with  caution!!!   It  is
                        significantly  faster than linear/quadratic and better
                        than constant model. But it is less robust (especially
                        in the presence of noise).
        NQ  -   number of points used to calculate  nodal  functions  (ignored
                for constant models). NQ should be LARGER than:
                * max(1.5*(1+NX),2^NX+1) for linear model,
                * max(3/4*(NX+2)*(NX+1),2^NX+1) for quadratic model.
                Values less than this threshold will be silently increased.
        NW  -   number of points used to calculate weights and to interpolate.
                Required: >=2^NX+1, values less than this  threshold  will  be
                silently increased.
                Recommended value: about 2*NQ

    OUTPUT PARAMETERS:
        Z   -   IDW interpolant.

    NOTES:
      * best results are obtained with quadratic models, worst - with constant
        models
      * when N is large, NQ and NW must be significantly smaller than  N  both
        to obtain optimal performance and to obtain optimal accuracy. In 2  or
        3-dimensional tasks NQ=15 and NW=25 are good values to start with.
      * NQ  and  NW  may  be  greater  than  N.  In  such  cases  they will be
        automatically decreased.
      * this subroutine is always succeeds (as long as correct parameters  are
        passed).
      * see  'Multivariate  Interpolation  of Large Sets of Scattered Data' by
        Robert J. Renka for more information on this algorithm.
      * this subroutine assumes that point distribution is uniform at the small
        scales.  If  it  isn't  -  for  example,  points are concentrated along
        "lines", but "lines" distribution is uniform at the larger scale - then
        you should use IDWBuildModifiedShepardR()

      -- ALGLIB PROJECT --
         Copyright 02.03.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void idwbuildmodifiedshepard(
        double[,] xy,
        int n,
        int nx,
        int d,
        int nq,
        int nw,
        out idwinterpolant z)

### `idwbuildmodifiedshepardr` function

    /*************************************************************************
    IDW interpolant using modified Shepard method for non-uniform datasets.

    This type of model uses  constant  nodal  functions and interpolates using
    all nodes which are closer than user-specified radius R. It  may  be  used
    when points distribution is non-uniform at the small scale, but it  is  at
    the distances as large as R.

    INPUT PARAMETERS:
        XY  -   X and Y values, array[0..N-1,0..NX].
                First NX columns contain X-values, last column contain
                Y-values.
        N   -   number of nodes, N>0.
        NX  -   space dimension, NX>=1.
        R   -   radius, R>0

    OUTPUT PARAMETERS:
        Z   -   IDW interpolant.

    NOTES:
    * if there is less than IDWKMin points within  R-ball,  algorithm  selects
      IDWKMin closest ones, so that continuity properties of  interpolant  are
      preserved even far from points.

      -- ALGLIB PROJECT --
         Copyright 11.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void idwbuildmodifiedshepardr(
        double[,] xy,
        int n,
        int nx,
        double r,
        out idwinterpolant z)

### `idwbuildnoisy` function

    /*************************************************************************
    IDW model for noisy data.

    This subroutine may be used to handle noisy data, i.e. data with noise  in
    OUTPUT values.  It differs from IDWBuildModifiedShepard() in the following
    aspects:
    * nodal functions are not constrained to pass through  nodes:  Qi(xi)<>yi,
      i.e. we have fitting  instead  of  interpolation.
    * weights which are used during least  squares fitting stage are all equal
      to 1.0 (independently of distance)
    * "fast"-linear or constant nodal functions are not supported (either  not
      robust enough or too rigid)

    This problem require far more complex tuning than interpolation  problems.
    Below you can find some recommendations regarding this problem:
    * focus on tuning NQ; it controls noise reduction. As for NW, you can just
      make it equal to 2*NQ.
    * you can use cross-validation to determine optimal NQ.
    * optimal NQ is a result of complex tradeoff  between  noise  level  (more
      noise = larger NQ required) and underlying  function  complexity  (given
      fixed N, larger NQ means smoothing of compex features in the data).  For
      example, NQ=N will reduce noise to the minimum level possible,  but  you
      will end up with just constant/linear/quadratic (depending on  D)  least
      squares model for the whole dataset.

    INPUT PARAMETERS:
        XY  -   X and Y values, array[0..N-1,0..NX].
                First NX columns contain X-values, last column contain
                Y-values.
        N   -   number of nodes, N>0.
        NX  -   space dimension, NX>=1.
        D   -   nodal function degree, either:
                * 1     linear model, least squares fitting. Simpe  model  for
                        datasets too small for quadratic models (or  for  very
                        noisy problems).
                * 2     quadratic  model,  least  squares  fitting. Best model
                        available (if your dataset is large enough).
        NQ  -   number of points used to calculate nodal functions.  NQ should
                be  significantly   larger   than  1.5  times  the  number  of
                coefficients in a nodal function to overcome effects of noise:
                * larger than 1.5*(1+NX) for linear model,
                * larger than 3/4*(NX+2)*(NX+1) for quadratic model.
                Values less than this threshold will be silently increased.
        NW  -   number of points used to calculate weights and to interpolate.
                Required: >=2^NX+1, values less than this  threshold  will  be
                silently increased.
                Recommended value: about 2*NQ or larger

    OUTPUT PARAMETERS:
        Z   -   IDW interpolant.

    NOTES:
      * best results are obtained with quadratic models, linear models are not
        recommended to use unless you are pretty sure that it is what you want
      * this subroutine is always succeeds (as long as correct parameters  are
        passed).
      * see  'Multivariate  Interpolation  of Large Sets of Scattered Data' by
        Robert J. Renka for more information on this algorithm.

      -- ALGLIB PROJECT --
         Copyright 02.03.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void idwbuildnoisy(
        double[,] xy,
        int n,
        int nx,
        int d,
        int nq,
        int nw,
        out idwinterpolant z)

### `idwcalc` function

    /*************************************************************************
    IDW interpolation

    INPUT PARAMETERS:
        Z   -   IDW interpolant built with one of model building
                subroutines.
        X   -   array[0..NX-1], interpolation point

    Result:
        IDW interpolant Z(X)

      -- ALGLIB --
         Copyright 02.03.2010 by Bochkanov Sergey
    *************************************************************************/
    public static double idwcalc(idwinterpolant z, double[] x)

## `igammaf` subpackage

### Functions

incompletegamma
incompletegammac
invincompletegammac

### Examples

### `incompletegamma` function

    /*************************************************************************
    Incomplete gamma integral

    The function is defined by

                              x
                               -
                      1       | |  -t  a-1
     igam(a,x)  =   -----     |   e   t   dt.
                     -      | |
                    | (a)    -
                              0

    In this implementation both arguments must be positive.
    The integral is evaluated by either a power series or
    continued fraction expansion, depending on the relative
    values of a and x.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE      0,30       200000       3.6e-14     2.9e-15
       IEEE      0,100      300000       9.9e-14     1.5e-14

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1985, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double incompletegamma(double a, double x)

### `incompletegammac` function

    /*************************************************************************
    Complemented incomplete gamma integral

    The function is defined by

     igamc(a,x)   =   1 - igam(a,x)

                               inf.
                                 -
                        1       | |  -t  a-1
                  =   -----     |   e   t   dt.
                       -      | |
                      | (a)    -
                                x

    In this implementation both arguments must be positive.
    The integral is evaluated by either a power series or
    continued fraction expansion, depending on the relative
    values of a and x.

    ACCURACY:

    Tested at random a, x.
                   a         x                      Relative error:
    arithmetic   domain   domain     # trials      peak         rms
       IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
       IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1985, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double incompletegammac(double a, double x)

### `invincompletegammac` function

    /*************************************************************************
    Inverse of complemented imcomplete gamma integral

    Given p, the function finds x such that

     igamc( a, x ) = p.

    Starting with the approximate value

            3
     x = a t

     where

     t = 1 - d - ndtri(p) sqrt(d)

    and

     d = 1/9a,

    the routine performs up to 10 Newton iterations to find the
    root of igamc(a,x) - p = 0.

    ACCURACY:

    Tested at random a, p in the intervals indicated.

                   a        p                      Relative error:
    arithmetic   domain   domain     # trials      peak         rms
       IEEE     0.5,100   0,0.5       100000       1.0e-14     1.7e-15
       IEEE     0.01,0.5  0,0.5       100000       9.0e-14     3.4e-15
       IEEE    0.5,10000  0,0.5        20000       2.3e-13     3.8e-14

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double invincompletegammac(double a, double y0)

## `inverseupdate` subpackage

### Functions

rmatrixinvupdatecolumn
rmatrixinvupdaterow
rmatrixinvupdatesimple
rmatrixinvupdateuv

### Examples

### `rmatrixinvupdatecolumn` function

    /*************************************************************************
    Inverse matrix update by the Sherman-Morrison formula

    The algorithm updates matrix A^-1 when adding a vector to a column
    of matrix A.

    Input parameters:
        InvA        -   inverse of matrix A.
                        Array whose indexes range within [0..N-1, 0..N-1].
        N           -   size of matrix A.
        UpdColumn   -   the column of A whose vector U was added.
                        0 <= UpdColumn <= N-1
        U           -   the vector to be added to a column.
                        Array whose index ranges within [0..N-1].

    Output parameters:
        InvA        -   inverse of modified matrix A.

      -- ALGLIB --
         Copyright 2005 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixinvupdatecolumn(
        ref double[,] inva,
        int n,
        int updcolumn,
        double[] u)

### `rmatrixinvupdaterow` function

    /*************************************************************************
    Inverse matrix update by the Sherman-Morrison formula

    The algorithm updates matrix A^-1 when adding a vector to a row
    of matrix A.

    Input parameters:
        InvA    -   inverse of matrix A.
                    Array whose indexes range within [0..N-1, 0..N-1].
        N       -   size of matrix A.
        UpdRow  -   the row of A whose vector V was added.
                    0 <= Row <= N-1
        V       -   the vector to be added to a row.
                    Array whose index ranges within [0..N-1].

    Output parameters:
        InvA    -   inverse of modified matrix A.

      -- ALGLIB --
         Copyright 2005 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixinvupdaterow(
        ref double[,] inva,
        int n,
        int updrow,
        double[] v)

### `rmatrixinvupdatesimple` function

    /*************************************************************************
    Inverse matrix update by the Sherman-Morrison formula

    The algorithm updates matrix A^-1 when adding a number to an element
    of matrix A.

    Input parameters:
        InvA    -   inverse of matrix A.
                    Array whose indexes range within [0..N-1, 0..N-1].
        N       -   size of matrix A.
        UpdRow  -   row where the element to be updated is stored.
        UpdColumn - column where the element to be updated is stored.
        UpdVal  -   a number to be added to the element.

    Output parameters:
        InvA    -   inverse of modified matrix A.

      -- ALGLIB --
         Copyright 2005 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixinvupdatesimple(
        ref double[,] inva,
        int n,
        int updrow,
        int updcolumn,
        double updval)

### `rmatrixinvupdateuv` function

    /*************************************************************************
    Inverse matrix update by the Sherman-Morrison formula

    The algorithm computes the inverse of matrix A+u*v’ by using the given matrix
    A^-1 and the vectors u and v.

    Input parameters:
        InvA    -   inverse of matrix A.
                    Array whose indexes range within [0..N-1, 0..N-1].
        N       -   size of matrix A.
        U       -   the vector modifying the matrix.
                    Array whose index ranges within [0..N-1].
        V       -   the vector modifying the matrix.
                    Array whose index ranges within [0..N-1].

    Output parameters:
        InvA - inverse of matrix A + u*v'.

      -- ALGLIB --
         Copyright 2005 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixinvupdateuv(
        ref double[,] inva,
        int n,
        double[] u,
        double[] v)

## `jacobianelliptic` subpackage

### Functions

jacobianellipticfunctions

### Examples

### `jacobianellipticfunctions` function

    /*************************************************************************
    Jacobian Elliptic Functions

    Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
    and dn(u|m) of parameter m between 0 and 1, and real
    argument u.

    These functions are periodic, with quarter-period on the
    real axis equal to the complete elliptic integral
    ellpk(1.0-m).

    Relation to incomplete elliptic integral:
    If u = ellik(phi,m), then sn(u|m) = sin(phi),
    and cn(u|m) = cos(phi).  Phi is called the amplitude of u.

    Computation is by means of the arithmetic-geometric mean
    algorithm, except when m is within 1e-9 of 0 or 1.  In the
    latter case with m close to 1, the approximation applies
    only for phi < pi/2.

    ACCURACY:

    Tested at random points with u between 0 and 10, m between
    0 and 1.

               Absolute error (* = relative error):
    arithmetic   function   # trials      peak         rms
       IEEE      phi         10000       9.2e-16*    1.4e-16*
       IEEE      sn          50000       4.1e-15     4.6e-16
       IEEE      cn          40000       3.6e-15     4.4e-16
       IEEE      dn          10000       1.3e-12     1.8e-14

     Peak error observed in consistency check using addition
    theorem for sn(u+v) was 4e-16 (absolute).  Also tested by
    the above relation to the incomplete elliptic integral.
    Accuracy deteriorates when u is large.

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static void jacobianellipticfunctions(
        double u,
        double m,
        out double sn,
        out double cn,
        out double dn,
        out double ph)

## `jarquebera` subpackage

### Functions

jarqueberatest

### Examples

### `jarqueberatest` function

    /*************************************************************************
    Jarque-Bera test

    This test checks hypotheses about the fact that a  given  sample  X  is  a
    sample of normal random variable.

    Requirements:
        * the number of elements in the sample is not less than 5.

    Input parameters:
        X   -   sample. Array whose index goes from 0 to N-1.
        N   -   size of the sample. N>=5

    Output parameters:
        BothTails   -   p-value for two-tailed test.
                        If BothTails is less than the given significance level
                        the null hypothesis is rejected.
        LeftTail    -   p-value for left-tailed test.
                        If LeftTail is less than the given significance level,
                        the null hypothesis is rejected.
        RightTail   -   p-value for right-tailed test.
                        If RightTail is less than the given significance level
                        the null hypothesis is rejected.

    Accuracy of the approximation used (5<=N<=1951):

    p-value  	    relative error (5<=N<=1951)
    [1, 0.1]            < 1%
    [0.1, 0.01]         < 2%
    [0.01, 0.001]       < 6%
    [0.001, 0]          wasn't measured

    For N>1951 accuracy wasn't measured but it shouldn't be sharply  different
    from table values.

      -- ALGLIB --
         Copyright 09.04.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void jarqueberatest(double[] x, int n, out double p)

## `kmeans` subpackage

### Functions

kmeansgenerate

### Examples

### `kmeansgenerate` function

    /*************************************************************************
    k-means++ clusterization

    INPUT PARAMETERS:
        XY          -   dataset, array [0..NPoints-1,0..NVars-1].
        NPoints     -   dataset size, NPoints>=K
        NVars       -   number of variables, NVars>=1
        K           -   desired number of clusters, K>=1
        Restarts    -   number of restarts, Restarts>=1

    OUTPUT PARAMETERS:
        Info        -   return code:
                        * -3, if task is degenerate (number of distinct points is
                              less than K)
                        * -1, if incorrect NPoints/NFeatures/K/Restarts was passed
                        *  1, if subroutine finished successfully
        C           -   array[0..NVars-1,0..K-1].matrix whose columns store
                        cluster's centers
        XYC         -   array[NPoints], which contains cluster indexes

      -- ALGLIB --
         Copyright 21.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void kmeansgenerate(
        double[,] xy,
        int npoints,
        int nvars,
        int k,
        int restarts,
        out int info,
        out double[,] c,
        out int[] xyc)

## `laguerre` subpackage

### Functions

laguerrecalculate
laguerrecoefficients
laguerresum

### Examples

### `laguerrecalculate` function

    /*************************************************************************
    Calculation of the value of the Laguerre polynomial.

    Parameters:
        n   -   degree, n>=0
        x   -   argument

    Result:
        the value of the Laguerre polynomial Ln at x
    *************************************************************************/
    public static double laguerrecalculate(int n, double x)

### `laguerrecoefficients` function

    /*************************************************************************
    Representation of Ln as C[0] + C[1]*X + ... + C[N]*X^N

    Input parameters:
        N   -   polynomial degree, n>=0

    Output parameters:
        C   -   coefficients
    *************************************************************************/
    public static void laguerrecoefficients(int n, out double[] c)

### `laguerresum` function

    /*************************************************************************
    Summation of Laguerre polynomials using Clenshaw’s recurrence formula.

    This routine calculates c[0]*L0(x) + c[1]*L1(x) + ... + c[N]*LN(x)

    Parameters:
        n   -   degree, n>=0
        x   -   argument

    Result:
        the value of the Laguerre polynomial at x
    *************************************************************************/
    public static double laguerresum(double[] c, int n, double x)

## `lda` subpackage

### Functions

fisherlda
fisherldan

### Examples

### `fisherlda` function

    /*************************************************************************
    Multiclass Fisher LDA

    Subroutine finds coefficients of linear combination which optimally separates
    training set on classes.

    INPUT PARAMETERS:
        XY          -   training set, array[0..NPoints-1,0..NVars].
                        First NVars columns store values of independent
                        variables, next column stores number of class (from 0
                        to NClasses-1) which dataset element belongs to. Fractional
                        values are rounded to nearest integer.
        NPoints     -   training set size, NPoints>=0
        NVars       -   number of independent variables, NVars>=1
        NClasses    -   number of classes, NClasses>=2

    OUTPUT PARAMETERS:
        Info        -   return code:
                        * -4, if internal EVD subroutine hasn't converged
                        * -2, if there is a point with class number
                              outside of [0..NClasses-1].
                        * -1, if incorrect parameters was passed (NPoints<0,
                              NVars<1, NClasses<2)
                        *  1, if task has been solved
                        *  2, if there was a multicollinearity in training set,
                              but task has been solved.
        W           -   linear combination coefficients, array[0..NVars-1]

      -- ALGLIB --
         Copyright 31.05.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void fisherlda(
        double[,] xy,
        int npoints,
        int nvars,
        int nclasses,
        out int info,
        out double[] w)

### `fisherldan` function

    /*************************************************************************
    N-dimensional multiclass Fisher LDA

    Subroutine finds coefficients of linear combinations which optimally separates
    training set on classes. It returns N-dimensional basis whose vector are sorted
    by quality of training set separation (in descending order).

    INPUT PARAMETERS:
        XY          -   training set, array[0..NPoints-1,0..NVars].
                        First NVars columns store values of independent
                        variables, next column stores number of class (from 0
                        to NClasses-1) which dataset element belongs to. Fractional
                        values are rounded to nearest integer.
        NPoints     -   training set size, NPoints>=0
        NVars       -   number of independent variables, NVars>=1
        NClasses    -   number of classes, NClasses>=2

    OUTPUT PARAMETERS:
        Info        -   return code:
                        * -4, if internal EVD subroutine hasn't converged
                        * -2, if there is a point with class number
                              outside of [0..NClasses-1].
                        * -1, if incorrect parameters was passed (NPoints<0,
                              NVars<1, NClasses<2)
                        *  1, if task has been solved
                        *  2, if there was a multicollinearity in training set,
                              but task has been solved.
        W           -   basis, array[0..NVars-1,0..NVars-1]
                        columns of matrix stores basis vectors, sorted by
                        quality of training set separation (in descending order)

      -- ALGLIB --
         Copyright 31.05.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void fisherldan(
        double[,] xy,
        int npoints,
        int nvars,
        int nclasses,
        out int info,
        out double[,] w)

## `legendre` subpackage

### Functions

legendrecalculate
legendrecoefficients
legendresum

### Examples

### `legendrecalculate` function

    /*************************************************************************
    Calculation of the value of the Legendre polynomial Pn.

    Parameters:
        n   -   degree, n>=0
        x   -   argument

    Result:
        the value of the Legendre polynomial Pn at x
    *************************************************************************/
    public static double legendrecalculate(int n, double x)

### `legendrecoefficients` function

    /*************************************************************************
    Representation of Pn as C[0] + C[1]*X + ... + C[N]*X^N

    Input parameters:
        N   -   polynomial degree, n>=0

    Output parameters:
        C   -   coefficients
    *************************************************************************/
    public static void legendrecoefficients(int n, out double[] c)

### `legendresum` function

    /*************************************************************************
    Summation of Legendre polynomials using Clenshaw’s recurrence formula.

    This routine calculates
        c[0]*P0(x) + c[1]*P1(x) + ... + c[N]*PN(x)

    Parameters:
        n   -   degree, n>=0
        x   -   argument

    Result:
        the value of the Legendre polynomial at x
    *************************************************************************/
    public static double legendresum(double[] c, int n, double x)

## `linreg` subpackage

### Classes

linearmodel
lrreport

### Functions

lravgerror
lravgrelerror
lrbuild
lrbuilds
lrbuildz
lrbuildzs
lrpack
lrprocess
lrrmserror
lrunpack

### Examples

### `linearmodel` class

    /*************************************************************************

    *************************************************************************/
    public class linearmodel
    {
    }

### `lrreport` class

    /*************************************************************************
    LRReport structure contains additional information about linear model:
    * C             -   covariation matrix,  array[0..NVars,0..NVars].
                        C[i,j] = Cov(A[i],A[j])
    * RMSError      -   root mean square error on a training set
    * AvgError      -   average error on a training set
    * AvgRelError   -   average relative error on a training set (excluding
                        observations with zero function value).
    * CVRMSError    -   leave-one-out cross-validation estimate of
                        generalization error. Calculated using fast algorithm
                        with O(NVars*NPoints) complexity.
    * CVAvgError    -   cross-validation estimate of average error
    * CVAvgRelError -   cross-validation estimate of average relative error

    All other fields of the structure are intended for internal use and should
    not be used outside ALGLIB.
    *************************************************************************/
    public class lrreport
    {
        public double[,]            c;
        public double               rmserror;
        public double               avgerror;
        public double               avgrelerror;
        public double               cvrmserror;
        public double               cvavgerror;
        public double               cvavgrelerror;
        public int                  ncvdefects;
        public int[]                cvdefects;
    }

### `lravgerror` function

    /*************************************************************************
    Average error on the test set

    INPUT PARAMETERS:
        LM      -   linear model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        average error.

      -- ALGLIB --
         Copyright 30.08.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double lravgerror(linearmodel lm, double[,] xy, int npoints)

### `lravgrelerror` function

    /*************************************************************************
    RMS error on the test set

    INPUT PARAMETERS:
        LM      -   linear model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        average relative error.

      -- ALGLIB --
         Copyright 30.08.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double lravgrelerror(
        linearmodel lm,
        double[,] xy,
        int npoints)

### `lrbuild` function

    /*************************************************************************
    Linear regression

    Subroutine builds model:

        Y = A(0)*X[0] + ... + A(N-1)*X[N-1] + A(N)

    and model found in ALGLIB format, covariation matrix, training set  errors
    (rms,  average,  average  relative)   and  leave-one-out  cross-validation
    estimate of the generalization error. CV  estimate calculated  using  fast
    algorithm with O(NPoints*NVars) complexity.

    When  covariation  matrix  is  calculated  standard deviations of function
    values are assumed to be equal to RMS error on the training set.

    INPUT PARAMETERS:
        XY          -   training set, array [0..NPoints-1,0..NVars]:
                        * NVars columns - independent variables
                        * last column - dependent variable
        NPoints     -   training set size, NPoints>NVars+1
        NVars       -   number of independent variables

    OUTPUT PARAMETERS:
        Info        -   return code:
                        * -255, in case of unknown internal error
                        * -4, if internal SVD subroutine haven't converged
                        * -1, if incorrect parameters was passed (NPoints<NVars+2, NVars<1).
                        *  1, if subroutine successfully finished
        LM          -   linear model in the ALGLIB format. Use subroutines of
                        this unit to work with the model.
        AR          -   additional results

      -- ALGLIB --
         Copyright 02.08.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void lrbuild(
        double[,] xy,
        int npoints,
        int nvars,
        out int info,
        out linearmodel lm,
        out lrreport ar)

### `lrbuilds` function

    /*************************************************************************
    Linear regression

    Variant of LRBuild which uses vector of standatd deviations (errors in
    function values).

    INPUT PARAMETERS:
        XY          -   training set, array [0..NPoints-1,0..NVars]:
                        * NVars columns - independent variables
                        * last column - dependent variable
        S           -   standard deviations (errors in function values)
                        array[0..NPoints-1], S[i]>0.
        NPoints     -   training set size, NPoints>NVars+1
        NVars       -   number of independent variables

    OUTPUT PARAMETERS:
        Info        -   return code:
                        * -255, in case of unknown internal error
                        * -4, if internal SVD subroutine haven't converged
                        * -1, if incorrect parameters was passed (NPoints<NVars+2, NVars<1).
                        * -2, if S[I]<=0
                        *  1, if subroutine successfully finished
        LM          -   linear model in the ALGLIB format. Use subroutines of
                        this unit to work with the model.
        AR          -   additional results

      -- ALGLIB --
         Copyright 02.08.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void lrbuilds(
        double[,] xy,
        double[] s,
        int npoints,
        int nvars,
        out int info,
        out linearmodel lm,
        out lrreport ar)

### `lrbuildz` function

    /*************************************************************************
    Like LRBuild but builds model

        Y = A(0)*X[0] + ... + A(N-1)*X[N-1]

    i.e. with zero constant term.

      -- ALGLIB --
         Copyright 30.10.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void lrbuildz(
        double[,] xy,
        int npoints,
        int nvars,
        out int info,
        out linearmodel lm,
        out lrreport ar)

### `lrbuildzs` function

    /*************************************************************************
    Like LRBuildS, but builds model

        Y = A(0)*X[0] + ... + A(N-1)*X[N-1]

    i.e. with zero constant term.

      -- ALGLIB --
         Copyright 30.10.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void lrbuildzs(
        double[,] xy,
        double[] s,
        int npoints,
        int nvars,
        out int info,
        out linearmodel lm,
        out lrreport ar)

### `lrpack` function

    /*************************************************************************
    "Packs" coefficients and creates linear model in ALGLIB format (LRUnpack
    reversed).

    INPUT PARAMETERS:
        V           -   coefficients, array[0..NVars]
        NVars       -   number of independent variables

    OUTPUT PAREMETERS:
        LM          -   linear model.

      -- ALGLIB --
         Copyright 30.08.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void lrpack(double[] v, int nvars, out linearmodel lm)

### `lrprocess` function

    /*************************************************************************
    Procesing

    INPUT PARAMETERS:
        LM      -   linear model
        X       -   input vector,  array[0..NVars-1].

    Result:
        value of linear model regression estimate

      -- ALGLIB --
         Copyright 03.09.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double lrprocess(linearmodel lm, double[] x)

### `lrrmserror` function

    /*************************************************************************
    RMS error on the test set

    INPUT PARAMETERS:
        LM      -   linear model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        root mean square error.

      -- ALGLIB --
         Copyright 30.08.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double lrrmserror(linearmodel lm, double[,] xy, int npoints)

### `lrunpack` function

    /*************************************************************************
    Unpacks coefficients of linear model.

    INPUT PARAMETERS:
        LM          -   linear model in ALGLIB format

    OUTPUT PARAMETERS:
        V           -   coefficients, array[0..NVars]
                        constant term (intercept) is stored in the V[NVars].
        NVars       -   number of independent variables (one less than number
                        of coefficients)

      -- ALGLIB --
         Copyright 30.08.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void lrunpack(linearmodel lm, out double[] v, out int nvars)

## `logit` subpackage

### Classes

logitmodel
mnlreport

### Functions

mnlavgce
mnlavgerror
mnlavgrelerror
mnlclserror
mnlpack
mnlprocess
mnlprocessi
mnlrelclserror
mnlrmserror
mnltrainh
mnlunpack

### Examples

### `logitmodel` class

    /*************************************************************************

    *************************************************************************/
    public class logitmodel
    {
    }

### `mnlreport` class

    /*************************************************************************
    MNLReport structure contains information about training process:
    * NGrad     -   number of gradient calculations
    * NHess     -   number of Hessian calculations
    *************************************************************************/
    public class mnlreport
    {
        public int                  ngrad;
        public int                  nhess;
    }

### `mnlavgce` function

    /*************************************************************************
    Average cross-entropy (in bits per element) on the test set

    INPUT PARAMETERS:
        LM      -   logit model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        CrossEntropy/(NPoints*ln(2)).

      -- ALGLIB --
         Copyright 10.09.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double mnlavgce(logitmodel lm, double[,] xy, int npoints)

### `mnlavgerror` function

    /*************************************************************************
    Average error on the test set

    INPUT PARAMETERS:
        LM      -   logit model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        average error (error when estimating posterior probabilities).

      -- ALGLIB --
         Copyright 30.08.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double mnlavgerror(logitmodel lm, double[,] xy, int npoints)

### `mnlavgrelerror` function

    /*************************************************************************
    Average relative error on the test set

    INPUT PARAMETERS:
        LM      -   logit model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        average relative error (error when estimating posterior probabilities).

      -- ALGLIB --
         Copyright 30.08.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double mnlavgrelerror(
        logitmodel lm,
        double[,] xy,
        int ssize)

### `mnlclserror` function

    /*************************************************************************
    Classification error on test set = MNLRelClsError*NPoints

      -- ALGLIB --
         Copyright 10.09.2008 by Bochkanov Sergey
    *************************************************************************/
    public static int mnlclserror(logitmodel lm, double[,] xy, int npoints)

### `mnlpack` function

    /*************************************************************************
    "Packs" coefficients and creates logit model in ALGLIB format (MNLUnpack
    reversed).

    INPUT PARAMETERS:
        A           -   model (see MNLUnpack)
        NVars       -   number of independent variables
        NClasses    -   number of classes

    OUTPUT PARAMETERS:
        LM          -   logit model.

      -- ALGLIB --
         Copyright 10.09.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mnlpack(
        double[,] a,
        int nvars,
        int nclasses,
        out logitmodel lm)

### `mnlprocess` function

    /*************************************************************************
    Procesing

    INPUT PARAMETERS:
        LM      -   logit model, passed by non-constant reference
                    (some fields of structure are used as temporaries
                    when calculating model output).
        X       -   input vector,  array[0..NVars-1].
        Y       -   (possibly) preallocated buffer; if size of Y is less than
                    NClasses, it will be reallocated.If it is large enough, it
                    is NOT reallocated, so we can save some time on reallocation.

    OUTPUT PARAMETERS:
        Y       -   result, array[0..NClasses-1]
                    Vector of posterior probabilities for classification task.

      -- ALGLIB --
         Copyright 10.09.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mnlprocess(logitmodel lm, double[] x, ref double[] y)

### `mnlprocessi` function

    /*************************************************************************
    'interactive'  variant  of  MNLProcess  for  languages  like  Python which
    support constructs like "Y = MNLProcess(LM,X)" and interactive mode of the
    interpreter

    This function allocates new array on each call,  so  it  is  significantly
    slower than its 'non-interactive' counterpart, but it is  more  convenient
    when you call it from command line.

      -- ALGLIB --
         Copyright 10.09.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mnlprocessi(logitmodel lm, double[] x, out double[] y)

### `mnlrelclserror` function

    /*************************************************************************
    Relative classification error on the test set

    INPUT PARAMETERS:
        LM      -   logit model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        percent of incorrectly classified cases.

      -- ALGLIB --
         Copyright 10.09.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double mnlrelclserror(
        logitmodel lm,
        double[,] xy,
        int npoints)

### `mnlrmserror` function

    /*************************************************************************
    RMS error on the test set

    INPUT PARAMETERS:
        LM      -   logit model
        XY      -   test set
        NPoints -   test set size

    RESULT:
        root mean square error (error when estimating posterior probabilities).

      -- ALGLIB --
         Copyright 30.08.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double mnlrmserror(logitmodel lm, double[,] xy, int npoints)

### `mnltrainh` function

    /*************************************************************************
    This subroutine trains logit model.

    INPUT PARAMETERS:
        XY          -   training set, array[0..NPoints-1,0..NVars]
                        First NVars columns store values of independent
                        variables, next column stores number of class (from 0
                        to NClasses-1) which dataset element belongs to. Fractional
                        values are rounded to nearest integer.
        NPoints     -   training set size, NPoints>=1
        NVars       -   number of independent variables, NVars>=1
        NClasses    -   number of classes, NClasses>=2

    OUTPUT PARAMETERS:
        Info        -   return code:
                        * -2, if there is a point with class number
                              outside of [0..NClasses-1].
                        * -1, if incorrect parameters was passed
                              (NPoints<NVars+2, NVars<1, NClasses<2).
                        *  1, if task has been solved
        LM          -   model built
        Rep         -   training report

      -- ALGLIB --
         Copyright 10.09.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mnltrainh(
        double[,] xy,
        int npoints,
        int nvars,
        int nclasses,
        out int info,
        out logitmodel lm,
        out mnlreport rep)

### `mnlunpack` function

    /*************************************************************************
    Unpacks coefficients of logit model. Logit model have form:

        P(class=i) = S(i) / (S(0) + S(1) + ... +S(M-1))
              S(i) = Exp(A[i,0]*X[0] + ... + A[i,N-1]*X[N-1] + A[i,N]), when i<M-1
            S(M-1) = 1

    INPUT PARAMETERS:
        LM          -   logit model in ALGLIB format

    OUTPUT PARAMETERS:
        V           -   coefficients, array[0..NClasses-2,0..NVars]
        NVars       -   number of independent variables
        NClasses    -   number of classes

      -- ALGLIB --
         Copyright 10.09.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mnlunpack(
        logitmodel lm,
        out double[,] a,
        out int nvars,
        out int nclasses)

## `lsfit` subpackage

### Classes

barycentricfitreport
lsfitreport
lsfitstate
polynomialfitreport
spline1dfitreport

### Functions

barycentricfitfloaterhormann
barycentricfitfloaterhormannwc
lsfitcreatef
lsfitcreatefg
lsfitcreatefgh
lsfitcreatewf
lsfitcreatewfg
lsfitcreatewfgh
lsfitfit
lsfitlinear
lsfitlinearc
lsfitlinearw
lsfitlinearwc
lsfitresults
lsfitsetbc
lsfitsetcond
lsfitsetscale
lsfitsetstpmax
lsfitsetxrep
polynomialfit
polynomialfitwc
spline1dfitcubic
spline1dfitcubicwc
spline1dfithermite
spline1dfithermitewc
spline1dfitpenalized
spline1dfitpenalizedw

### Examples

lsfit_d_lin |  | Unconstrained (general) linear least squares fitting with and without weights
lsfit_d_linc |  | Constrained (general) linear least squares fitting with and without weights
lsfit_d_nlf |  | Nonlinear fitting using function value only
lsfit_d_nlfb |  | Bound contstrained nonlinear fitting using function value only
lsfit_d_nlfg |  | Nonlinear fitting using gradient
lsfit_d_nlfgh |  | Nonlinear fitting using gradient and Hessian
lsfit_d_nlscale |  | Nonlinear fitting with custom scaling and bound constraints
lsfit_d_pol |  | Unconstrained polynomial fitting
lsfit_d_polc |  | Constrained polynomial fitting
lsfit_d_spline |  | Unconstrained fitting by penalized regression spline

### `barycentricfitreport` class

    /*************************************************************************
    Barycentric fitting report:
        RMSError        RMS error
        AvgError        average error
        AvgRelError     average relative error (for non-zero Y[I])
        MaxError        maximum error
        TaskRCond       reciprocal of task's condition number
    *************************************************************************/
    public class barycentricfitreport
    {
        public double               taskrcond;
        public int                  dbest;
        public double               rmserror;
        public double               avgerror;
        public double               avgrelerror;
        public double               maxerror;
    }

### `lsfitreport` class

    /*************************************************************************
    Least squares fitting report:
        TaskRCond       reciprocal of task's condition number
        IterationsCount number of internal iterations

        RMSError        RMS error
        AvgError        average error
        AvgRelError     average relative error (for non-zero Y[I])
        MaxError        maximum error

        WRMSError       weighted RMS error
    *************************************************************************/
    public class lsfitreport
    {
        public double               taskrcond;
        public int                  iterationscount;
        public double               rmserror;
        public double               avgerror;
        public double               avgrelerror;
        public double               maxerror;
        public double               wrmserror;
    }

### `lsfitstate` class

    /*************************************************************************
    Nonlinear fitter.

    You should use ALGLIB functions to work with fitter.
    Never try to access its fields directly!
    *************************************************************************/
    public class lsfitstate
    {
    }

### `polynomialfitreport` class

    /*************************************************************************
    Polynomial fitting report:
        TaskRCond       reciprocal of task's condition number
        RMSError        RMS error
        AvgError        average error
        AvgRelError     average relative error (for non-zero Y[I])
        MaxError        maximum error
    *************************************************************************/
    public class polynomialfitreport
    {
        public double               taskrcond;
        public double               rmserror;
        public double               avgerror;
        public double               avgrelerror;
        public double               maxerror;
    }

### `spline1dfitreport` class

    /*************************************************************************
    Spline fitting report:
        RMSError        RMS error
        AvgError        average error
        AvgRelError     average relative error (for non-zero Y[I])
        MaxError        maximum error

    Fields  below are  filled  by   obsolete    functions   (Spline1DFitCubic,
    Spline1DFitHermite). Modern fitting functions do NOT fill these fields:
        TaskRCond       reciprocal of task's condition number
    *************************************************************************/
    public class spline1dfitreport
    {
        public double               taskrcond;
        public double               rmserror;
        public double               avgerror;
        public double               avgrelerror;
        public double               maxerror;
    }

### `barycentricfitfloaterhormann` function

    /*************************************************************************
    Rational least squares fitting using  Floater-Hormann  rational  functions
    with optimal D chosen from [0,9].

    Equidistant  grid  with M node on [min(x),max(x)]  is  used to build basis
    functions. Different values of D are tried, optimal  D  (least  root  mean
    square error) is chosen.  Task  is  linear, so linear least squares solver
    is used. Complexity  of  this  computational  scheme is  O(N*M^2)  (mostly
    dominated by the least squares solver).

    INPUT PARAMETERS:
        X   -   points, array[0..N-1].
        Y   -   function values, array[0..N-1].
        N   -   number of points, N>0.
        M   -   number of basis functions ( = number_of_nodes), M>=2.

    OUTPUT PARAMETERS:
        Info-   same format as in LSFitLinearWC() subroutine.
                * Info>0    task is solved
                * Info<=0   an error occured:
                            -4 means inconvergence of internal SVD
                            -3 means inconsistent constraints
        B   -   barycentric interpolant.
        Rep -   report, same format as in LSFitLinearWC() subroutine.
                Following fields are set:
                * DBest         best value of the D parameter
                * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

      -- ALGLIB PROJECT --
         Copyright 18.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void barycentricfitfloaterhormann(
        double[] x,
        double[] y,
        int n,
        int m,
        out int info,
        out barycentricinterpolant b,
        out barycentricfitreport rep)

### `barycentricfitfloaterhormannwc` function

    /*************************************************************************
    Weghted rational least  squares  fitting  using  Floater-Hormann  rational
    functions  with  optimal  D  chosen  from  [0,9],  with  constraints   and
    individual weights.

    Equidistant  grid  with M node on [min(x),max(x)]  is  used to build basis
    functions. Different values of D are tried, optimal D (least WEIGHTED root
    mean square error) is chosen.  Task  is  linear,  so  linear least squares
    solver  is  used.  Complexity  of  this  computational  scheme is O(N*M^2)
    (mostly dominated by the least squares solver).

    SEE ALSO
    * BarycentricFitFloaterHormann(), "lightweight" fitting without invididual
      weights and constraints.

    INPUT PARAMETERS:
        X   -   points, array[0..N-1].
        Y   -   function values, array[0..N-1].
        W   -   weights, array[0..N-1]
                Each summand in square  sum  of  approximation deviations from
                given  values  is  multiplied  by  the square of corresponding
                weight. Fill it by 1's if you don't  want  to  solve  weighted
                task.
        N   -   number of points, N>0.
        XC  -   points where function values/derivatives are constrained,
                array[0..K-1].
        YC  -   values of constraints, array[0..K-1]
        DC  -   array[0..K-1], types of constraints:
                * DC[i]=0   means that S(XC[i])=YC[i]
                * DC[i]=1   means that S'(XC[i])=YC[i]
                SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
        K   -   number of constraints, 0<=K<M.
                K=0 means no constraints (XC/YC/DC are not used in such cases)
        M   -   number of basis functions ( = number_of_nodes), M>=2.

    OUTPUT PARAMETERS:
        Info-   same format as in LSFitLinearWC() subroutine.
                * Info>0    task is solved
                * Info<=0   an error occured:
                            -4 means inconvergence of internal SVD
                            -3 means inconsistent constraints
                            -1 means another errors in parameters passed
                               (N<=0, for example)
        B   -   barycentric interpolant.
        Rep -   report, same format as in LSFitLinearWC() subroutine.
                Following fields are set:
                * DBest         best value of the D parameter
                * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

    IMPORTANT:
        this subroutine doesn't calculate task's condition number for K<>0.

    SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

    Setting constraints can lead  to undesired  results,  like ill-conditioned
    behavior, or inconsistency being detected. From the other side,  it allows
    us to improve quality of the fit. Here we summarize  our  experience  with
    constrained barycentric interpolants:
    * excessive  constraints  can  be  inconsistent.   Floater-Hormann   basis
      functions aren't as flexible as splines (although they are very smooth).
    * the more evenly constraints are spread across [min(x),max(x)],  the more
      chances that they will be consistent
    * the  greater  is  M (given  fixed  constraints),  the  more chances that
      constraints will be consistent
    * in the general case, consistency of constraints IS NOT GUARANTEED.
    * in the several special cases, however, we CAN guarantee consistency.
    * one of this cases is constraints on the function  VALUES at the interval
      boundaries. Note that consustency of the  constraints  on  the  function
      DERIVATIVES is NOT guaranteed (you can use in such cases  cubic  splines
      which are more flexible).
    * another  special  case  is ONE constraint on the function value (OR, but
      not AND, derivative) anywhere in the interval

    Our final recommendation is to use constraints  WHEN  AND  ONLY  WHEN  you
    can't solve your task without them. Anything beyond  special  cases  given
    above is not guaranteed and may result in inconsistency.

      -- ALGLIB PROJECT --
         Copyright 18.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void barycentricfitfloaterhormannwc(
        double[] x,
        double[] y,
        double[] w,
        int n,
        double[] xc,
        double[] yc,
        int[] dc,
        int k,
        int m,
        out int info,
        out barycentricinterpolant b,
        out barycentricfitreport rep)

### `lsfitcreatef` function

    /*************************************************************************
    Nonlinear least squares fitting using function values only.

    Combination of numerical differentiation and secant updates is used to
    obtain function Jacobian.

    Nonlinear task min(F(c)) is solved, where

        F(c) = (f(c,x[0])-y[0])^2 + ... + (f(c,x[n-1])-y[n-1])^2,

        * N is a number of points,
        * M is a dimension of a space points belong to,
        * K is a dimension of a space of parameters being fitted,
        * w is an N-dimensional vector of weight coefficients,
        * x is a set of N points, each of them is an M-dimensional vector,
        * c is a K-dimensional vector of parameters being fitted

    This subroutine uses only f(c,x[i]).

    INPUT PARAMETERS:
        X       -   array[0..N-1,0..M-1], points (one row = one point)
        Y       -   array[0..N-1], function values.
        C       -   array[0..K-1], initial approximation to the solution,
        N       -   number of points, N>1
        M       -   dimension of space
        K       -   number of parameters being fitted
        DiffStep-   numerical differentiation step;
                    should not be very small or large;
                    large = loss of accuracy
                    small = growth of round-off errors

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

      -- ALGLIB --
         Copyright 18.10.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitcreatef(
        double[,] x,
        double[] y,
        double[] c,
        double diffstep,
        out lsfitstate state)
    public static void lsfitcreatef(
        double[,] x,
        double[] y,
        double[] c,
        int n,
        int m,
        int k,
        double diffstep,
        out lsfitstate state)

Examples: [1] [2] [3]

### `lsfitcreatefg` function

    /*************************************************************************
    Nonlinear least squares fitting using gradient only, without individual
    weights.

    Nonlinear task min(F(c)) is solved, where

        F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2,

        * N is a number of points,
        * M is a dimension of a space points belong to,
        * K is a dimension of a space of parameters being fitted,
        * x is a set of N points, each of them is an M-dimensional vector,
        * c is a K-dimensional vector of parameters being fitted

    This subroutine uses only f(c,x[i]) and its gradient.

    INPUT PARAMETERS:
        X       -   array[0..N-1,0..M-1], points (one row = one point)
        Y       -   array[0..N-1], function values.
        C       -   array[0..K-1], initial approximation to the solution,
        N       -   number of points, N>1
        M       -   dimension of space
        K       -   number of parameters being fitted
        CheapFG -   boolean flag, which is:
                    * True  if both function and gradient calculation complexity
                            are less than O(M^2).  An improved  algorithm  can
                            be  used  which corresponds  to  FGJ  scheme  from
                            MINLM unit.
                    * False otherwise.
                            Standard Jacibian-bases  Levenberg-Marquardt  algo
                            will be used (FJ scheme).

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitcreatefg(
        double[,] x,
        double[] y,
        double[] c,
        bool cheapfg,
        out lsfitstate state)
    public static void lsfitcreatefg(
        double[,] x,
        double[] y,
        double[] c,
        int n,
        int m,
        int k,
        bool cheapfg,
        out lsfitstate state)

Examples: [1]

### `lsfitcreatefgh` function

    /*************************************************************************
    Nonlinear least squares fitting using gradient/Hessian, without individial
    weights.

    Nonlinear task min(F(c)) is solved, where

        F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2,

        * N is a number of points,
        * M is a dimension of a space points belong to,
        * K is a dimension of a space of parameters being fitted,
        * x is a set of N points, each of them is an M-dimensional vector,
        * c is a K-dimensional vector of parameters being fitted

    This subroutine uses f(c,x[i]), its gradient and its Hessian.

    INPUT PARAMETERS:
        X       -   array[0..N-1,0..M-1], points (one row = one point)
        Y       -   array[0..N-1], function values.
        C       -   array[0..K-1], initial approximation to the solution,
        N       -   number of points, N>1
        M       -   dimension of space
        K       -   number of parameters being fitted

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitcreatefgh(
        double[,] x,
        double[] y,
        double[] c,
        out lsfitstate state)
    public static void lsfitcreatefgh(
        double[,] x,
        double[] y,
        double[] c,
        int n,
        int m,
        int k,
        out lsfitstate state)

Examples: [1]

### `lsfitcreatewf` function

    /*************************************************************************
    Weighted nonlinear least squares fitting using function values only.

    Combination of numerical differentiation and secant updates is used to
    obtain function Jacobian.

    Nonlinear task min(F(c)) is solved, where

        F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,

        * N is a number of points,
        * M is a dimension of a space points belong to,
        * K is a dimension of a space of parameters being fitted,
        * w is an N-dimensional vector of weight coefficients,
        * x is a set of N points, each of them is an M-dimensional vector,
        * c is a K-dimensional vector of parameters being fitted

    This subroutine uses only f(c,x[i]).

    INPUT PARAMETERS:
        X       -   array[0..N-1,0..M-1], points (one row = one point)
        Y       -   array[0..N-1], function values.
        W       -   weights, array[0..N-1]
        C       -   array[0..K-1], initial approximation to the solution,
        N       -   number of points, N>1
        M       -   dimension of space
        K       -   number of parameters being fitted
        DiffStep-   numerical differentiation step;
                    should not be very small or large;
                    large = loss of accuracy
                    small = growth of round-off errors

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

      -- ALGLIB --
         Copyright 18.10.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitcreatewf(
        double[,] x,
        double[] y,
        double[] w,
        double[] c,
        double diffstep,
        out lsfitstate state)
    public static void lsfitcreatewf(
        double[,] x,
        double[] y,
        double[] w,
        double[] c,
        int n,
        int m,
        int k,
        double diffstep,
        out lsfitstate state)

Examples: [1] [2]

### `lsfitcreatewfg` function

    /*************************************************************************
    Weighted nonlinear least squares fitting using gradient only.

    Nonlinear task min(F(c)) is solved, where

        F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,

        * N is a number of points,
        * M is a dimension of a space points belong to,
        * K is a dimension of a space of parameters being fitted,
        * w is an N-dimensional vector of weight coefficients,
        * x is a set of N points, each of them is an M-dimensional vector,
        * c is a K-dimensional vector of parameters being fitted

    This subroutine uses only f(c,x[i]) and its gradient.

    INPUT PARAMETERS:
        X       -   array[0..N-1,0..M-1], points (one row = one point)
        Y       -   array[0..N-1], function values.
        W       -   weights, array[0..N-1]
        C       -   array[0..K-1], initial approximation to the solution,
        N       -   number of points, N>1
        M       -   dimension of space
        K       -   number of parameters being fitted
        CheapFG -   boolean flag, which is:
                    * True  if both function and gradient calculation complexity
                            are less than O(M^2).  An improved  algorithm  can
                            be  used  which corresponds  to  FGJ  scheme  from
                            MINLM unit.
                    * False otherwise.
                            Standard Jacibian-bases  Levenberg-Marquardt  algo
                            will be used (FJ scheme).

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

    See also:
        LSFitResults
        LSFitCreateFG (fitting without weights)
        LSFitCreateWFGH (fitting using Hessian)
        LSFitCreateFGH (fitting using Hessian, without weights)

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitcreatewfg(
        double[,] x,
        double[] y,
        double[] w,
        double[] c,
        bool cheapfg,
        out lsfitstate state)
    public static void lsfitcreatewfg(
        double[,] x,
        double[] y,
        double[] w,
        double[] c,
        int n,
        int m,
        int k,
        bool cheapfg,
        out lsfitstate state)

Examples: [1]

### `lsfitcreatewfgh` function

    /*************************************************************************
    Weighted nonlinear least squares fitting using gradient/Hessian.

    Nonlinear task min(F(c)) is solved, where

        F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,

        * N is a number of points,
        * M is a dimension of a space points belong to,
        * K is a dimension of a space of parameters being fitted,
        * w is an N-dimensional vector of weight coefficients,
        * x is a set of N points, each of them is an M-dimensional vector,
        * c is a K-dimensional vector of parameters being fitted

    This subroutine uses f(c,x[i]), its gradient and its Hessian.

    INPUT PARAMETERS:
        X       -   array[0..N-1,0..M-1], points (one row = one point)
        Y       -   array[0..N-1], function values.
        W       -   weights, array[0..N-1]
        C       -   array[0..K-1], initial approximation to the solution,
        N       -   number of points, N>1
        M       -   dimension of space
        K       -   number of parameters being fitted

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitcreatewfgh(
        double[,] x,
        double[] y,
        double[] w,
        double[] c,
        out lsfitstate state)
    public static void lsfitcreatewfgh(
        double[,] x,
        double[] y,
        double[] w,
        double[] c,
        int n,
        int m,
        int k,
        out lsfitstate state)

Examples: [1]

### `lsfitfit` function

    /*************************************************************************
    This family of functions is used to launcn iterations of nonlinear fitter

    These functions accept following parameters:
        state   -   algorithm state
        func    -   callback which calculates function (or merit function)
                    value func at given point x
        grad    -   callback which calculates function (or merit function)
                    value func and gradient grad at given point x
        hess    -   callback which calculates function (or merit function)
                    value func, gradient grad and Hessian hess at given point x
        rep     -   optional callback which is called after each iteration
                    can be NULL
        ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                    can be NULL

    NOTES:

    1. this algorithm is somewhat unusual because it works with  parameterized
       function f(C,X), where X is a function argument (we  have  many  points
       which are characterized by different  argument  values),  and  C  is  a
       parameter to fit.

       For example, if we want to do linear fit by f(c0,c1,x) = c0*x+c1,  then
       x will be argument, and {c0,c1} will be parameters.

       It is important to understand that this algorithm finds minimum in  the
       space of function PARAMETERS (not arguments), so it  needs  derivatives
       of f() with respect to C, not X.

       In the example above it will need f=c0*x+c1 and {df/dc0,df/dc1} = {x,1}
       instead of {df/dx} = {c0}.

    2. Callback functions accept C as the first parameter, and X as the second

    3. If  state  was  created  with  LSFitCreateFG(),  algorithm  needs  just
       function   and   its   gradient,   but   if   state   was  created with
       LSFitCreateFGH(), algorithm will need function, gradient and Hessian.

       According  to  the  said  above,  there  ase  several  versions of this
       function, which accept different sets of callbacks.

       This flexibility opens way to subtle errors - you may create state with
       LSFitCreateFGH() (optimization using Hessian), but call function  which
       does not accept Hessian. So when algorithm will request Hessian,  there
       will be no callback to call. In this case exception will be thrown.

       Be careful to avoid such errors because there is no way to find them at
       compile time - you can see them at runtime only.

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitfit(lsfitstate state, ndimensional_pfunc func, ndimensional_rep rep, object obj)
    public static void lsfitfit(lsfitstate state, ndimensional_pfunc func, ndimensional_pgrad grad, ndimensional_rep rep, object obj)
    public static void lsfitfit(lsfitstate state, ndimensional_pfunc func, ndimensional_pgrad grad, ndimensional_phess hess, ndimensional_rep rep, object obj)

Examples: [1] [2] [3] [4] [5]

### `lsfitlinear` function

    /*************************************************************************
    Linear least squares fitting.

    QR decomposition is used to reduce task to MxM, then triangular solver  or
    SVD-based solver is used depending on condition number of the  system.  It
    allows to maximize speed and retain decent accuracy.

    INPUT PARAMETERS:
        Y       -   array[0..N-1] Function values in  N  points.
        FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                    FMatrix[I, J] - value of J-th basis function in I-th point.
        N       -   number of points used. N>=1.
        M       -   number of basis functions, M>=1.

    OUTPUT PARAMETERS:
        Info    -   error code:
                    * -4    internal SVD decomposition subroutine failed (very
                            rare and for degenerate systems only)
                    *  1    task is solved
        C       -   decomposition coefficients, array[0..M-1]
        Rep     -   fitting report. Following fields are set:
                    * Rep.TaskRCond     reciprocal of condition number
                    * RMSError          rms error on the (X,Y).
                    * AvgError          average error on the (X,Y).
                    * AvgRelError       average relative error on the non-zero Y
                    * MaxError          maximum error
                                        NON-WEIGHTED ERRORS ARE CALCULATED

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitlinear(
        double[] y,
        double[,] fmatrix,
        out int info,
        out double[] c,
        out lsfitreport rep)
    public static void lsfitlinear(
        double[] y,
        double[,] fmatrix,
        int n,
        int m,
        out int info,
        out double[] c,
        out lsfitreport rep)

Examples: [1]

### `lsfitlinearc` function

    /*************************************************************************
    Constained linear least squares fitting.

    This  is  variation  of LSFitLinear(),  which searchs for min|A*x=b| given
    that  K  additional  constaints  C*x=bc are satisfied. It reduces original
    task to modified one: min|B*y-d| WITHOUT constraints,  then  LSFitLinear()
    is called.

    INPUT PARAMETERS:
        Y       -   array[0..N-1] Function values in  N  points.
        FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                    FMatrix[I,J] - value of J-th basis function in I-th point.
        CMatrix -   a table of constaints, array[0..K-1,0..M].
                    I-th row of CMatrix corresponds to I-th linear constraint:
                    CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
        N       -   number of points used. N>=1.
        M       -   number of basis functions, M>=1.
        K       -   number of constraints, 0 <= K < M
                    K=0 corresponds to absence of constraints.

    OUTPUT PARAMETERS:
        Info    -   error code:
                    * -4    internal SVD decomposition subroutine failed (very
                            rare and for degenerate systems only)
                    * -3    either   too   many  constraints  (M   or   more),
                            degenerate  constraints   (some   constraints  are
                            repetead twice) or inconsistent  constraints  were
                            specified.
                    *  1    task is solved
        C       -   decomposition coefficients, array[0..M-1]
        Rep     -   fitting report. Following fields are set:
                    * RMSError          rms error on the (X,Y).
                    * AvgError          average error on the (X,Y).
                    * AvgRelError       average relative error on the non-zero Y
                    * MaxError          maximum error
                                        NON-WEIGHTED ERRORS ARE CALCULATED

    IMPORTANT:
        this subroitine doesn't calculate task's condition number for K<>0.

      -- ALGLIB --
         Copyright 07.09.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitlinearc(
        double[] y,
        double[,] fmatrix,
        double[,] cmatrix,
        out int info,
        out double[] c,
        out lsfitreport rep)
    public static void lsfitlinearc(
        double[] y,
        double[,] fmatrix,
        double[,] cmatrix,
        int n,
        int m,
        int k,
        out int info,
        out double[] c,
        out lsfitreport rep)

Examples: [1]

### `lsfitlinearw` function

    /*************************************************************************
    Weighted linear least squares fitting.

    QR decomposition is used to reduce task to MxM, then triangular solver  or
    SVD-based solver is used depending on condition number of the  system.  It
    allows to maximize speed and retain decent accuracy.

    INPUT PARAMETERS:
        Y       -   array[0..N-1] Function values in  N  points.
        W       -   array[0..N-1]  Weights  corresponding to function  values.
                    Each summand in square  sum  of  approximation  deviations
                    from  given  values  is  multiplied  by  the   square   of
                    corresponding weight.
        FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                    FMatrix[I, J] - value of J-th basis function in I-th point.
        N       -   number of points used. N>=1.
        M       -   number of basis functions, M>=1.

    OUTPUT PARAMETERS:
        Info    -   error code:
                    * -4    internal SVD decomposition subroutine failed (very
                            rare and for degenerate systems only)
                    * -1    incorrect N/M were specified
                    *  1    task is solved
        C       -   decomposition coefficients, array[0..M-1]
        Rep     -   fitting report. Following fields are set:
                    * Rep.TaskRCond     reciprocal of condition number
                    * RMSError          rms error on the (X,Y).
                    * AvgError          average error on the (X,Y).
                    * AvgRelError       average relative error on the non-zero Y
                    * MaxError          maximum error
                                        NON-WEIGHTED ERRORS ARE CALCULATED

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitlinearw(
        double[] y,
        double[] w,
        double[,] fmatrix,
        out int info,
        out double[] c,
        out lsfitreport rep)
    public static void lsfitlinearw(
        double[] y,
        double[] w,
        double[,] fmatrix,
        int n,
        int m,
        out int info,
        out double[] c,
        out lsfitreport rep)

Examples: [1]

### `lsfitlinearwc` function

    /*************************************************************************
    Weighted constained linear least squares fitting.

    This  is  variation  of LSFitLinearW(), which searchs for min|A*x=b| given
    that  K  additional  constaints  C*x=bc are satisfied. It reduces original
    task to modified one: min|B*y-d| WITHOUT constraints,  then LSFitLinearW()
    is called.

    INPUT PARAMETERS:
        Y       -   array[0..N-1] Function values in  N  points.
        W       -   array[0..N-1]  Weights  corresponding to function  values.
                    Each summand in square  sum  of  approximation  deviations
                    from  given  values  is  multiplied  by  the   square   of
                    corresponding weight.
        FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                    FMatrix[I,J] - value of J-th basis function in I-th point.
        CMatrix -   a table of constaints, array[0..K-1,0..M].
                    I-th row of CMatrix corresponds to I-th linear constraint:
                    CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
        N       -   number of points used. N>=1.
        M       -   number of basis functions, M>=1.
        K       -   number of constraints, 0 <= K < M
                    K=0 corresponds to absence of constraints.

    OUTPUT PARAMETERS:
        Info    -   error code:
                    * -4    internal SVD decomposition subroutine failed (very
                            rare and for degenerate systems only)
                    * -3    either   too   many  constraints  (M   or   more),
                            degenerate  constraints   (some   constraints  are
                            repetead twice) or inconsistent  constraints  were
                            specified.
                    *  1    task is solved
        C       -   decomposition coefficients, array[0..M-1]
        Rep     -   fitting report. Following fields are set:
                    * RMSError          rms error on the (X,Y).
                    * AvgError          average error on the (X,Y).
                    * AvgRelError       average relative error on the non-zero Y
                    * MaxError          maximum error
                                        NON-WEIGHTED ERRORS ARE CALCULATED

    IMPORTANT:
        this subroitine doesn't calculate task's condition number for K<>0.

      -- ALGLIB --
         Copyright 07.09.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitlinearwc(
        double[] y,
        double[] w,
        double[,] fmatrix,
        double[,] cmatrix,
        out int info,
        out double[] c,
        out lsfitreport rep)
    public static void lsfitlinearwc(
        double[] y,
        double[] w,
        double[,] fmatrix,
        double[,] cmatrix,
        int n,
        int m,
        int k,
        out int info,
        out double[] c,
        out lsfitreport rep)

Examples: [1]

### `lsfitresults` function

    /*************************************************************************
    Nonlinear least squares fitting results.

    Called after return from LSFitFit().

    INPUT PARAMETERS:
        State   -   algorithm state

    OUTPUT PARAMETERS:
        Info    -   completetion code:
                        *  1    relative function improvement is no more than
                                EpsF.
                        *  2    relative step is no more than EpsX.
                        *  4    gradient norm is no more than EpsG
                        *  5    MaxIts steps was taken
                        *  7    stopping conditions are too stringent,
                                further improvement is impossible
        C       -   array[0..K-1], solution
        Rep     -   optimization report. Following fields are set:
                    * Rep.TerminationType completetion code:
                    * RMSError          rms error on the (X,Y).
                    * AvgError          average error on the (X,Y).
                    * AvgRelError       average relative error on the non-zero Y
                    * MaxError          maximum error
                                        NON-WEIGHTED ERRORS ARE CALCULATED
                    * WRMSError         weighted rms error on the (X,Y).

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitresults(
        lsfitstate state,
        out int info,
        out double[] c,
        out lsfitreport rep)

Examples: [1] [2] [3] [4] [5]

### `lsfitsetbc` function

    /*************************************************************************
    This function sets boundary constraints for underlying optimizer

    Boundary constraints are inactive by default (after initial creation).
    They are preserved until explicitly turned off with another SetBC() call.

    INPUT PARAMETERS:
        State   -   structure stores algorithm state
        BndL    -   lower bounds, array[K].
                    If some (all) variables are unbounded, you may specify
                    very small number or -INF (latter is recommended because
                    it will allow solver to use better algorithm).
        BndU    -   upper bounds, array[K].
                    If some (all) variables are unbounded, you may specify
                    very large number or +INF (latter is recommended because
                    it will allow solver to use better algorithm).

    NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
    variable will be "frozen" at X[i]=BndL[i]=BndU[i].

    NOTE 2: unlike other constrained optimization algorithms, this solver  has
    following useful properties:
    * bound constraints are always satisfied exactly
    * function is evaluated only INSIDE area specified by bound constraints

      -- ALGLIB --
         Copyright 14.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitsetbc(
        lsfitstate state,
        double[] bndl,
        double[] bndu)

### `lsfitsetcond` function

    /*************************************************************************
    Stopping conditions for nonlinear least squares fitting.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        EpsF    -   stopping criterion. Algorithm stops if
                    |F(k+1)-F(k)| <= EpsF*max{|F(k)|, |F(k+1)|, 1}
        EpsX    -   >=0
                    The subroutine finishes its work if  on  k+1-th  iteration
                    the condition |v|<=EpsX is fulfilled, where:
                    * |.| means Euclidian norm
                    * v - scaled step vector, v[i]=dx[i]/s[i]
                    * dx - ste pvector, dx=X(k+1)-X(k)
                    * s - scaling coefficients set by LSFitSetScale()
        MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                    iterations   is    unlimited.   Only   Levenberg-Marquardt
                    iterations  are  counted  (L-BFGS/CG  iterations  are  NOT
                    counted because their cost is very low compared to that of
                    LM).

    NOTE

    Passing EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic
    stopping criterion selection (according to the scheme used by MINLM unit).

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitsetcond(
        lsfitstate state,
        double epsf,
        double epsx,
        int maxits)

Examples: [1] [2] [3] [4] [5]

### `lsfitsetscale` function

    /*************************************************************************
    This function sets scaling coefficients for underlying optimizer.

    ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
    size and gradient are scaled before comparison with tolerances).  Scale of
    the I-th variable is a translation invariant measure of:
    a) "how large" the variable is
    b) how large the step should be to make significant changes in the function

    Generally, scale is NOT considered to be a form of preconditioner.  But LM
    optimizer is unique in that it uses scaling matrix both  in  the  stopping
    condition tests and as Marquardt damping factor.

    Proper scaling is very important for the algorithm performance. It is less
    important for the quality of results, but still has some influence (it  is
    easier  to  converge  when  variables  are  properly  scaled, so premature
    stopping is possible when very badly scalled variables are  combined  with
    relaxed stopping conditions).

    INPUT PARAMETERS:
        State   -   structure stores algorithm state
        S       -   array[N], non-zero scaling coefficients
                    S[i] may be negative, sign doesn't matter.

      -- ALGLIB --
         Copyright 14.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitsetscale(lsfitstate state, double[] s)

Examples: [1]

### `lsfitsetstpmax` function

    /*************************************************************************
    This function sets maximum step length

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                    want to limit step length.

    Use this subroutine when you optimize target function which contains exp()
    or  other  fast  growing  functions,  and optimization algorithm makes too
    large  steps  which  leads  to overflow. This function allows us to reject
    steps  that  are  too  large  (and  therefore  expose  us  to the possible
    overflow) without actually calculating function value at the x+stp*d.

    NOTE: non-zero StpMax leads to moderate  performance  degradation  because
    intermediate  step  of  preconditioned L-BFGS optimization is incompatible
    with limits on step size.

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitsetstpmax(lsfitstate state, double stpmax)

### `lsfitsetxrep` function

    /*************************************************************************
    This function turns on/off reporting.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        NeedXRep-   whether iteration reports are needed or not

    When reports are needed, State.C (current parameters) and State.F (current
    value of fitting function) are reported.

      -- ALGLIB --
         Copyright 15.08.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void lsfitsetxrep(lsfitstate state, bool needxrep)

### `polynomialfit` function

    /*************************************************************************
    Fitting by polynomials in barycentric form. This function provides  simple
    unterface for unconstrained unweighted fitting. See  PolynomialFitWC()  if
    you need constrained fitting.

    Task is linear, so linear least squares solver is used. Complexity of this
    computational scheme is O(N*M^2), mostly dominated by least squares solver

    SEE ALSO:
        PolynomialFitWC()

    INPUT PARAMETERS:
        X   -   points, array[0..N-1].
        Y   -   function values, array[0..N-1].
        N   -   number of points, N>0
                * if given, only leading N elements of X/Y are used
                * if not given, automatically determined from sizes of X/Y
        M   -   number of basis functions (= polynomial_degree + 1), M>=1

    OUTPUT PARAMETERS:
        Info-   same format as in LSFitLinearW() subroutine:
                * Info>0    task is solved
                * Info<=0   an error occured:
                            -4 means inconvergence of internal SVD
        P   -   interpolant in barycentric form.
        Rep -   report, same format as in LSFitLinearW() subroutine.
                Following fields are set:
                * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

    NOTES:
        you can convert P from barycentric form  to  the  power  or  Chebyshev
        basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions  from
        POLINT subpackage.

      -- ALGLIB PROJECT --
         Copyright 10.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void polynomialfit(
        double[] x,
        double[] y,
        int m,
        out int info,
        out barycentricinterpolant p,
        out polynomialfitreport rep)
    public static void polynomialfit(
        double[] x,
        double[] y,
        int n,
        int m,
        out int info,
        out barycentricinterpolant p,
        out polynomialfitreport rep)

Examples: [1]

### `polynomialfitwc` function

    /*************************************************************************
    Weighted  fitting by polynomials in barycentric form, with constraints  on
    function values or first derivatives.

    Small regularizing term is used when solving constrained tasks (to improve
    stability).

    Task is linear, so linear least squares solver is used. Complexity of this
    computational scheme is O(N*M^2), mostly dominated by least squares solver

    SEE ALSO:
        PolynomialFit()

    INPUT PARAMETERS:
        X   -   points, array[0..N-1].
        Y   -   function values, array[0..N-1].
        W   -   weights, array[0..N-1]
                Each summand in square  sum  of  approximation deviations from
                given  values  is  multiplied  by  the square of corresponding
                weight. Fill it by 1's if you don't  want  to  solve  weighted
                task.
        N   -   number of points, N>0.
                * if given, only leading N elements of X/Y/W are used
                * if not given, automatically determined from sizes of X/Y/W
        XC  -   points where polynomial values/derivatives are constrained,
                array[0..K-1].
        YC  -   values of constraints, array[0..K-1]
        DC  -   array[0..K-1], types of constraints:
                * DC[i]=0   means that P(XC[i])=YC[i]
                * DC[i]=1   means that P'(XC[i])=YC[i]
                SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
        K   -   number of constraints, 0<=K<M.
                K=0 means no constraints (XC/YC/DC are not used in such cases)
        M   -   number of basis functions (= polynomial_degree + 1), M>=1

    OUTPUT PARAMETERS:
        Info-   same format as in LSFitLinearW() subroutine:
                * Info>0    task is solved
                * Info<=0   an error occured:
                            -4 means inconvergence of internal SVD
                            -3 means inconsistent constraints
        P   -   interpolant in barycentric form.
        Rep -   report, same format as in LSFitLinearW() subroutine.
                Following fields are set:
                * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

    IMPORTANT:
        this subroitine doesn't calculate task's condition number for K<>0.

    NOTES:
        you can convert P from barycentric form  to  the  power  or  Chebyshev
        basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions  from
        POLINT subpackage.

    SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

    Setting constraints can lead  to undesired  results,  like ill-conditioned
    behavior, or inconsistency being detected. From the other side,  it allows
    us to improve quality of the fit. Here we summarize  our  experience  with
    constrained regression splines:
    * even simple constraints can be inconsistent, see  Wikipedia  article  on
      this subject: http://en.wikipedia.org/wiki/Birkhoff_interpolation
    * the  greater  is  M (given  fixed  constraints),  the  more chances that
      constraints will be consistent
    * in the general case, consistency of constraints is NOT GUARANTEED.
    * in the one special cases, however, we can  guarantee  consistency.  This
      case  is:  M>1  and constraints on the function values (NOT DERIVATIVES)

    Our final recommendation is to use constraints  WHEN  AND  ONLY  when  you
    can't solve your task without them. Anything beyond  special  cases  given
    above is not guaranteed and may result in inconsistency.

      -- ALGLIB PROJECT --
         Copyright 10.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void polynomialfitwc(
        double[] x,
        double[] y,
        double[] w,
        double[] xc,
        double[] yc,
        int[] dc,
        int m,
        out int info,
        out barycentricinterpolant p,
        out polynomialfitreport rep)
    public static void polynomialfitwc(
        double[] x,
        double[] y,
        double[] w,
        int n,
        double[] xc,
        double[] yc,
        int[] dc,
        int k,
        int m,
        out int info,
        out barycentricinterpolant p,
        out polynomialfitreport rep)

Examples: [1]

### `spline1dfitcubic` function

    /*************************************************************************
    Least squares fitting by cubic spline.

    This subroutine is "lightweight" alternative for more complex and feature-
    rich Spline1DFitCubicWC().  See  Spline1DFitCubicWC() for more information
    about subroutine parameters (we don't duplicate it here because of length)

      -- ALGLIB PROJECT --
         Copyright 18.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dfitcubic(
        double[] x,
        double[] y,
        int m,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)
    public static void spline1dfitcubic(
        double[] x,
        double[] y,
        int n,
        int m,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)

### `spline1dfitcubicwc` function

    /*************************************************************************
    Weighted fitting by cubic  spline,  with constraints on function values or
    derivatives.

    Equidistant grid with M-2 nodes on [min(x,xc),max(x,xc)] is  used to build
    basis functions. Basis functions are cubic splines with continuous  second
    derivatives  and  non-fixed first  derivatives  at  interval  ends.  Small
    regularizing term is used  when  solving  constrained  tasks  (to  improve
    stability).

    Task is linear, so linear least squares solver is used. Complexity of this
    computational scheme is O(N*M^2), mostly dominated by least squares solver

    SEE ALSO
        Spline1DFitHermiteWC()  -   fitting by Hermite splines (more flexible,
                                    less smooth)
        Spline1DFitCubic()      -   "lightweight" fitting  by  cubic  splines,
                                    without invididual weights and constraints

    INPUT PARAMETERS:
        X   -   points, array[0..N-1].
        Y   -   function values, array[0..N-1].
        W   -   weights, array[0..N-1]
                Each summand in square  sum  of  approximation deviations from
                given  values  is  multiplied  by  the square of corresponding
                weight. Fill it by 1's if you don't  want  to  solve  weighted
                task.
        N   -   number of points (optional):
                * N>0
                * if given, only first N elements of X/Y/W are processed
                * if not given, automatically determined from X/Y/W sizes
        XC  -   points where spline values/derivatives are constrained,
                array[0..K-1].
        YC  -   values of constraints, array[0..K-1]
        DC  -   array[0..K-1], types of constraints:
                * DC[i]=0   means that S(XC[i])=YC[i]
                * DC[i]=1   means that S'(XC[i])=YC[i]
                SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
        K   -   number of constraints (optional):
                * 0<=K<M.
                * K=0 means no constraints (XC/YC/DC are not used)
                * if given, only first K elements of XC/YC/DC are used
                * if not given, automatically determined from XC/YC/DC
        M   -   number of basis functions ( = number_of_nodes+2), M>=4.

    OUTPUT PARAMETERS:
        Info-   same format as in LSFitLinearWC() subroutine.
                * Info>0    task is solved
                * Info<=0   an error occured:
                            -4 means inconvergence of internal SVD
                            -3 means inconsistent constraints
        S   -   spline interpolant.
        Rep -   report, same format as in LSFitLinearWC() subroutine.
                Following fields are set:
                * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

    IMPORTANT:
        this subroitine doesn't calculate task's condition number for K<>0.

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller may pass unsorted array.

    SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

    Setting constraints can lead  to undesired  results,  like ill-conditioned
    behavior, or inconsistency being detected. From the other side,  it allows
    us to improve quality of the fit. Here we summarize  our  experience  with
    constrained regression splines:
    * excessive constraints can be inconsistent. Splines are  piecewise  cubic
      functions, and it is easy to create an example, where  large  number  of
      constraints  concentrated  in  small  area will result in inconsistency.
      Just because spline is not flexible enough to satisfy all of  them.  And
      same constraints spread across the  [min(x),max(x)]  will  be  perfectly
      consistent.
    * the more evenly constraints are spread across [min(x),max(x)],  the more
      chances that they will be consistent
    * the  greater  is  M (given  fixed  constraints),  the  more chances that
      constraints will be consistent
    * in the general case, consistency of constraints IS NOT GUARANTEED.
    * in the several special cases, however, we CAN guarantee consistency.
    * one of this cases is constraints  on  the  function  values  AND/OR  its
      derivatives at the interval boundaries.
    * another  special  case  is ONE constraint on the function value (OR, but
      not AND, derivative) anywhere in the interval

    Our final recommendation is to use constraints  WHEN  AND  ONLY  WHEN  you
    can't solve your task without them. Anything beyond  special  cases  given
    above is not guaranteed and may result in inconsistency.

      -- ALGLIB PROJECT --
         Copyright 18.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dfitcubicwc(
        double[] x,
        double[] y,
        double[] w,
        double[] xc,
        double[] yc,
        int[] dc,
        int m,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)
    public static void spline1dfitcubicwc(
        double[] x,
        double[] y,
        double[] w,
        int n,
        double[] xc,
        double[] yc,
        int[] dc,
        int k,
        int m,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)

### `spline1dfithermite` function

    /*************************************************************************
    Least squares fitting by Hermite spline.

    This subroutine is "lightweight" alternative for more complex and feature-
    rich Spline1DFitHermiteWC().  See Spline1DFitHermiteWC()  description  for
    more information about subroutine parameters (we don't duplicate  it  here
    because of length).

      -- ALGLIB PROJECT --
         Copyright 18.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dfithermite(
        double[] x,
        double[] y,
        int m,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)
    public static void spline1dfithermite(
        double[] x,
        double[] y,
        int n,
        int m,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)

### `spline1dfithermitewc` function

    /*************************************************************************
    Weighted  fitting  by Hermite spline,  with constraints on function values
    or first derivatives.

    Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is  used  to  build
    basis functions. Basis functions are Hermite splines.  Small  regularizing
    term is used when solving constrained tasks (to improve stability).

    Task is linear, so linear least squares solver is used. Complexity of this
    computational scheme is O(N*M^2), mostly dominated by least squares solver

    SEE ALSO
        Spline1DFitCubicWC()    -   fitting by Cubic splines (less flexible,
                                    more smooth)
        Spline1DFitHermite()    -   "lightweight" Hermite fitting, without
                                    invididual weights and constraints

    INPUT PARAMETERS:
        X   -   points, array[0..N-1].
        Y   -   function values, array[0..N-1].
        W   -   weights, array[0..N-1]
                Each summand in square  sum  of  approximation deviations from
                given  values  is  multiplied  by  the square of corresponding
                weight. Fill it by 1's if you don't  want  to  solve  weighted
                task.
        N   -   number of points (optional):
                * N>0
                * if given, only first N elements of X/Y/W are processed
                * if not given, automatically determined from X/Y/W sizes
        XC  -   points where spline values/derivatives are constrained,
                array[0..K-1].
        YC  -   values of constraints, array[0..K-1]
        DC  -   array[0..K-1], types of constraints:
                * DC[i]=0   means that S(XC[i])=YC[i]
                * DC[i]=1   means that S'(XC[i])=YC[i]
                SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
        K   -   number of constraints (optional):
                * 0<=K<M.
                * K=0 means no constraints (XC/YC/DC are not used)
                * if given, only first K elements of XC/YC/DC are used
                * if not given, automatically determined from XC/YC/DC
        M   -   number of basis functions (= 2 * number of nodes),
                M>=4,
                M IS EVEN!

    OUTPUT PARAMETERS:
        Info-   same format as in LSFitLinearW() subroutine:
                * Info>0    task is solved
                * Info<=0   an error occured:
                            -4 means inconvergence of internal SVD
                            -3 means inconsistent constraints
                            -2 means odd M was passed (which is not supported)
                            -1 means another errors in parameters passed
                               (N<=0, for example)
        S   -   spline interpolant.
        Rep -   report, same format as in LSFitLinearW() subroutine.
                Following fields are set:
                * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

    IMPORTANT:
        this subroitine doesn't calculate task's condition number for K<>0.

    IMPORTANT:
        this subroitine supports only even M's

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller may pass unsorted array.

    SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

    Setting constraints can lead  to undesired  results,  like ill-conditioned
    behavior, or inconsistency being detected. From the other side,  it allows
    us to improve quality of the fit. Here we summarize  our  experience  with
    constrained regression splines:
    * excessive constraints can be inconsistent. Splines are  piecewise  cubic
      functions, and it is easy to create an example, where  large  number  of
      constraints  concentrated  in  small  area will result in inconsistency.
      Just because spline is not flexible enough to satisfy all of  them.  And
      same constraints spread across the  [min(x),max(x)]  will  be  perfectly
      consistent.
    * the more evenly constraints are spread across [min(x),max(x)],  the more
      chances that they will be consistent
    * the  greater  is  M (given  fixed  constraints),  the  more chances that
      constraints will be consistent
    * in the general case, consistency of constraints is NOT GUARANTEED.
    * in the several special cases, however, we can guarantee consistency.
    * one of this cases is  M>=4  and   constraints  on   the  function  value
      (AND/OR its derivative) at the interval boundaries.
    * another special case is M>=4  and  ONE  constraint on the function value
      (OR, BUT NOT AND, derivative) anywhere in [min(x),max(x)]

    Our final recommendation is to use constraints  WHEN  AND  ONLY  when  you
    can't solve your task without them. Anything beyond  special  cases  given
    above is not guaranteed and may result in inconsistency.

      -- ALGLIB PROJECT --
         Copyright 18.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dfithermitewc(
        double[] x,
        double[] y,
        double[] w,
        double[] xc,
        double[] yc,
        int[] dc,
        int m,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)
    public static void spline1dfithermitewc(
        double[] x,
        double[] y,
        double[] w,
        int n,
        double[] xc,
        double[] yc,
        int[] dc,
        int k,
        int m,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)

### `spline1dfitpenalized` function

    /*************************************************************************
    Rational least squares fitting using  Floater-Hormann  rational  functions
    with optimal D chosen from [0,9].

    Equidistant  grid  with M node on [min(x),max(x)]  is  used to build basis
    functions. Different values of D are tried, optimal  D  (least  root  mean
    square error) is chosen.  Task  is  linear, so linear least squares solver
    is used. Complexity  of  this  computational  scheme is  O(N*M^2)  (mostly
    dominated by the least squares solver).

    INPUT PARAMETERS:
        X   -   points, array[0..N-1].
        Y   -   function values, array[0..N-1].
        N   -   number of points, N>0.
        M   -   number of basis functions ( = number_of_nodes), M>=2.

    OUTPUT PARAMETERS:
        Info-   same format as in LSFitLinearWC() subroutine.
                * Info>0    task is solved
                * Info<=0   an error occured:
                            -4 means inconvergence of internal SVD
                            -3 means inconsistent constraints
        B   -   barycentric interpolant.
        Rep -   report, same format as in LSFitLinearWC() subroutine.
                Following fields are set:
                * DBest         best value of the D parameter
                * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

      -- ALGLIB PROJECT --
         Copyright 18.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dfitpenalized(
        double[] x,
        double[] y,
        int m,
        double rho,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)
    public static void spline1dfitpenalized(
        double[] x,
        double[] y,
        int n,
        int m,
        double rho,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)

Examples: [1]

### `spline1dfitpenalizedw` function

    /*************************************************************************
    Weighted fitting by penalized cubic spline.

    Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is  used  to  build
    basis functions. Basis functions are cubic splines with  natural  boundary
    conditions. Problem is regularized by  adding non-linearity penalty to the
    usual least squares penalty function:

        S(x) = arg min { LS + P }, where
        LS   = SUM { w[i]^2*(y[i] - S(x[i]))^2 } - least squares penalty
        P    = C*10^rho*integral{ S''(x)^2*dx } - non-linearity penalty
        rho  - tunable constant given by user
        C    - automatically determined scale parameter,
               makes penalty invariant with respect to scaling of X, Y, W.

    INPUT PARAMETERS:
        X   -   points, array[0..N-1].
        Y   -   function values, array[0..N-1].
        W   -   weights, array[0..N-1]
                Each summand in square  sum  of  approximation deviations from
                given  values  is  multiplied  by  the square of corresponding
                weight. Fill it by 1's if you don't  want  to  solve  weighted
                problem.
        N   -   number of points (optional):
                * N>0
                * if given, only first N elements of X/Y/W are processed
                * if not given, automatically determined from X/Y/W sizes
        M   -   number of basis functions ( = number_of_nodes), M>=4.
        Rho -   regularization  constant  passed   by   user.   It   penalizes
                nonlinearity in the regression spline. It  is  logarithmically
                scaled,  i.e.  actual  value  of  regularization  constant  is
                calculated as 10^Rho. It is automatically scaled so that:
                * Rho=2.0 corresponds to moderate amount of nonlinearity
                * generally, it should be somewhere in the [-8.0,+8.0]
                If you do not want to penalize nonlineary,
                pass small Rho. Values as low as -15 should work.

    OUTPUT PARAMETERS:
        Info-   same format as in LSFitLinearWC() subroutine.
                * Info>0    task is solved
                * Info<=0   an error occured:
                            -4 means inconvergence of internal SVD or
                               Cholesky decomposition; problem may be
                               too ill-conditioned (very rare)
        S   -   spline interpolant.
        Rep -   Following fields are set:
                * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

    IMPORTANT:
        this subroitine doesn't calculate task's condition number for K<>0.

    NOTE 1: additional nodes are added to the spline outside  of  the  fitting
    interval to force linearity when x<min(x,xc) or x>max(x,xc).  It  is  done
    for consistency - we penalize non-linearity  at [min(x,xc),max(x,xc)],  so
    it is natural to force linearity outside of this interval.

    NOTE 2: function automatically sorts points,  so  caller may pass unsorted
    array.

      -- ALGLIB PROJECT --
         Copyright 19.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dfitpenalizedw(
        double[] x,
        double[] y,
        double[] w,
        int m,
        double rho,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)
    public static void spline1dfitpenalizedw(
        double[] x,
        double[] y,
        double[] w,
        int n,
        int m,
        double rho,
        out int info,
        out spline1dinterpolant s,
        out spline1dfitreport rep)

Examples: [1]

### lsfit_d_lin example

    public static int Main(string[] args)
    {
        //
        // In this example we demonstrate linear fitting by f(x|a) = a*exp(0.5*x).
        //
        // We have:
        // * y - vector of experimental data
        // * fmatrix -  matrix of basis functions calculated at sample points
        //              Actually, we have only one basis function F0 = exp(0.5*x).
        //
        double[,] fmatrix = new double[,]{{0.606531},{0.670320},{0.740818},{0.818731},{0.904837},{1.000000},{1.105171},{1.221403},{1.349859},{1.491825},{1.648721}};
        double[] y = new double[]{1.133719,1.306522,1.504604,1.554663,1.884638,2.072436,2.257285,2.534068,2.622017,2.897713,3.219371};
        int info;
        double[] c;
        alglib.lsfitreport rep;

        //
        // Linear fitting without weights
        //
        alglib.lsfitlinear(y, fmatrix, out info, out c, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 1
        System.Console.WriteLine("{0}", alglib.ap.format(c,4)); // EXPECTED: [1.98650]

        //
        // Linear fitting with individual weights.
        // Slightly different result is returned.
        //
        double[] w = new double[]{1.414213,1,1,1,1,1,1,1,1,1,1};
        alglib.lsfitlinearw(y, w, fmatrix, out info, out c, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 1
        System.Console.WriteLine("{0}", alglib.ap.format(c,4)); // EXPECTED: [1.983354]
        System.Console.ReadLine();
        return 0;
    }

### lsfit_d_linc example

    public static int Main(string[] args)
    {
        //
        // In this example we demonstrate linear fitting by f(x|a,b) = a*x+b
        // with simple constraint f(0)=0.
        //
        // We have:
        // * y - vector of experimental data
        // * fmatrix -  matrix of basis functions sampled at [0,1] with step 0.2:
        //                  [ 1.0   0.0 ]
        //                  [ 1.0   0.2 ]
        //                  [ 1.0   0.4 ]
        //                  [ 1.0   0.6 ]
        //                  [ 1.0   0.8 ]
        //                  [ 1.0   1.0 ]
        //              first column contains value of first basis function (constant term)
        //              second column contains second basis function (linear term)
        // * cmatrix -  matrix of linear constraints:
        //                  [ 1.0  0.0  0.0 ]
        //              first two columns contain coefficients before basis functions,
        //              last column contains desired value of their sum.
        //              So [1,0,0] means "1*constant_term + 0*linear_term = 0"
        //
        double[] y = new double[]{0.072436,0.246944,0.491263,0.522300,0.714064,0.921929};
        double[,] fmatrix = new double[,]{{1,0.0},{1,0.2},{1,0.4},{1,0.6},{1,0.8},{1,1.0}};
        double[,] cmatrix = new double[,]{{1,0,0}};
        int info;
        double[] c;
        alglib.lsfitreport rep;

        //
        // Constrained fitting without weights
        //
        alglib.lsfitlinearc(y, fmatrix, cmatrix, out info, out c, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 1
        System.Console.WriteLine("{0}", alglib.ap.format(c,3)); // EXPECTED: [0,0.932933]

        //
        // Constrained fitting with individual weights
        //
        double[] w = new double[]{1,1.414213,1,1,1,1};
        alglib.lsfitlinearwc(y, w, fmatrix, cmatrix, out info, out c, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 1
        System.Console.WriteLine("{0}", alglib.ap.format(c,3)); // EXPECTED: [0,0.938322]
        System.Console.ReadLine();
        return 0;
    }

### lsfit_d_nlf example

    public static void function_cx_1_func(double[] c, double[] x, ref double func, object obj)
    {
        // this callback calculates f(c,x)=exp(-c0*sqr(x0))
        // where x is a position on X-axis and c is adjustable parameter
        func = System.Math.Exp(-c[0]*x[0]*x[0]);
    }
    public static int Main(string[] args)
    {
        //
        // In this example we demonstrate exponential fitting
        // by f(x) = exp(-c*x^2)
        // using function value only.
        //
        // Gradient is estimated using combination of numerical differences
        // and secant updates. diffstep variable stores differentiation step
        // (we have to tell algorithm what step to use).
        //
        double[,] x = new double[,]{{-1},{-0.8},{-0.6},{-0.4},{-0.2},{0},{0.2},{0.4},{0.6},{0.8},{1.0}};
        double[] y = new double[]{0.223130,0.382893,0.582748,0.786628,0.941765,1.000000,0.941765,0.786628,0.582748,0.382893,0.223130};
        double[] c = new double[]{0.3};
        double epsf = 0;
        double epsx = 0.000001;
        int maxits = 0;
        int info;
        alglib.lsfitstate state;
        alglib.lsfitreport rep;
        double diffstep = 0.0001;

        //
        // Fitting without weights
        //
        alglib.lsfitcreatef(x, y, c, diffstep, out state);
        alglib.lsfitsetcond(state, epsf, epsx, maxits);
        alglib.lsfitfit(state, function_cx_1_func, null, null);
        alglib.lsfitresults(state, out info, out c, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 2
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [1.5]

        //
        // Fitting with weights
        // (you can change weights and see how it changes result)
        //
        double[] w = new double[]{1,1,1,1,1,1,1,1,1,1,1};
        alglib.lsfitcreatewf(x, y, w, c, diffstep, out state);
        alglib.lsfitsetcond(state, epsf, epsx, maxits);
        alglib.lsfitfit(state, function_cx_1_func, null, null);
        alglib.lsfitresults(state, out info, out c, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 2
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [1.5]
        System.Console.ReadLine();
        return 0;
    }

### lsfit_d_nlfb example

    public static void function_cx_1_func(double[] c, double[] x, ref double func, object obj)
    {
        // this callback calculates f(c,x)=exp(-c0*sqr(x0))
        // where x is a position on X-axis and c is adjustable parameter
        func = System.Math.Exp(-c[0]*x[0]*x[0]);
    }
    public static int Main(string[] args)
    {
        //
        // In this example we demonstrate exponential fitting by
        //     f(x) = exp(-c*x^2)
        // subject to bound constraints
        //     0.0 <= c <= 1.0
        // using function value only.
        //
        // Gradient is estimated using combination of numerical differences
        // and secant updates. diffstep variable stores differentiation step
        // (we have to tell algorithm what step to use).
        //
        // Unconstrained solution is c=1.5, but because of constraints we should
        // get c=1.0 (at the boundary).
        //
        double[,] x = new double[,]{{-1},{-0.8},{-0.6},{-0.4},{-0.2},{0},{0.2},{0.4},{0.6},{0.8},{1.0}};
        double[] y = new double[]{0.223130,0.382893,0.582748,0.786628,0.941765,1.000000,0.941765,0.786628,0.582748,0.382893,0.223130};
        double[] c = new double[]{0.3};
        double[] bndl = new double[]{0.0};
        double[] bndu = new double[]{1.0};
        double epsf = 0;
        double epsx = 0.000001;
        int maxits = 0;
        int info;
        alglib.lsfitstate state;
        alglib.lsfitreport rep;
        double diffstep = 0.0001;

        alglib.lsfitcreatef(x, y, c, diffstep, out state);
        alglib.lsfitsetbc(state, bndl, bndu);
        alglib.lsfitsetcond(state, epsf, epsx, maxits);
        alglib.lsfitfit(state, function_cx_1_func, null, null);
        alglib.lsfitresults(state, out info, out c, out rep);
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [1.0]
        System.Console.ReadLine();
        return 0;
    }

### lsfit_d_nlfg example

    public static void function_cx_1_func(double[] c, double[] x, ref double func, object obj)
    {
        // this callback calculates f(c,x)=exp(-c0*sqr(x0))
        // where x is a position on X-axis and c is adjustable parameter
        func = System.Math.Exp(-c[0]*x[0]*x[0]);
    }
    public static void function_cx_1_grad(double[] c, double[] x, ref double func, double[] grad, object obj)
    {
        // this callback calculates f(c,x)=exp(-c0*sqr(x0)) and gradient G={df/dc[i]}
        // where x is a position on X-axis and c is adjustable parameter.
        // IMPORTANT: gradient is calculated with respect to C, not to X
        func = System.Math.Exp(-c[0]*System.Math.Pow(x[0],2));
        grad[0] = -System.Math.Pow(x[0],2)*func;
    }
    public static int Main(string[] args)
    {
        //
        // In this example we demonstrate exponential fitting
        // by f(x) = exp(-c*x^2)
        // using function value and gradient (with respect to c).
        //
        double[,] x = new double[,]{{-1},{-0.8},{-0.6},{-0.4},{-0.2},{0},{0.2},{0.4},{0.6},{0.8},{1.0}};
        double[] y = new double[]{0.223130,0.382893,0.582748,0.786628,0.941765,1.000000,0.941765,0.786628,0.582748,0.382893,0.223130};
        double[] c = new double[]{0.3};
        double epsf = 0;
        double epsx = 0.000001;
        int maxits = 0;
        int info;
        alglib.lsfitstate state;
        alglib.lsfitreport rep;

        //
        // Fitting without weights
        //
        alglib.lsfitcreatefg(x, y, c, true, out state);
        alglib.lsfitsetcond(state, epsf, epsx, maxits);
        alglib.lsfitfit(state, function_cx_1_func, function_cx_1_grad, null, null);
        alglib.lsfitresults(state, out info, out c, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 2
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [1.5]

        //
        // Fitting with weights
        // (you can change weights and see how it changes result)
        //
        double[] w = new double[]{1,1,1,1,1,1,1,1,1,1,1};
        alglib.lsfitcreatewfg(x, y, w, c, true, out state);
        alglib.lsfitsetcond(state, epsf, epsx, maxits);
        alglib.lsfitfit(state, function_cx_1_func, function_cx_1_grad, null, null);
        alglib.lsfitresults(state, out info, out c, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 2
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [1.5]
        System.Console.ReadLine();
        return 0;
    }

### lsfit_d_nlfgh example

    public static void function_cx_1_func(double[] c, double[] x, ref double func, object obj)
    {
        // this callback calculates f(c,x)=exp(-c0*sqr(x0))
        // where x is a position on X-axis and c is adjustable parameter
        func = System.Math.Exp(-c[0]*x[0]*x[0]);
    }
    public static void function_cx_1_grad(double[] c, double[] x, ref double func, double[] grad, object obj)
    {
        // this callback calculates f(c,x)=exp(-c0*sqr(x0)) and gradient G={df/dc[i]}
        // where x is a position on X-axis and c is adjustable parameter.
        // IMPORTANT: gradient is calculated with respect to C, not to X
        func = System.Math.Exp(-c[0]*System.Math.Pow(x[0],2));
        grad[0] = -System.Math.Pow(x[0],2)*func;
    }
    public static void function_cx_1_hess(double[] c, double[] x, ref double func, double[] grad, double[,] hess, object obj)
    {
        // this callback calculates f(c,x)=exp(-c0*sqr(x0)), gradient G={df/dc[i]} and Hessian H={d2f/(dc[i]*dc[j])}
        // where x is a position on X-axis and c is adjustable parameter.
        // IMPORTANT: gradient/Hessian are calculated with respect to C, not to X
        func = System.Math.Exp(-c[0]*System.Math.Pow(x[0],2));
        grad[0] = -System.Math.Pow(x[0],2)*func;
        hess[0,0] = System.Math.Pow(x[0],4)*func;
    }
    public static int Main(string[] args)
    {
        //
        // In this example we demonstrate exponential fitting
        // by f(x) = exp(-c*x^2)
        // using function value, gradient and Hessian (with respect to c)
        //
        double[,] x = new double[,]{{-1},{-0.8},{-0.6},{-0.4},{-0.2},{0},{0.2},{0.4},{0.6},{0.8},{1.0}};
        double[] y = new double[]{0.223130,0.382893,0.582748,0.786628,0.941765,1.000000,0.941765,0.786628,0.582748,0.382893,0.223130};
        double[] c = new double[]{0.3};
        double epsf = 0;
        double epsx = 0.000001;
        int maxits = 0;
        int info;
        alglib.lsfitstate state;
        alglib.lsfitreport rep;

        //
        // Fitting without weights
        //
        alglib.lsfitcreatefgh(x, y, c, out state);
        alglib.lsfitsetcond(state, epsf, epsx, maxits);
        alglib.lsfitfit(state, function_cx_1_func, function_cx_1_grad, function_cx_1_hess, null, null);
        alglib.lsfitresults(state, out info, out c, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 2
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [1.5]

        //
        // Fitting with weights
        // (you can change weights and see how it changes result)
        //
        double[] w = new double[]{1,1,1,1,1,1,1,1,1,1,1};
        alglib.lsfitcreatewfgh(x, y, w, c, out state);
        alglib.lsfitsetcond(state, epsf, epsx, maxits);
        alglib.lsfitfit(state, function_cx_1_func, function_cx_1_grad, function_cx_1_hess, null, null);
        alglib.lsfitresults(state, out info, out c, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 2
        System.Console.WriteLine("{0}", alglib.ap.format(c,1)); // EXPECTED: [1.5]
        System.Console.ReadLine();
        return 0;
    }

### lsfit_d_nlscale example

    public static void function_debt_func(double[] c, double[] x, ref double func, object obj)
    {
        //
        // this callback calculates f(c,x)=c[0]*(1+c[1]*(pow(x[0]-1999,c[2])-1))
        //
        func = c[0]*(1+c[1]*(System.Math.Pow(x[0]-1999,c[2])-1));
    }
    public static int Main(string[] args)
    {
        //
        // In this example we demonstrate fitting by
        //     f(x) = c[0]*(1+c[1]*((x-1999)^c[2]-1))
        // subject to bound constraints
        //     -INF  < c[0] < +INF
        //      -10 <= c[1] <= +10
        //      0.1 <= c[2] <= 2.0
        // Data we want to fit are time series of Japan national debt
        // collected from 2000 to 2008 measured in USD (dollars, not
        // millions of dollars).
        //
        // Our variables are:
        //     c[0] - debt value at initial moment (2000),
        //     c[1] - direction coefficient (growth or decrease),
        //     c[2] - curvature coefficient.
        // You may see that our variables are badly scaled - first one
        // is order of 10^12, and next two are somewhere about 1 in
        // magnitude. Such problem is difficult to solve without some
        // kind of scaling.
        // That is exactly where lsfitsetscale() function can be used.
        // We set scale of our variables to [1.0E12, 1, 1], which allows
        // us to easily solve this problem.
        //
        // You can try commenting out lsfitsetscale() call - and you will
        // see that algorithm will fail to converge.
        //
        double[,] x = new double[,]{{2000},{2001},{2002},{2003},{2004},{2005},{2006},{2007},{2008}};
        double[] y = new double[]{4323239600000.0,4560913100000.0,5564091500000.0,6743189300000.0,7284064600000.0,7050129600000.0,7092221500000.0,8483907600000.0,8625804400000.0};
        double[] c = new double[]{1.0e+13,1,1};
        double epsf = 0;
        double epsx = 1.0e-5;
        double[] bndl = new double[]{-System.Double.PositiveInfinity,-10,0.1};
        double[] bndu = new double[]{System.Double.PositiveInfinity,+10,2.0};
        double[] s = new double[]{1.0e+12,1,1};
        int maxits = 0;
        int info;
        alglib.lsfitstate state;
        alglib.lsfitreport rep;
        double diffstep = 1.0e-5;

        alglib.lsfitcreatef(x, y, c, diffstep, out state);
        alglib.lsfitsetcond(state, epsf, epsx, maxits);
        alglib.lsfitsetbc(state, bndl, bndu);
        alglib.lsfitsetscale(state, s);
        alglib.lsfitfit(state, function_debt_func, null, null);
        alglib.lsfitresults(state, out info, out c, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 2
        System.Console.WriteLine("{0}", alglib.ap.format(c,-2)); // EXPECTED: [4.142560E+12, 0.434240, 0.565376]
        System.Console.ReadLine();
        return 0;
    }

### lsfit_d_pol example

    public static int Main(string[] args)
    {
        //
        // This example demonstrates polynomial fitting.
        //
        // Fitting is done by two (M=2) functions from polynomial basis:
        //     f0 = 1
        //     f1 = x
        // Basically, it just a linear fit; more complex polynomials may be used
        // (e.g. parabolas with M=3, cubic with M=4), but even such simple fit allows
        // us to demonstrate polynomialfit() function in action.
        //
        // We have:
        // * x      set of abscissas
        // * y      experimental data
        //
        // Additionally we demonstrate weighted fitting, where second point has
        // more weight than other ones.
        //
        double[] x = new double[]{0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0};
        double[] y = new double[]{0.00,0.05,0.26,0.32,0.33,0.43,0.60,0.60,0.77,0.98,1.02};
        int m = 2;
        double t = 2;
        int info;
        alglib.barycentricinterpolant p;
        alglib.polynomialfitreport rep;
        double v;

        //
        // Fitting without individual weights
        //
        // NOTE: result is returned as barycentricinterpolant structure.
        //       if you want to get representation in the power basis,
        //       you can use barycentricbar2pow() function to convert
        //       from barycentric to power representation (see docs for
        //       POLINT subpackage for more info).
        //
        alglib.polynomialfit(x, y, m, out info, out p, out rep);
        v = alglib.barycentriccalc(p, t);
        System.Console.WriteLine("{0:F2", v); // EXPECTED: 2.011

        //
        // Fitting with individual weights
        //
        // NOTE: slightly different result is returned
        //
        double[] w = new double[]{1,1.414213562,1,1,1,1,1,1,1,1,1};
        double[] xc = new double[0];
        double[] yc = new double[0];
        int[] dc = new int[0];
        alglib.polynomialfitwc(x, y, w, xc, yc, dc, m, out info, out p, out rep);
        v = alglib.barycentriccalc(p, t);
        System.Console.WriteLine("{0:F2", v); // EXPECTED: 2.023
        System.Console.ReadLine();
        return 0;
    }

### lsfit_d_polc example

    public static int Main(string[] args)
    {
        //
        // This example demonstrates polynomial fitting.
        //
        // Fitting is done by two (M=2) functions from polynomial basis:
        //     f0 = 1
        //     f1 = x
        // with simple constraint on function value
        //     f(0) = 0
        // Basically, it just a linear fit; more complex polynomials may be used
        // (e.g. parabolas with M=3, cubic with M=4), but even such simple fit allows
        // us to demonstrate polynomialfit() function in action.
        //
        // We have:
        // * x      set of abscissas
        // * y      experimental data
        // * xc     points where constraints are placed
        // * yc     constraints on derivatives
        // * dc     derivative indices
        //          (0 means function itself, 1 means first derivative)
        //
        double[] x = new double[]{1.0,1.0};
        double[] y = new double[]{0.9,1.1};
        double[] w = new double[]{1,1};
        double[] xc = new double[]{0};
        double[] yc = new double[]{0};
        int[] dc = new int[]{0};
        double t = 2;
        int m = 2;
        int info;
        alglib.barycentricinterpolant p;
        alglib.polynomialfitreport rep;
        double v;

        alglib.polynomialfitwc(x, y, w, xc, yc, dc, m, out info, out p, out rep);
        v = alglib.barycentriccalc(p, t);
        System.Console.WriteLine("{0:F2", v); // EXPECTED: 2.000
        System.Console.ReadLine();
        return 0;
    }

### lsfit_d_spline example

    public static int Main(string[] args)
    {
        //
        // In this example we demonstrate penalized spline fitting of noisy data
        //
        // We have:
        // * x - abscissas
        // * y - vector of experimental data, straight line with small noise
        //
        double[] x = new double[]{0.00,0.10,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90};
        double[] y = new double[]{0.10,0.00,0.30,0.40,0.30,0.40,0.62,0.68,0.75,0.95};
        int info;
        double v;
        alglib.spline1dinterpolant s;
        alglib.spline1dfitreport rep;
        double rho;

        //
        // Fit with VERY small amount of smoothing (rho = -5.0)
        // and large number of basis functions (M=50).
        //
        // With such small regularization penalized spline almost fully reproduces function values
        //
        rho = -5.0;
        alglib.spline1dfitpenalized(x, y, 50, rho, out info, out s, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 1
        v = alglib.spline1dcalc(s, 0.0);
        System.Console.WriteLine("{0:F1", v); // EXPECTED: 0.10

        //
        // Fit with VERY large amount of smoothing (rho = 10.0)
        // and large number of basis functions (M=50).
        //
        // With such regularization our spline should become close to the straight line fit.
        // We will compare its value in x=1.0 with results obtained from such fit.
        //
        rho = +10.0;
        alglib.spline1dfitpenalized(x, y, 50, rho, out info, out s, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 1
        v = alglib.spline1dcalc(s, 1.0);
        System.Console.WriteLine("{0:F2", v); // EXPECTED: 0.969

        //
        // In real life applications you may need some moderate degree of fitting,
        // so we try to fit once more with rho=3.0.
        //
        rho = +3.0;
        alglib.spline1dfitpenalized(x, y, 50, rho, out info, out s, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 1
        System.Console.ReadLine();
        return 0;
    }

## `mannwhitneyu` subpackage

### Functions

mannwhitneyutest

### Examples

### `mannwhitneyutest` function

    /*************************************************************************
    Mann-Whitney U-test

    This test checks hypotheses about whether X  and  Y  are  samples  of  two
    continuous distributions of the same shape  and  same  median  or  whether
    their medians are different.

    The following tests are performed:
        * two-tailed test (null hypothesis - the medians are equal)
        * left-tailed test (null hypothesis - the median of the  first  sample
          is greater than or equal to the median of the second sample)
        * right-tailed test (null hypothesis - the median of the first  sample
          is less than or equal to the median of the second sample).

    Requirements:
        * the samples are independent
        * X and Y are continuous distributions (or discrete distributions well-
          approximating continuous distributions)
        * distributions of X and Y have the  same  shape.  The  only  possible
          difference is their position (i.e. the value of the median)
        * the number of elements in each sample is not less than 5
        * the scale of measurement should be ordinal, interval or ratio  (i.e.
          the test could not be applied to nominal variables).

    The test is non-parametric and doesn't require distributions to be normal.

    Input parameters:
        X   -   sample 1. Array whose index goes from 0 to N-1.
        N   -   size of the sample. N>=5
        Y   -   sample 2. Array whose index goes from 0 to M-1.
        M   -   size of the sample. M>=5

    Output parameters:
        BothTails   -   p-value for two-tailed test.
                        If BothTails is less than the given significance level
                        the null hypothesis is rejected.
        LeftTail    -   p-value for left-tailed test.
                        If LeftTail is less than the given significance level,
                        the null hypothesis is rejected.
        RightTail   -   p-value for right-tailed test.
                        If RightTail is less than the given significance level
                        the null hypothesis is rejected.

    To calculate p-values, special approximation is used. This method lets  us
    calculate p-values with satisfactory  accuracy  in  interval  [0.0001, 1].
    There is no approximation outside the [0.0001, 1] interval. Therefore,  if
    the significance level outlies this interval, the test returns 0.0001.

    Relative precision of approximation of p-value:

    N          M          Max.err.   Rms.err.
    5..10      N..10      1.4e-02    6.0e-04
    5..10      N..100     2.2e-02    5.3e-06
    10..15     N..15      1.0e-02    3.2e-04
    10..15     N..100     1.0e-02    2.2e-05
    15..100    N..100     6.1e-03    2.7e-06

    For N,M>100 accuracy checks weren't put into  practice,  but  taking  into
    account characteristics of asymptotic approximation used, precision should
    not be sharply different from the values for interval [5, 100].

      -- ALGLIB --
         Copyright 09.04.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mannwhitneyutest(
        double[] x,
        int n,
        double[] y,
        int m,
        out double bothtails,
        out double lefttail,
        out double righttail)

## `matdet` subpackage

### Functions

cmatrixdet
cmatrixludet
rmatrixdet
rmatrixludet
spdmatrixcholeskydet
spdmatrixdet

### Examples

matdet_d_1 |  | Determinant calculation, real matrix, short form
matdet_d_2 |  | Determinant calculation, real matrix, full form
matdet_d_3 |  | Determinant calculation, complex matrix, short form
matdet_d_4 |  | Determinant calculation, complex matrix, full form
matdet_d_5 |  | Determinant calculation, complex matrix with zero imaginary part, short form

### `cmatrixdet` function

    /*************************************************************************
    Calculation of the determinant of a general matrix

    Input parameters:
        A       -   matrix, array[0..N-1, 0..N-1]
        N       -   (optional) size of matrix A:
                    * if given, only principal NxN submatrix is processed and
                      overwritten. other elements are unchanged.
                    * if not given, automatically determined from matrix size
                      (A must be square matrix)

    Result: determinant of matrix A.

      -- ALGLIB --
         Copyright 2005 by Bochkanov Sergey
    *************************************************************************/
    public static complex cmatrixdet(complex[,] a)
    public static complex cmatrixdet(complex[,] a, int n)

Examples: [1] [2] [3]

### `cmatrixludet` function

    /*************************************************************************
    Determinant calculation of the matrix given by its LU decomposition.

    Input parameters:
        A       -   LU decomposition of the matrix (output of
                    RMatrixLU subroutine).
        Pivots  -   table of permutations which were made during
                    the LU decomposition.
                    Output of RMatrixLU subroutine.
        N       -   (optional) size of matrix A:
                    * if given, only principal NxN submatrix is processed and
                      overwritten. other elements are unchanged.
                    * if not given, automatically determined from matrix size
                      (A must be square matrix)

    Result: matrix determinant.

      -- ALGLIB --
         Copyright 2005 by Bochkanov Sergey
    *************************************************************************/
    public static complex cmatrixludet(complex[,] a, int[] pivots)
    public static complex cmatrixludet(complex[,] a, int[] pivots, int n)

### `rmatrixdet` function

    /*************************************************************************
    Calculation of the determinant of a general matrix

    Input parameters:
        A       -   matrix, array[0..N-1, 0..N-1]
        N       -   (optional) size of matrix A:
                    * if given, only principal NxN submatrix is processed and
                      overwritten. other elements are unchanged.
                    * if not given, automatically determined from matrix size
                      (A must be square matrix)

    Result: determinant of matrix A.

      -- ALGLIB --
         Copyright 2005 by Bochkanov Sergey
    *************************************************************************/
    public static double rmatrixdet(double[,] a)
    public static double rmatrixdet(double[,] a, int n)

Examples: [1] [2]

### `rmatrixludet` function

    /*************************************************************************
    Determinant calculation of the matrix given by its LU decomposition.

    Input parameters:
        A       -   LU decomposition of the matrix (output of
                    RMatrixLU subroutine).
        Pivots  -   table of permutations which were made during
                    the LU decomposition.
                    Output of RMatrixLU subroutine.
        N       -   (optional) size of matrix A:
                    * if given, only principal NxN submatrix is processed and
                      overwritten. other elements are unchanged.
                    * if not given, automatically determined from matrix size
                      (A must be square matrix)

    Result: matrix determinant.

      -- ALGLIB --
         Copyright 2005 by Bochkanov Sergey
    *************************************************************************/
    public static double rmatrixludet(double[,] a, int[] pivots)
    public static double rmatrixludet(double[,] a, int[] pivots, int n)

### `spdmatrixcholeskydet` function

    /*************************************************************************
    Determinant calculation of the matrix given by the Cholesky decomposition.

    Input parameters:
        A       -   Cholesky decomposition,
                    output of SMatrixCholesky subroutine.
        N       -   (optional) size of matrix A:
                    * if given, only principal NxN submatrix is processed and
                      overwritten. other elements are unchanged.
                    * if not given, automatically determined from matrix size
                      (A must be square matrix)

    As the determinant is equal to the product of squares of diagonal elements,
    it’s not necessary to specify which triangle - lower or upper - the matrix
    is stored in.

    Result:
        matrix determinant.

      -- ALGLIB --
         Copyright 2005-2008 by Bochkanov Sergey
    *************************************************************************/
    public static double spdmatrixcholeskydet(double[,] a)
    public static double spdmatrixcholeskydet(double[,] a, int n)

### `spdmatrixdet` function

    /*************************************************************************
    Determinant calculation of the symmetric positive definite matrix.

    Input parameters:
        A       -   matrix. Array with elements [0..N-1, 0..N-1].
        N       -   (optional) size of matrix A:
                    * if given, only principal NxN submatrix is processed and
                      overwritten. other elements are unchanged.
                    * if not given, automatically determined from matrix size
                      (A must be square matrix)
        IsUpper -   (optional) storage type:
                    * if True, symmetric matrix  A  is  given  by  its  upper
                      triangle, and the lower triangle isn’t used/changed  by
                      function
                    * if False, symmetric matrix  A  is  given  by  its lower
                      triangle, and the upper triangle isn’t used/changed  by
                      function
                    * if not given, both lower and upper  triangles  must  be
                      filled.

    Result:
        determinant of matrix A.
        If matrix A is not positive definite, exception is thrown.

      -- ALGLIB --
         Copyright 2005-2008 by Bochkanov Sergey
    *************************************************************************/
    public static double spdmatrixdet(double[,] a)
    public static double spdmatrixdet(double[,] a, int n, bool isupper)

### matdet_d_1 example

    public static int Main(string[] args)
    {
        double[,] b = new double[,]{{1,2},{2,1}};
        double a;
        a = alglib.rmatrixdet(b);
        System.Console.WriteLine("{0:F3", a); // EXPECTED: -3
        System.Console.ReadLine();
        return 0;
    }

### matdet_d_2 example

    public static int Main(string[] args)
    {
        double[,] b = new double[,]{{5,4},{4,5}};
        double a;
        a = alglib.rmatrixdet(b, 2);
        System.Console.WriteLine("{0:F3", a); // EXPECTED: 9
        System.Console.ReadLine();
        return 0;
    }

### matdet_d_3 example

    public static int Main(string[] args)
    {
        alglib.complex[,] b = new alglib.complex[,]{{new alglib.complex(1,+1),2},{2,new alglib.complex(1,-1)}};
        alglib.complex a;
        a = alglib.cmatrixdet(b);
        System.Console.WriteLine("{0}", alglib.ap.format(a,3)); // EXPECTED: -2
        System.Console.ReadLine();
        return 0;
    }

### matdet_d_4 example

    public static int Main(string[] args)
    {
        alglib.complex a;
        alglib.complex[,] b = new alglib.complex[,]{{new alglib.complex(0,5),4},{new alglib.complex(0,4),5}};
        a = alglib.cmatrixdet(b, 2);
        System.Console.WriteLine("{0}", alglib.ap.format(a,3)); // EXPECTED: 9i
        System.Console.ReadLine();
        return 0;
    }

### matdet_d_5 example

    public static int Main(string[] args)
    {
        alglib.complex a;
        alglib.complex[,] b = new alglib.complex[,]{{9,1},{2,1}};
        a = alglib.cmatrixdet(b);
        System.Console.WriteLine("{0}", alglib.ap.format(a,3)); // EXPECTED: 7
        System.Console.ReadLine();
        return 0;
    }

## `matgen` subpackage

### Functions

cmatrixrndcond
cmatrixrndorthogonal
cmatrixrndorthogonalfromtheleft
cmatrixrndorthogonalfromtheright
hmatrixrndcond
hmatrixrndmultiply
hpdmatrixrndcond
rmatrixrndcond
rmatrixrndorthogonal
rmatrixrndorthogonalfromtheleft
rmatrixrndorthogonalfromtheright
smatrixrndcond
smatrixrndmultiply
spdmatrixrndcond

### Examples

### `cmatrixrndcond` function

    /*************************************************************************
    Generation of random NxN complex matrix with given condition number C and
    norm2(A)=1

    INPUT PARAMETERS:
        N   -   matrix size
        C   -   condition number (in 2-norm)

    OUTPUT PARAMETERS:
        A   -   random matrix with norm2(A)=1 and cond(A)=C

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixrndcond(int n, double c, out complex[,] a)

### `cmatrixrndorthogonal` function

    /*************************************************************************
    Generation of a random Haar distributed orthogonal complex matrix

    INPUT PARAMETERS:
        N   -   matrix size, N>=1

    OUTPUT PARAMETERS:
        A   -   orthogonal NxN matrix, array[0..N-1,0..N-1]

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixrndorthogonal(int n, out complex[,] a)

### `cmatrixrndorthogonalfromtheleft` function

    /*************************************************************************
    Multiplication of MxN complex matrix by MxM random Haar distributed
    complex orthogonal matrix

    INPUT PARAMETERS:
        A   -   matrix, array[0..M-1, 0..N-1]
        M, N-   matrix size

    OUTPUT PARAMETERS:
        A   -   Q*A, where Q is random MxM orthogonal matrix

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixrndorthogonalfromtheleft(
        ref complex[,] a,
        int m,
        int n)

### `cmatrixrndorthogonalfromtheright` function

    /*************************************************************************
    Multiplication of MxN complex matrix by NxN random Haar distributed
    complex orthogonal matrix

    INPUT PARAMETERS:
        A   -   matrix, array[0..M-1, 0..N-1]
        M, N-   matrix size

    OUTPUT PARAMETERS:
        A   -   A*Q, where Q is random NxN orthogonal matrix

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixrndorthogonalfromtheright(
        ref complex[,] a,
        int m,
        int n)

### `hmatrixrndcond` function

    /*************************************************************************
    Generation of random NxN Hermitian matrix with given condition number  and
    norm2(A)=1

    INPUT PARAMETERS:
        N   -   matrix size
        C   -   condition number (in 2-norm)

    OUTPUT PARAMETERS:
        A   -   random matrix with norm2(A)=1 and cond(A)=C

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void hmatrixrndcond(int n, double c, out complex[,] a)

### `hmatrixrndmultiply` function

    /*************************************************************************
    Hermitian multiplication of NxN matrix by random Haar distributed
    complex orthogonal matrix

    INPUT PARAMETERS:
        A   -   matrix, array[0..N-1, 0..N-1]
        N   -   matrix size

    OUTPUT PARAMETERS:
        A   -   Q^H*A*Q, where Q is random NxN orthogonal matrix

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void hmatrixrndmultiply(ref complex[,] a, int n)

### `hpdmatrixrndcond` function

    /*************************************************************************
    Generation of random NxN Hermitian positive definite matrix with given
    condition number and norm2(A)=1

    INPUT PARAMETERS:
        N   -   matrix size
        C   -   condition number (in 2-norm)

    OUTPUT PARAMETERS:
        A   -   random HPD matrix with norm2(A)=1 and cond(A)=C

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void hpdmatrixrndcond(int n, double c, out complex[,] a)

### `rmatrixrndcond` function

    /*************************************************************************
    Generation of random NxN matrix with given condition number and norm2(A)=1

    INPUT PARAMETERS:
        N   -   matrix size
        C   -   condition number (in 2-norm)

    OUTPUT PARAMETERS:
        A   -   random matrix with norm2(A)=1 and cond(A)=C

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixrndcond(int n, double c, out double[,] a)

### `rmatrixrndorthogonal` function

    /*************************************************************************
    Generation of a random uniformly distributed (Haar) orthogonal matrix

    INPUT PARAMETERS:
        N   -   matrix size, N>=1

    OUTPUT PARAMETERS:
        A   -   orthogonal NxN matrix, array[0..N-1,0..N-1]

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixrndorthogonal(int n, out double[,] a)

### `rmatrixrndorthogonalfromtheleft` function

    /*************************************************************************
    Multiplication of MxN matrix by MxM random Haar distributed orthogonal matrix

    INPUT PARAMETERS:
        A   -   matrix, array[0..M-1, 0..N-1]
        M, N-   matrix size

    OUTPUT PARAMETERS:
        A   -   Q*A, where Q is random MxM orthogonal matrix

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixrndorthogonalfromtheleft(
        ref double[,] a,
        int m,
        int n)

### `rmatrixrndorthogonalfromtheright` function

    /*************************************************************************
    Multiplication of MxN matrix by NxN random Haar distributed orthogonal matrix

    INPUT PARAMETERS:
        A   -   matrix, array[0..M-1, 0..N-1]
        M, N-   matrix size

    OUTPUT PARAMETERS:
        A   -   A*Q, where Q is random NxN orthogonal matrix

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixrndorthogonalfromtheright(
        ref double[,] a,
        int m,
        int n)

### `smatrixrndcond` function

    /*************************************************************************
    Generation of random NxN symmetric matrix with given condition number  and
    norm2(A)=1

    INPUT PARAMETERS:
        N   -   matrix size
        C   -   condition number (in 2-norm)

    OUTPUT PARAMETERS:
        A   -   random matrix with norm2(A)=1 and cond(A)=C

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void smatrixrndcond(int n, double c, out double[,] a)

### `smatrixrndmultiply` function

    /*************************************************************************
    Symmetric multiplication of NxN matrix by random Haar distributed
    orthogonal  matrix

    INPUT PARAMETERS:
        A   -   matrix, array[0..N-1, 0..N-1]
        N   -   matrix size

    OUTPUT PARAMETERS:
        A   -   Q'*A*Q, where Q is random NxN orthogonal matrix

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void smatrixrndmultiply(ref double[,] a, int n)

### `spdmatrixrndcond` function

    /*************************************************************************
    Generation of random NxN symmetric positive definite matrix with given
    condition number and norm2(A)=1

    INPUT PARAMETERS:
        N   -   matrix size
        C   -   condition number (in 2-norm)

    OUTPUT PARAMETERS:
        A   -   random SPD matrix with norm2(A)=1 and cond(A)=C

      -- ALGLIB routine --
         04.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static void spdmatrixrndcond(int n, double c, out double[,] a)

## `matinv` subpackage

### Classes

matinvreport

### Functions

cmatrixinverse
cmatrixluinverse
cmatrixtrinverse
hpdmatrixcholeskyinverse
hpdmatrixinverse
rmatrixinverse
rmatrixluinverse
rmatrixtrinverse
spdmatrixcholeskyinverse
spdmatrixinverse

### Examples

matinv_d_c1 |  | Complex matrix inverse
matinv_d_hpd1 |  | HPD matrix inverse
matinv_d_r1 |  | Real matrix inverse
matinv_d_spd1 |  | SPD matrix inverse

### `matinvreport` class

    /*************************************************************************
    Matrix inverse report:
    * R1    reciprocal of condition number in 1-norm
    * RInf  reciprocal of condition number in inf-norm
    *************************************************************************/
    public class matinvreport
    {
        public double               r1;
        public double               rinf;
    }

### `cmatrixinverse` function

    /*************************************************************************
    Inversion of a general matrix.

    Input parameters:
        A       -   matrix
        N       -   size of matrix A (optional) :
                    * if given, only principal NxN submatrix is processed  and
                      overwritten. other elements are unchanged.
                    * if not given,  size  is  automatically  determined  from
                      matrix size (A must be square matrix)

    Output parameters:
        Info    -   return code, same as in RMatrixLUInverse
        Rep     -   solver report, same as in RMatrixLUInverse
        A       -   inverse of matrix A, same as in RMatrixLUInverse

      -- ALGLIB --
         Copyright 2005 by Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixinverse(
        ref complex[,] a,
        out int info,
        out matinvreport rep)
    public static void cmatrixinverse(
        ref complex[,] a,
        int n,
        out int info,
        out matinvreport rep)

Examples: [1]

### `cmatrixluinverse` function

    /*************************************************************************
    Inversion of a matrix given by its LU decomposition.

    INPUT PARAMETERS:
        A       -   LU decomposition of the matrix
                    (output of CMatrixLU subroutine).
        Pivots  -   table of permutations
                    (the output of CMatrixLU subroutine).
        N       -   size of matrix A (optional) :
                    * if given, only principal NxN submatrix is processed  and
                      overwritten. other elements are unchanged.
                    * if not given,  size  is  automatically  determined  from
                      matrix size (A must be square matrix)

    OUTPUT PARAMETERS:
        Info    -   return code, same as in RMatrixLUInverse
        Rep     -   solver report, same as in RMatrixLUInverse
        A       -   inverse of matrix A, same as in RMatrixLUInverse

      -- ALGLIB routine --
         05.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixluinverse(
        ref complex[,] a,
        int[] pivots,
        out int info,
        out matinvreport rep)
    public static void cmatrixluinverse(
        ref complex[,] a,
        int[] pivots,
        int n,
        out int info,
        out matinvreport rep)

### `cmatrixtrinverse` function

    /*************************************************************************
    Triangular matrix inverse (complex)

    The subroutine inverts the following types of matrices:
        * upper triangular
        * upper triangular with unit diagonal
        * lower triangular
        * lower triangular with unit diagonal

    In case of an upper (lower) triangular matrix,  the  inverse  matrix  will
    also be upper (lower) triangular, and after the end of the algorithm,  the
    inverse matrix replaces the source matrix. The elements  below (above) the
    main diagonal are not changed by the algorithm.

    If  the matrix  has a unit diagonal, the inverse matrix also  has  a  unit
    diagonal, and the diagonal elements are not passed to the algorithm.

    Input parameters:
        A       -   matrix, array[0..N-1, 0..N-1].
        N       -   size of matrix A (optional) :
                    * if given, only principal NxN submatrix is processed  and
                      overwritten. other elements are unchanged.
                    * if not given,  size  is  automatically  determined  from
                      matrix size (A must be square matrix)
        IsUpper -   True, if the matrix is upper triangular.
        IsUnit  -   diagonal type (optional):
                    * if True, matrix has unit diagonal (a[i,i] are NOT used)
                    * if False, matrix diagonal is arbitrary
                    * if not given, False is assumed

    Output parameters:
        Info    -   same as for RMatrixLUInverse
        Rep     -   same as for RMatrixLUInverse
        A       -   same as for RMatrixLUInverse.

      -- ALGLIB --
         Copyright 05.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixtrinverse(
        ref complex[,] a,
        bool isupper,
        out int info,
        out matinvreport rep)
    public static void cmatrixtrinverse(
        ref complex[,] a,
        int n,
        bool isupper,
        bool isunit,
        out int info,
        out matinvreport rep)

### `hpdmatrixcholeskyinverse` function

    /*************************************************************************
    Inversion of a Hermitian positive definite matrix which is given
    by Cholesky decomposition.

    Input parameters:
        A       -   Cholesky decomposition of the matrix to be inverted:
                    A=U’*U or A = L*L'.
                    Output of  HPDMatrixCholesky subroutine.
        N       -   size of matrix A (optional) :
                    * if given, only principal NxN submatrix is processed  and
                      overwritten. other elements are unchanged.
                    * if not given,  size  is  automatically  determined  from
                      matrix size (A must be square matrix)
        IsUpper -   storage type (optional):
                    * if True, symmetric  matrix  A  is  given  by  its  upper
                      triangle, and the lower triangle isn’t  used/changed  by
                      function
                    * if False,  symmetric matrix  A  is  given  by  its lower
                      triangle, and the  upper triangle isn’t used/changed  by
                      function
                    * if not given, lower half is used.

    Output parameters:
        Info    -   return code, same as in RMatrixLUInverse
        Rep     -   solver report, same as in RMatrixLUInverse
        A       -   inverse of matrix A, same as in RMatrixLUInverse

      -- ALGLIB routine --
         10.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void hpdmatrixcholeskyinverse(
        ref complex[,] a,
        out int info,
        out matinvreport rep)
    public static void hpdmatrixcholeskyinverse(
        ref complex[,] a,
        int n,
        bool isupper,
        out int info,
        out matinvreport rep)

### `hpdmatrixinverse` function

    /*************************************************************************
    Inversion of a Hermitian positive definite matrix.

    Given an upper or lower triangle of a Hermitian positive definite matrix,
    the algorithm generates matrix A^-1 and saves the upper or lower triangle
    depending on the input.

    Input parameters:
        A       -   matrix to be inverted (upper or lower triangle).
                    Array with elements [0..N-1,0..N-1].
        N       -   size of matrix A (optional) :
                    * if given, only principal NxN submatrix is processed  and
                      overwritten. other elements are unchanged.
                    * if not given,  size  is  automatically  determined  from
                      matrix size (A must be square matrix)
        IsUpper -   storage type (optional):
                    * if True, symmetric  matrix  A  is  given  by  its  upper
                      triangle, and the lower triangle isn’t  used/changed  by
                      function
                    * if False,  symmetric matrix  A  is  given  by  its lower
                      triangle, and the  upper triangle isn’t used/changed  by
                      function
                    * if not given,  both lower and upper  triangles  must  be
                      filled.

    Output parameters:
        Info    -   return code, same as in RMatrixLUInverse
        Rep     -   solver report, same as in RMatrixLUInverse
        A       -   inverse of matrix A, same as in RMatrixLUInverse

      -- ALGLIB routine --
         10.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void hpdmatrixinverse(
        ref complex[,] a,
        out int info,
        out matinvreport rep)
    public static void hpdmatrixinverse(
        ref complex[,] a,
        int n,
        bool isupper,
        out int info,
        out matinvreport rep)

Examples: [1]

### `rmatrixinverse` function

    /*************************************************************************
    Inversion of a general matrix.

    Input parameters:
        A       -   matrix.
        N       -   size of matrix A (optional) :
                    * if given, only principal NxN submatrix is processed  and
                      overwritten. other elements are unchanged.
                    * if not given,  size  is  automatically  determined  from
                      matrix size (A must be square matrix)

    Output parameters:
        Info    -   return code, same as in RMatrixLUInverse
        Rep     -   solver report, same as in RMatrixLUInverse
        A       -   inverse of matrix A, same as in RMatrixLUInverse

    Result:
        True, if the matrix is not singular.
        False, if the matrix is singular.

      -- ALGLIB --
         Copyright 2005-2010 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixinverse(
        ref double[,] a,
        out int info,
        out matinvreport rep)
    public static void rmatrixinverse(
        ref double[,] a,
        int n,
        out int info,
        out matinvreport rep)

Examples: [1]

### `rmatrixluinverse` function

    /*************************************************************************
    Inversion of a matrix given by its LU decomposition.

    INPUT PARAMETERS:
        A       -   LU decomposition of the matrix
                    (output of RMatrixLU subroutine).
        Pivots  -   table of permutations
                    (the output of RMatrixLU subroutine).
        N       -   size of matrix A (optional) :
                    * if given, only principal NxN submatrix is processed  and
                      overwritten. other elements are unchanged.
                    * if not given,  size  is  automatically  determined  from
                      matrix size (A must be square matrix)

    OUTPUT PARAMETERS:
        Info    -   return code:
                    * -3    A is singular, or VERY close to singular.
                            it is filled by zeros in such cases.
                    *  1    task is solved (but matrix A may be ill-conditioned,
                            check R1/RInf parameters for condition numbers).
        Rep     -   solver report, see below for more info
        A       -   inverse of matrix A.
                    Array whose indexes range within [0..N-1, 0..N-1].

    SOLVER REPORT

    Subroutine sets following fields of the Rep structure:
    * R1        reciprocal of condition number: 1/cond(A), 1-norm.
    * RInf      reciprocal of condition number: 1/cond(A), inf-norm.

      -- ALGLIB routine --
         05.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixluinverse(
        ref double[,] a,
        int[] pivots,
        out int info,
        out matinvreport rep)
    public static void rmatrixluinverse(
        ref double[,] a,
        int[] pivots,
        int n,
        out int info,
        out matinvreport rep)

### `rmatrixtrinverse` function

    /*************************************************************************
    Triangular matrix inverse (real)

    The subroutine inverts the following types of matrices:
        * upper triangular
        * upper triangular with unit diagonal
        * lower triangular
        * lower triangular with unit diagonal

    In case of an upper (lower) triangular matrix,  the  inverse  matrix  will
    also be upper (lower) triangular, and after the end of the algorithm,  the
    inverse matrix replaces the source matrix. The elements  below (above) the
    main diagonal are not changed by the algorithm.

    If  the matrix  has a unit diagonal, the inverse matrix also  has  a  unit
    diagonal, and the diagonal elements are not passed to the algorithm.

    Input parameters:
        A       -   matrix, array[0..N-1, 0..N-1].
        N       -   size of matrix A (optional) :
                    * if given, only principal NxN submatrix is processed  and
                      overwritten. other elements are unchanged.
                    * if not given,  size  is  automatically  determined  from
                      matrix size (A must be square matrix)
        IsUpper -   True, if the matrix is upper triangular.
        IsUnit  -   diagonal type (optional):
                    * if True, matrix has unit diagonal (a[i,i] are NOT used)
                    * if False, matrix diagonal is arbitrary
                    * if not given, False is assumed

    Output parameters:
        Info    -   same as for RMatrixLUInverse
        Rep     -   same as for RMatrixLUInverse
        A       -   same as for RMatrixLUInverse.

      -- ALGLIB --
         Copyright 05.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixtrinverse(
        ref double[,] a,
        bool isupper,
        out int info,
        out matinvreport rep)
    public static void rmatrixtrinverse(
        ref double[,] a,
        int n,
        bool isupper,
        bool isunit,
        out int info,
        out matinvreport rep)

### `spdmatrixcholeskyinverse` function

    /*************************************************************************
    Inversion of a symmetric positive definite matrix which is given
    by Cholesky decomposition.

    Input parameters:
        A       -   Cholesky decomposition of the matrix to be inverted:
                    A=U’*U or A = L*L'.
                    Output of  SPDMatrixCholesky subroutine.
        N       -   size of matrix A (optional) :
                    * if given, only principal NxN submatrix is processed  and
                      overwritten. other elements are unchanged.
                    * if not given,  size  is  automatically  determined  from
                      matrix size (A must be square matrix)
        IsUpper -   storage type (optional):
                    * if True, symmetric  matrix  A  is  given  by  its  upper
                      triangle, and the lower triangle isn’t  used/changed  by
                      function
                    * if False,  symmetric matrix  A  is  given  by  its lower
                      triangle, and the  upper triangle isn’t used/changed  by
                      function
                    * if not given, lower half is used.

    Output parameters:
        Info    -   return code, same as in RMatrixLUInverse
        Rep     -   solver report, same as in RMatrixLUInverse
        A       -   inverse of matrix A, same as in RMatrixLUInverse

      -- ALGLIB routine --
         10.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void spdmatrixcholeskyinverse(
        ref double[,] a,
        out int info,
        out matinvreport rep)
    public static void spdmatrixcholeskyinverse(
        ref double[,] a,
        int n,
        bool isupper,
        out int info,
        out matinvreport rep)

### `spdmatrixinverse` function

    /*************************************************************************
    Inversion of a symmetric positive definite matrix.

    Given an upper or lower triangle of a symmetric positive definite matrix,
    the algorithm generates matrix A^-1 and saves the upper or lower triangle
    depending on the input.

    Input parameters:
        A       -   matrix to be inverted (upper or lower triangle).
                    Array with elements [0..N-1,0..N-1].
        N       -   size of matrix A (optional) :
                    * if given, only principal NxN submatrix is processed  and
                      overwritten. other elements are unchanged.
                    * if not given,  size  is  automatically  determined  from
                      matrix size (A must be square matrix)
        IsUpper -   storage type (optional):
                    * if True, symmetric  matrix  A  is  given  by  its  upper
                      triangle, and the lower triangle isn’t  used/changed  by
                      function
                    * if False,  symmetric matrix  A  is  given  by  its lower
                      triangle, and the  upper triangle isn’t used/changed  by
                      function
                    * if not given,  both lower and upper  triangles  must  be
                      filled.

    Output parameters:
        Info    -   return code, same as in RMatrixLUInverse
        Rep     -   solver report, same as in RMatrixLUInverse
        A       -   inverse of matrix A, same as in RMatrixLUInverse

      -- ALGLIB routine --
         10.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void spdmatrixinverse(
        ref double[,] a,
        out int info,
        out matinvreport rep)
    public static void spdmatrixinverse(
        ref double[,] a,
        int n,
        bool isupper,
        out int info,
        out matinvreport rep)

Examples: [1]

### matinv_d_c1 example

    public static int Main(string[] args)
    {
        alglib.complex[,] a = new alglib.complex[,]{{new alglib.complex(0,1),-1},{new alglib.complex(0,1),1}};
        int info;
        alglib.matinvreport rep;
        alglib.cmatrixinverse(ref a, out info, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 1
        System.Console.WriteLine("{0}", alglib.ap.format(a,4)); // EXPECTED: [[-0.5i,-0.5i],[-0.5,0.5]]
        System.Console.WriteLine("{0:F4", rep.r1); // EXPECTED: 0.5
        System.Console.WriteLine("{0:F4", rep.rinf); // EXPECTED: 0.5
        System.Console.ReadLine();
        return 0;
    }

### matinv_d_hpd1 example

    public static int Main(string[] args)
    {
        alglib.complex[,] a = new alglib.complex[,]{{2,1},{1,2}};
        int info;
        alglib.matinvreport rep;
        alglib.hpdmatrixinverse(ref a, out info, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 1
        System.Console.WriteLine("{0}", alglib.ap.format(a,4)); // EXPECTED: [[0.666666,-0.333333],[-0.333333,0.666666]]
        System.Console.ReadLine();
        return 0;
    }

### matinv_d_r1 example

    public static int Main(string[] args)
    {
        double[,] a = new double[,]{{1,-1},{1,1}};
        int info;
        alglib.matinvreport rep;
        alglib.rmatrixinverse(ref a, out info, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 1
        System.Console.WriteLine("{0}", alglib.ap.format(a,4)); // EXPECTED: [[0.5,0.5],[-0.5,0.5]]
        System.Console.WriteLine("{0:F4", rep.r1); // EXPECTED: 0.5
        System.Console.WriteLine("{0:F4", rep.rinf); // EXPECTED: 0.5
        System.Console.ReadLine();
        return 0;
    }

### matinv_d_spd1 example

    public static int Main(string[] args)
    {
        double[,] a = new double[,]{{2,1},{1,2}};
        int info;
        alglib.matinvreport rep;
        alglib.spdmatrixinverse(ref a, out info, out rep);
        System.Console.WriteLine("{0}", info); // EXPECTED: 1
        System.Console.WriteLine("{0}", alglib.ap.format(a,4)); // EXPECTED: [[0.666666,-0.333333],[-0.333333,0.666666]]
        System.Console.ReadLine();
        return 0;
    }

## `mcpd` subpackage

### Classes

mcpdreport
mcpdstate

### Functions

mcpdaddbc
mcpdaddec
mcpdaddtrack
mcpdcreate
mcpdcreateentry
mcpdcreateentryexit
mcpdcreateexit
mcpdresults
mcpdsetbc
mcpdsetec
mcpdsetlc
mcpdsetpredictionweights
mcpdsetprior
mcpdsettikhonovregularizer
mcpdsolve

### Examples

mcpd_simple1 |  | Simple unconstrained MCPD model (no entry/exit states)
mcpd_simple2 |  | Simple MCPD model (no entry/exit states) with equality constraints

### `mcpdreport` class

    /*************************************************************************
    This structure is a MCPD training report:
        InnerIterationsCount    -   number of inner iterations of the
                                    underlying optimization algorithm
        OuterIterationsCount    -   number of outer iterations of the
                                    underlying optimization algorithm
        NFEV                    -   number of merit function evaluations
        TerminationType         -   termination type
                                    (same as for MinBLEIC optimizer, positive
                                    values denote success, negative ones -
                                    failure)

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public class mcpdreport
    {
        public int                  inneriterationscount;
        public int                  outeriterationscount;
        public int                  nfev;
        public int                  terminationtype;
    }

### `mcpdstate` class

    /*************************************************************************
    This structure is a MCPD (Markov Chains for Population Data) solver.

    You should use ALGLIB functions in order to work with this object.

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public class mcpdstate
    {
    }

### `mcpdaddbc` function

    /*************************************************************************
    This function is used to add bound constraints  on  the  elements  of  the
    transition matrix P.

    MCPD solver has four types of constraints which can be placed on P:
    * user-specified equality constraints (optional)
    * user-specified bound constraints (optional)
    * user-specified general linear constraints (optional)
    * basic constraints (always present):
      * non-negativity: P[i,j]>=0
      * consistency: every column of P sums to 1.0

    Final  constraints  which  are  passed  to  the  underlying  optimizer are
    calculated  as  intersection  of all present constraints. For example, you
    may specify boundary constraint on P[0,0] and equality one:
        0.1<=P[0,0]<=0.9
        P[0,0]=0.5
    Such  combination  of  constraints  will  be  silently  reduced  to  their
    intersection, which is P[0,0]=0.5.

    This  function  can  be  used to ADD bound constraint for one element of P
    without changing constraints for other elements.

    You  can  also  use  MCPDSetBC()  function  which  allows to  place  bound
    constraints  on arbitrary subset of elements of P.   Set of constraints is
    specified  by  BndL/BndU matrices, which may contain arbitrary combination
    of finite numbers or infinities (like -INF<x<=0.5 or 0.1<=x<+INF).

    These functions (MCPDSetBC and MCPDAddBC) interact as follows:
    * there is internal matrix of bound constraints which is stored in the
      MCPD solver
    * MCPDSetBC() replaces this matrix by another one (SET)
    * MCPDAddBC() modifies one element of this matrix and  leaves  other  ones
      unchanged (ADD)
    * thus  MCPDAddBC()  call  preserves  all  modifications  done by previous
      calls,  while  MCPDSetBC()  completely discards all changes  done to the
      equality constraints.

    INPUT PARAMETERS:
        S       -   solver
        I       -   row index of element being constrained
        J       -   column index of element being constrained
        BndL    -   lower bound
        BndU    -   upper bound

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdaddbc(
        mcpdstate s,
        int i,
        int j,
        double bndl,
        double bndu)

### `mcpdaddec` function

    /*************************************************************************
    This function is used to add equality constraints on the elements  of  the
    transition matrix P.

    MCPD solver has four types of constraints which can be placed on P:
    * user-specified equality constraints (optional)
    * user-specified bound constraints (optional)
    * user-specified general linear constraints (optional)
    * basic constraints (always present):
      * non-negativity: P[i,j]>=0
      * consistency: every column of P sums to 1.0

    Final  constraints  which  are  passed  to  the  underlying  optimizer are
    calculated  as  intersection  of all present constraints. For example, you
    may specify boundary constraint on P[0,0] and equality one:
        0.1<=P[0,0]<=0.9
        P[0,0]=0.5
    Such  combination  of  constraints  will  be  silently  reduced  to  their
    intersection, which is P[0,0]=0.5.

    This function can be used to ADD equality constraint for one element of  P
    without changing constraints for other elements.

    You  can  also  use  MCPDSetEC()  function  which  allows  you  to specify
    arbitrary set of equality constraints in one call.

    These functions (MCPDSetEC and MCPDAddEC) interact as follows:
    * there is internal matrix of equality constraints which is stored in the
      MCPD solver
    * MCPDSetEC() replaces this matrix by another one (SET)
    * MCPDAddEC() modifies one element of this matrix and leaves  other  ones
      unchanged (ADD)
    * thus  MCPDAddEC()  call  preserves  all  modifications done by previous
      calls,  while  MCPDSetEC()  completely discards all changes done to the
      equality constraints.

    INPUT PARAMETERS:
        S       -   solver
        I       -   row index of element being constrained
        J       -   column index of element being constrained
        C       -   value (constraint for P[I,J]).  Can  be  either  NAN  (no
                    constraint) or finite value from [0,1].

    NOTES:

    1. infinite values of C  will lead to exception being thrown. Values  less
    than 0.0 or greater than 1.0 will lead to error code being returned  after
    call to MCPDSolve().

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdaddec(mcpdstate s, int i, int j, double c)

Examples: [1]

### `mcpdaddtrack` function

    /*************************************************************************
    This  function  is  used to add a track - sequence of system states at the
    different moments of its evolution.

    You  may  add  one  or several tracks to the MCPD solver. In case you have
    several tracks, they won't overwrite each other. For example,  if you pass
    two tracks, A1-A2-A3 (system at t=A+1, t=A+2 and t=A+3) and B1-B2-B3, then
    solver will try to model transitions from t=A+1 to t=A+2, t=A+2 to  t=A+3,
    t=B+1 to t=B+2, t=B+2 to t=B+3. But it WONT mix these two tracks - i.e. it
    wont try to model transition from t=A+3 to t=B+1.

    INPUT PARAMETERS:
        S       -   solver
        XY      -   track, array[K,N]:
                    * I-th row is a state at t=I
                    * elements of XY must be non-negative (exception will be
                      thrown on negative elements)
        K       -   number of points in a track
                    * if given, only leading K rows of XY are used
                    * if not given, automatically determined from size of XY

    NOTES:

    1. Track may contain either proportional or population data:
       * with proportional data all rows of XY must sum to 1.0, i.e. we have
         proportions instead of absolute population values
       * with population data rows of XY contain population counts and generally
         do not sum to 1.0 (although they still must be non-negative)

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdaddtrack(mcpdstate s, double[,] xy)
    public static void mcpdaddtrack(mcpdstate s, double[,] xy, int k)

Examples: [1] [2]

### `mcpdcreate` function

    /*************************************************************************
    DESCRIPTION:

    This function creates MCPD (Markov Chains for Population Data) solver.

    This  solver  can  be  used  to find transition matrix P for N-dimensional
    prediction  problem  where transition from X[i] to X[i+1] is  modelled  as
        X[i+1] = P*X[i]
    where X[i] and X[i+1] are N-dimensional population vectors (components  of
    each X are non-negative), and P is a N*N transition matrix (elements of  P
    are non-negative, each column sums to 1.0).

    Such models arise when when:
    * there is some population of individuals
    * individuals can have different states
    * individuals can transit from one state to another
    * population size is constant, i.e. there is no new individuals and no one
      leaves population
    * you want to model transitions of individuals from one state into another

    USAGE:

    Here we give very brief outline of the MCPD. We strongly recommend you  to
    read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide
    on data analysis which is available at http://www.alglib.net/dataanalysis/

    1. User initializes algorithm state with MCPDCreate() call

    2. User  adds  one  or  more  tracks -  sequences of states which describe
       evolution of a system being modelled from different starting conditions

    3. User may add optional boundary, equality  and/or  linear constraints on
       the coefficients of P by calling one of the following functions:
       * MCPDSetEC() to set equality constraints
       * MCPDSetBC() to set bound constraints
       * MCPDSetLC() to set linear constraints

    4. Optionally,  user  may  set  custom  weights  for prediction errors (by
       default, algorithm assigns non-equal, automatically chosen weights  for
       errors in the prediction of different components of X). It can be  done
       with a call of MCPDSetPredictionWeights() function.

    5. User calls MCPDSolve() function which takes algorithm  state and
       pointer (delegate, etc.) to callback function which calculates F/G.

    6. User calls MCPDResults() to get solution

    INPUT PARAMETERS:
        N       -   problem dimension, N>=1

    OUTPUT PARAMETERS:
        State   -   structure stores algorithm state

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdcreate(int n, out mcpdstate s)

Examples: [1] [2]

### `mcpdcreateentry` function

    /*************************************************************************
    DESCRIPTION:

    This function is a specialized version of MCPDCreate()  function,  and  we
    recommend  you  to read comments for this function for general information
    about MCPD solver.

    This  function  creates  MCPD (Markov Chains for Population  Data)  solver
    for "Entry-state" model,  i.e. model  where transition from X[i] to X[i+1]
    is modelled as
        X[i+1] = P*X[i]
    where
        X[i] and X[i+1] are N-dimensional state vectors
        P is a N*N transition matrix
    and  one  selected component of X[] is called "entry" state and is treated
    in a special way:
        system state always transits from "entry" state to some another state
        system state can not transit from any state into "entry" state
    Such conditions basically mean that row of P which corresponds to  "entry"
    state is zero.

    Such models arise when:
    * there is some population of individuals
    * individuals can have different states
    * individuals can transit from one state to another
    * population size is NOT constant -  at every moment of time there is some
      (unpredictable) amount of "new" individuals, which can transit into  one
      of the states at the next turn, but still no one leaves population
    * you want to model transitions of individuals from one state into another
    * but you do NOT want to predict amount of "new"  individuals  because  it
      does not depends on individuals already present (hence  system  can  not
      transit INTO entry state - it can only transit FROM it).

    This model is discussed  in  more  details  in  the ALGLIB User Guide (see
    http://www.alglib.net/dataanalysis/ for more data).

    INPUT PARAMETERS:
        N       -   problem dimension, N>=2
        EntryState- index of entry state, in 0..N-1

    OUTPUT PARAMETERS:
        State   -   structure stores algorithm state

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdcreateentry(int n, int entrystate, out mcpdstate s)

### `mcpdcreateentryexit` function

    /*************************************************************************
    DESCRIPTION:

    This function is a specialized version of MCPDCreate()  function,  and  we
    recommend  you  to read comments for this function for general information
    about MCPD solver.

    This  function  creates  MCPD (Markov Chains for Population  Data)  solver
    for "Entry-Exit-states" model, i.e. model where  transition  from  X[i] to
    X[i+1] is modelled as
        X[i+1] = P*X[i]
    where
        X[i] and X[i+1] are N-dimensional state vectors
        P is a N*N transition matrix
    one selected component of X[] is called "entry" state and is treated in  a
    special way:
        system state always transits from "entry" state to some another state
        system state can not transit from any state into "entry" state
    and another one component of X[] is called "exit" state and is treated  in
    a special way too:
        system state can transit from any state into "exit" state
        system state can not transit from "exit" state into any other state
        transition operator discards "exit" state (makes it zero at each turn)
    Such conditions basically mean that:
        row of P which corresponds to "entry" state is zero
        column of P which corresponds to "exit" state is zero
    Multiplication by such P may decrease sum of vector components.

    Such models arise when:
    * there is some population of individuals
    * individuals can have different states
    * individuals can transit from one state to another
    * population size is NOT constant
    * at every moment of time there is some (unpredictable)  amount  of  "new"
      individuals, which can transit into one of the states at the next turn
    * some  individuals  can  move  (predictably)  into "exit" state and leave
      population at the next turn
    * you want to model transitions of individuals from one state into another,
      including transitions from the "entry" state and into the "exit" state.
    * but you do NOT want to predict amount of "new"  individuals  because  it
      does not depends on individuals already present (hence  system  can  not
      transit INTO entry state - it can only transit FROM it).

    This model is discussed  in  more  details  in  the ALGLIB User Guide (see
    http://www.alglib.net/dataanalysis/ for more data).

    INPUT PARAMETERS:
        N       -   problem dimension, N>=2
        EntryState- index of entry state, in 0..N-1
        ExitState-  index of exit state, in 0..N-1

    OUTPUT PARAMETERS:
        State   -   structure stores algorithm state

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdcreateentryexit(
        int n,
        int entrystate,
        int exitstate,
        out mcpdstate s)

### `mcpdcreateexit` function

    /*************************************************************************
    DESCRIPTION:

    This function is a specialized version of MCPDCreate()  function,  and  we
    recommend  you  to read comments for this function for general information
    about MCPD solver.

    This  function  creates  MCPD (Markov Chains for Population  Data)  solver
    for "Exit-state" model,  i.e. model  where  transition from X[i] to X[i+1]
    is modelled as
        X[i+1] = P*X[i]
    where
        X[i] and X[i+1] are N-dimensional state vectors
        P is a N*N transition matrix
    and  one  selected component of X[] is called "exit"  state and is treated
    in a special way:
        system state can transit from any state into "exit" state
        system state can not transit from "exit" state into any other state
        transition operator discards "exit" state (makes it zero at each turn)
    Such  conditions  basically  mean  that  column  of P which corresponds to
    "exit" state is zero. Multiplication by such P may decrease sum of  vector
    components.

    Such models arise when:
    * there is some population of individuals
    * individuals can have different states
    * individuals can transit from one state to another
    * population size is NOT constant - individuals can move into "exit" state
      and leave population at the next turn, but there are no new individuals
    * amount of individuals which leave population can be predicted
    * you want to model transitions of individuals from one state into another
      (including transitions into the "exit" state)

    This model is discussed  in  more  details  in  the ALGLIB User Guide (see
    http://www.alglib.net/dataanalysis/ for more data).

    INPUT PARAMETERS:
        N       -   problem dimension, N>=2
        ExitState-  index of exit state, in 0..N-1

    OUTPUT PARAMETERS:
        State   -   structure stores algorithm state

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdcreateexit(int n, int exitstate, out mcpdstate s)

### `mcpdresults` function

    /*************************************************************************
    MCPD results

    INPUT PARAMETERS:
        State   -   algorithm state

    OUTPUT PARAMETERS:
        P       -   array[N,N], transition matrix
        Rep     -   optimization report. You should check Rep.TerminationType
                    in  order  to  distinguish  successful  termination  from
                    unsuccessful one. Speaking short, positive values  denote
                    success, negative ones are failures.
                    More information about fields of this  structure  can  be
                    found in the comments on MCPDReport datatype.

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdresults(
        mcpdstate s,
        out double[,] p,
        out mcpdreport rep)

Examples: [1] [2]

### `mcpdsetbc` function

    /*************************************************************************
    This function is used to add bound constraints  on  the  elements  of  the
    transition matrix P.

    MCPD solver has four types of constraints which can be placed on P:
    * user-specified equality constraints (optional)
    * user-specified bound constraints (optional)
    * user-specified general linear constraints (optional)
    * basic constraints (always present):
      * non-negativity: P[i,j]>=0
      * consistency: every column of P sums to 1.0

    Final  constraints  which  are  passed  to  the  underlying  optimizer are
    calculated  as  intersection  of all present constraints. For example, you
    may specify boundary constraint on P[0,0] and equality one:
        0.1<=P[0,0]<=0.9
        P[0,0]=0.5
    Such  combination  of  constraints  will  be  silently  reduced  to  their
    intersection, which is P[0,0]=0.5.

    This  function  can  be  used  to  place bound   constraints  on arbitrary
    subset  of  elements  of  P.  Set of constraints is specified by BndL/BndU
    matrices, which may contain arbitrary combination  of  finite  numbers  or
    infinities (like -INF<x<=0.5 or 0.1<=x<+INF).

    You can also use MCPDAddBC() function which allows to ADD bound constraint
    for one element of P without changing constraints for other elements.

    These functions (MCPDSetBC and MCPDAddBC) interact as follows:
    * there is internal matrix of bound constraints which is stored in the
      MCPD solver
    * MCPDSetBC() replaces this matrix by another one (SET)
    * MCPDAddBC() modifies one element of this matrix and  leaves  other  ones
      unchanged (ADD)
    * thus  MCPDAddBC()  call  preserves  all  modifications  done by previous
      calls,  while  MCPDSetBC()  completely discards all changes  done to the
      equality constraints.

    INPUT PARAMETERS:
        S       -   solver
        BndL    -   lower bounds constraints, array[N,N]. Elements of BndL can
                    be finite numbers or -INF.
        BndU    -   upper bounds constraints, array[N,N]. Elements of BndU can
                    be finite numbers or +INF.

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdsetbc(mcpdstate s, double[,] bndl, double[,] bndu)

### `mcpdsetec` function

    /*************************************************************************
    This function is used to add equality constraints on the elements  of  the
    transition matrix P.

    MCPD solver has four types of constraints which can be placed on P:
    * user-specified equality constraints (optional)
    * user-specified bound constraints (optional)
    * user-specified general linear constraints (optional)
    * basic constraints (always present):
      * non-negativity: P[i,j]>=0
      * consistency: every column of P sums to 1.0

    Final  constraints  which  are  passed  to  the  underlying  optimizer are
    calculated  as  intersection  of all present constraints. For example, you
    may specify boundary constraint on P[0,0] and equality one:
        0.1<=P[0,0]<=0.9
        P[0,0]=0.5
    Such  combination  of  constraints  will  be  silently  reduced  to  their
    intersection, which is P[0,0]=0.5.

    This  function  can  be  used  to  place equality constraints on arbitrary
    subset of elements of P. Set of constraints is specified by EC, which  may
    contain either NAN's or finite numbers from [0,1]. NAN denotes absence  of
    constraint, finite number denotes equality constraint on specific  element
    of P.

    You can also  use  MCPDAddEC()  function  which  allows  to  ADD  equality
    constraint  for  one  element  of P without changing constraints for other
    elements.

    These functions (MCPDSetEC and MCPDAddEC) interact as follows:
    * there is internal matrix of equality constraints which is stored in  the
      MCPD solver
    * MCPDSetEC() replaces this matrix by another one (SET)
    * MCPDAddEC() modifies one element of this matrix and  leaves  other  ones
      unchanged (ADD)
    * thus  MCPDAddEC()  call  preserves  all  modifications  done by previous
      calls,  while  MCPDSetEC()  completely discards all changes  done to the
      equality constraints.

    INPUT PARAMETERS:
        S       -   solver
        EC      -   equality constraints, array[N,N]. Elements of  EC  can  be
                    either NAN's or finite  numbers from  [0,1].  NAN  denotes
                    absence  of  constraints,  while  finite  value    denotes
                    equality constraint on the corresponding element of P.

    NOTES:

    1. infinite values of EC will lead to exception being thrown. Values  less
    than 0.0 or greater than 1.0 will lead to error code being returned  after
    call to MCPDSolve().

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdsetec(mcpdstate s, double[,] ec)

### `mcpdsetlc` function

    /*************************************************************************
    This function is used to set linear equality/inequality constraints on the
    elements of the transition matrix P.

    This function can be used to set one or several general linear constraints
    on the elements of P. Two types of constraints are supported:
    * equality constraints
    * inequality constraints (both less-or-equal and greater-or-equal)

    Coefficients  of  constraints  are  specified  by  matrix  C (one  of  the
    parameters).  One  row  of  C  corresponds  to  one  constraint.   Because
    transition  matrix P has N*N elements,  we  need  N*N columns to store all
    coefficients  (they  are  stored row by row), and one more column to store
    right part - hence C has N*N+1 columns.  Constraint  kind is stored in the
    CT array.

    Thus, I-th linear constraint is
        P[0,0]*C[I,0] + P[0,1]*C[I,1] + .. + P[0,N-1]*C[I,N-1] +
            + P[1,0]*C[I,N] + P[1,1]*C[I,N+1] + ... +
            + P[N-1,N-1]*C[I,N*N-1]  ?=?  C[I,N*N]
    where ?=? can be either "=" (CT[i]=0), "<=" (CT[i]<0) or ">=" (CT[i]>0).

    Your constraint may involve only some subset of P (less than N*N elements).
    For example it can be something like
        P[0,0] + P[0,1] = 0.5
    In this case you still should pass matrix  with N*N+1 columns, but all its
    elements (except for C[0,0], C[0,1] and C[0,N*N-1]) will be zero.

    INPUT PARAMETERS:
        S       -   solver
        C       -   array[K,N*N+1] - coefficients of constraints
                    (see above for complete description)
        CT      -   array[K] - constraint types
                    (see above for complete description)
        K       -   number of equality/inequality constraints, K>=0:
                    * if given, only leading K elements of C/CT are used
                    * if not given, automatically determined from sizes of C/CT

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdsetlc(mcpdstate s, double[,] c, int[] ct)
    public static void mcpdsetlc(mcpdstate s, double[,] c, int[] ct, int k)

### `mcpdsetpredictionweights` function

    /*************************************************************************
    This function is used to change prediction weights

    MCPD solver scales prediction errors as follows
        Error(P) = ||W*(y-P*x)||^2
    where
        x is a system state at time t
        y is a system state at time t+1
        P is a transition matrix
        W is a diagonal scaling matrix

    By default, weights are chosen in order  to  minimize  relative prediction
    error instead of absolute one. For example, if one component of  state  is
    about 0.5 in magnitude and another one is about 0.05, then algorithm  will
    make corresponding weights equal to 2.0 and 20.0.

    INPUT PARAMETERS:
        S       -   solver
        PW      -   array[N], weights:
                    * must be non-negative values (exception will be thrown otherwise)
                    * zero values will be replaced by automatically chosen values

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdsetpredictionweights(mcpdstate s, double[] pw)

### `mcpdsetprior` function

    /*************************************************************************
    This  function  allows to set prior values used for regularization of your
    problem.

    By default, regularizing term is equal to r*||P-prior_P||^2, where r is  a
    small non-zero value,  P is transition matrix, prior_P is identity matrix,
    ||X||^2 is a sum of squared elements of X.

    This  function  allows  you to change prior values prior_P. You  can  also
    change r with MCPDSetTikhonovRegularizer() function.

    INPUT PARAMETERS:
        S       -   solver
        PP      -   array[N,N], matrix of prior values:
                    1. elements must be real numbers from [0,1]
                    2. columns must sum to 1.0.
                    First property is checked (exception is thrown otherwise),
                    while second one is not checked/enforced.

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdsetprior(mcpdstate s, double[,] pp)

### `mcpdsettikhonovregularizer` function

    /*************************************************************************
    This function allows to  tune  amount  of  Tikhonov  regularization  being
    applied to your problem.

    By default, regularizing term is equal to r*||P-prior_P||^2, where r is  a
    small non-zero value,  P is transition matrix, prior_P is identity matrix,
    ||X||^2 is a sum of squared elements of X.

    This  function  allows  you to change coefficient r. You can  also  change
    prior values with MCPDSetPrior() function.

    INPUT PARAMETERS:
        S       -   solver
        V       -   regularization  coefficient, finite non-negative value. It
                    is  not  recommended  to specify zero value unless you are
                    pretty sure that you want it.

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdsettikhonovregularizer(mcpdstate s, double v)

### `mcpdsolve` function

    /*************************************************************************
    This function is used to start solution of the MCPD problem.

    After return from this function, you can use MCPDResults() to get solution
    and completion code.

      -- ALGLIB --
         Copyright 23.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mcpdsolve(mcpdstate s)

Examples: [1] [2]

### mcpd_simple1 example

    public static int Main(string[] args)
    {
        //
        // The very simple MCPD example
        //
        // We have a loan portfolio. Our loans can be in one of two states:
        // * normal loans ("good" ones)
        // * past due loans ("bad" ones)
        //
        // We assume that:
        // * loans can transition from any state to any other state. In
        //   particular, past due loan can become "good" one at any moment
        //   with same (fixed) probability. Not realistic, but it is toy example :)
        // * portfolio size does not change over time
        //
        // Thus, we have following model
        //     state_new = P*state_old
        // where
        //         ( p00  p01 )
        //     P = (          )
        //         ( p10  p11 )
        //
        // We want to model transitions between these two states using MCPD
        // approach (Markov Chains for Proportional/Population Data), i.e.
        // to restore hidden transition matrix P using actual portfolio data.
        // We have:
        // * poportional data, i.e. proportion of loans in the normal and past
        //   due states (not portfolio size measured in some currency, although
        //   it is possible to work with population data too)
        // * two tracks, i.e. two sequences which describe portfolio
        //   evolution from two different starting states: [1,0] (all loans
        //   are "good") and [0.8,0.2] (only 80% of portfolio is in the "good"
        //   state)
        //
        alglib.mcpdstate s;
        alglib.mcpdreport rep;
        double[,] p;
        double[,] track0 = new double[,]{{1.00000,0.00000},{0.95000,0.05000},{0.92750,0.07250},{0.91738,0.08263},{0.91282,0.08718}};
        double[,] track1 = new double[,]{{0.80000,0.20000},{0.86000,0.14000},{0.88700,0.11300},{0.89915,0.10085}};

        alglib.mcpdcreate(2, out s);
        alglib.mcpdaddtrack(s, track0);
        alglib.mcpdaddtrack(s, track1);
        alglib.mcpdsolve(s);
        alglib.mcpdresults(s, out p, out rep);

        //
        // Hidden matrix P is equal to
        //         ( 0.95  0.50 )
        //         (            )
        //         ( 0.05  0.50 )
        // which means that "good" loans can become "bad" with 5% probability,
        // while "bad" loans will return to good state with 50% probability.
        //
        System.Console.WriteLine("{0}", alglib.ap.format(p,2)); // EXPECTED: [[0.95,0.50],[0.05,0.50]]
        System.Console.ReadLine();
        return 0;
    }

### mcpd_simple2 example

    public static int Main(string[] args)
    {
        //
        // Simple MCPD example
        //
        // We have a loan portfolio. Our loans can be in one of three states:
        // * normal loans
        // * past due loans
        // * charged off loans
        //
        // We assume that:
        // * normal loan can stay normal or become past due (but not charged off)
        // * past due loan can stay past due, become normal or charged off
        // * charged off loan will stay charged off for the rest of eternity
        // * portfolio size does not change over time
        // Not realistic, but it is toy example :)
        //
        // Thus, we have following model
        //     state_new = P*state_old
        // where
        //         ( p00  p01    )
        //     P = ( p10  p11    )
        //         (      p21  1 )
        // i.e. four elements of P are known a priori.
        //
        // Although it is possible (given enough data) to In order to enforce
        // this property we set equality constraints on these elements.
        //
        // We want to model transitions between these two states using MCPD
        // approach (Markov Chains for Proportional/Population Data), i.e.
        // to restore hidden transition matrix P using actual portfolio data.
        // We have:
        // * poportional data, i.e. proportion of loans in the current and past
        //   due states (not portfolio size measured in some currency, although
        //   it is possible to work with population data too)
        // * two tracks, i.e. two sequences which describe portfolio
        //   evolution from two different starting states: [1,0,0] (all loans
        //   are "good") and [0.8,0.2,0.0] (only 80% of portfolio is in the "good"
        //   state)
        //
        alglib.mcpdstate s;
        alglib.mcpdreport rep;
        double[,] p;
        double[,] track0 = new double[,]{{1.000000,0.000000,0.000000},{0.950000,0.050000,0.000000},{0.927500,0.060000,0.012500},{0.911125,0.061375,0.027500},{0.896256,0.060900,0.042844}};
        double[,] track1 = new double[,]{{0.800000,0.200000,0.000000},{0.860000,0.090000,0.050000},{0.862000,0.065500,0.072500},{0.851650,0.059475,0.088875},{0.838805,0.057451,0.103744}};

        alglib.mcpdcreate(3, out s);
        alglib.mcpdaddtrack(s, track0);
        alglib.mcpdaddtrack(s, track1);
        alglib.mcpdaddec(s, 0, 2, 0.0);
        alglib.mcpdaddec(s, 1, 2, 0.0);
        alglib.mcpdaddec(s, 2, 2, 1.0);
        alglib.mcpdaddec(s, 2, 0, 0.0);
        alglib.mcpdsolve(s);
        alglib.mcpdresults(s, out p, out rep);

        //
        // Hidden matrix P is equal to
        //         ( 0.95 0.50      )
        //         ( 0.05 0.25      )
        //         (      0.25 1.00 )
        // which means that "good" loans can become past due with 5% probability,
        // while past due loans will become charged off with 25% probability or
        // return back to normal state with 50% probability.
        //
        System.Console.WriteLine("{0}", alglib.ap.format(p,2)); // EXPECTED: [[0.95,0.50,0.00],[0.05,0.25,0.00],[0.00,0.25,1.00]]
        System.Console.ReadLine();
        return 0;
    }

## `minbleic` subpackage

### Classes

minbleicreport
minbleicstate

### Functions

minbleiccreate
minbleiccreatef
minbleicoptimize
minbleicrestartfrom
minbleicresults
minbleicresultsbuf
minbleicsetbc
minbleicsetinnercond
minbleicsetlc
minbleicsetmaxits
minbleicsetoutercond
minbleicsetprecdefault
minbleicsetprecdiag
minbleicsetprecscale
minbleicsetscale
minbleicsetstpmax
minbleicsetxrep

### Examples

minbleic_d_1 |  | Nonlinear optimization with bound constraints
minbleic_d_2 |  | Nonlinear optimization with linear inequality constraints
minbleic_ftrim |  | Nonlinear optimization by BLEIC, function with singularities
minbleic_numdiff |  | Nonlinear optimization with bound constraints and numerical differentiation

### `minbleicreport` class

    /*************************************************************************
    This structure stores optimization report:
    * InnerIterationsCount      number of inner iterations
    * OuterIterationsCount      number of outer iterations
    * NFEV                      number of gradient evaluations
    * TerminationType           termination type (see below)

    TERMINATION CODES

    TerminationType field contains completion code, which can be:
      -10   unsupported combination of algorithm settings:
            1) StpMax is set to non-zero value,
            AND 2) non-default preconditioner is used.
            You can't use both features at the same moment,
            so you have to choose one of them (and to turn
            off another one).
      -3    inconsistent constraints. Feasible point is
            either nonexistent or too hard to find. Try to
            restart optimizer with better initial
            approximation
       4    conditions on constraints are fulfilled
            with error less than or equal to EpsC
       5    MaxIts steps was taken
       7    stopping conditions are too stringent,
            further improvement is impossible,
            X contains best point found so far.

    ADDITIONAL FIELDS

    There are additional fields which can be used for debugging:
    * DebugEqErr                error in the equality constraints (2-norm)
    * DebugFS                   f, calculated at projection of initial point
                                to the feasible set
    * DebugFF                   f, calculated at the final point
    * DebugDX                   |X_start-X_final|
    *************************************************************************/
    public class minbleicreport
    {
        public int                  inneriterationscount;
        public int                  outeriterationscount;
        public int                  nfev;
        public int                  terminationtype;
        public double               debugeqerr;
        public double               debugfs;
        public double               debugff;
        public double               debugdx;
    }

### `minbleicstate` class

    /*************************************************************************
    This object stores nonlinear optimizer state.
    You should use functions provided by MinBLEIC subpackage to work with this
    object
    *************************************************************************/
    public class minbleicstate
    {
    }

### `minbleiccreate` function

    /*************************************************************************
                         BOUND CONSTRAINED OPTIMIZATION
           WITH ADDITIONAL LINEAR EQUALITY AND INEQUALITY CONSTRAINTS

    DESCRIPTION:
    The  subroutine  minimizes  function   F(x)  of N arguments subject to any
    combination of:
    * bound constraints
    * linear inequality constraints
    * linear equality constraints

    REQUIREMENTS:
    * user must provide function value and gradient
    * starting point X0 must be feasible or
      not too far away from the feasible set
    * grad(f) must be Lipschitz continuous on a level set:
      L = { x : f(x)<=f(x0) }
    * function must be defined everywhere on the feasible set F

    USAGE:

    Constrained optimization if far more complex than the unconstrained one.
    Here we give very brief outline of the BLEIC optimizer. We strongly recommend
    you to read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide
    on optimization, which is available at http://www.alglib.net/optimization/

    1. User initializes algorithm state with MinBLEICCreate() call

    2. USer adds boundary and/or linear constraints by calling
       MinBLEICSetBC() and MinBLEICSetLC() functions.

    3. User sets stopping conditions for underlying unconstrained solver
       with MinBLEICSetInnerCond() call.
       This function controls accuracy of underlying optimization algorithm.

    4. User sets stopping conditions for outer iteration by calling
       MinBLEICSetOuterCond() function.
       This function controls handling of boundary and inequality constraints.

    5. Additionally, user may set limit on number of internal iterations
       by MinBLEICSetMaxIts() call.
       This function allows to prevent algorithm from looping forever.

    6. User calls MinBLEICOptimize() function which takes algorithm  state and
       pointer (delegate, etc.) to callback function which calculates F/G.

    7. User calls MinBLEICResults() to get solution

    8. Optionally user may call MinBLEICRestartFrom() to solve another problem
       with same N but another starting point.
       MinBLEICRestartFrom() allows to reuse already initialized structure.

    INPUT PARAMETERS:
        N       -   problem dimension, N>0:
                    * if given, only leading N elements of X are used
                    * if not given, automatically determined from size ofX
        X       -   starting point, array[N]:
                    * it is better to set X to a feasible point
                    * but X can be infeasible, in which case algorithm will try
                      to find feasible point first, using X as initial
                      approximation.

    OUTPUT PARAMETERS:
        State   -   structure stores algorithm state

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleiccreate(double[] x, out minbleicstate state)
    public static void minbleiccreate(
        int n,
        double[] x,
        out minbleicstate state)

Examples: [1] [2] [3]

### `minbleiccreatef` function

    /*************************************************************************
    The subroutine is finite difference variant of MinBLEICCreate().  It  uses
    finite differences in order to differentiate target function.

    Description below contains information which is specific to  this function
    only. We recommend to read comments on MinBLEICCreate() in  order  to  get
    more information about creation of BLEIC optimizer.

    INPUT PARAMETERS:
        N       -   problem dimension, N>0:
                    * if given, only leading N elements of X are used
                    * if not given, automatically determined from size of X
        X       -   starting point, array[0..N-1].
        DiffStep-   differentiation step, >0

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

    NOTES:
    1. algorithm uses 4-point central formula for differentiation.
    2. differentiation step along I-th axis is equal to DiffStep*S[I] where
       S[] is scaling vector which can be set by MinBLEICSetScale() call.
    3. we recommend you to use moderate values of  differentiation  step.  Too
       large step will result in too large truncation  errors, while too small
       step will result in too large numerical  errors.  1.0E-6  can  be  good
       value to start with.
    4. Numerical  differentiation  is   very   inefficient  -   one   gradient
       calculation needs 4*N function evaluations. This function will work for
       any N - either small (1...10), moderate (10...100) or  large  (100...).
       However, performance penalty will be too severe for any N's except  for
       small ones.
       We should also say that code which relies on numerical  differentiation
       is  less  robust and precise. CG needs exact gradient values. Imprecise
       gradient may slow  down  convergence, especially  on  highly  nonlinear
       problems.
       Thus  we  recommend to use this function for fast prototyping on small-
       dimensional problems only, and to implement analytical gradient as soon
       as possible.

      -- ALGLIB --
         Copyright 16.05.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleiccreatef(
        double[] x,
        double diffstep,
        out minbleicstate state)
    public static void minbleiccreatef(
        int n,
        double[] x,
        double diffstep,
        out minbleicstate state)

Examples: [1]

### `minbleicoptimize` function

    /*************************************************************************
    This family of functions is used to launcn iterations of nonlinear optimizer

    These functions accept following parameters:
        state   -   algorithm state
        func    -   callback which calculates function (or merit function)
                    value func at given point x
        grad    -   callback which calculates function (or merit function)
                    value func and gradient grad at given point x
        rep     -   optional callback which is called after each iteration
                    can be NULL
        ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                    can be NULL

    NOTES:

    1. This function has two different implementations: one which  uses  exact
       (analytical) user-supplied gradient,  and one which uses function value
       only  and  numerically  differentiates  function  in  order  to  obtain
       gradient.

       Depending  on  the  specific  function  used to create optimizer object
       (either  MinBLEICCreate() for analytical gradient or  MinBLEICCreateF()
       for numerical differentiation) you should choose appropriate variant of
       MinBLEICOptimize() - one  which  accepts  function  AND gradient or one
       which accepts function ONLY.

       Be careful to choose variant of MinBLEICOptimize() which corresponds to
       your optimization scheme! Table below lists different  combinations  of
       callback (function/gradient) passed to MinBLEICOptimize()  and specific
       function used to create optimizer.

                         |         USER PASSED TO MinBLEICOptimize()
       CREATED WITH      |  function only   |  function and gradient
       ------------------------------------------------------------
       MinBLEICCreateF() |     work                FAIL
       MinBLEICCreate()  |     FAIL                work

       Here "FAIL" denotes inappropriate combinations  of  optimizer  creation
       function  and  MinBLEICOptimize()  version.   Attemps   to   use   such
       combination (for  example,  to  create optimizer with MinBLEICCreateF()
       and  to  pass  gradient  information  to  MinCGOptimize()) will lead to
       exception being thrown. Either  you  did  not pass gradient when it WAS
       needed or you passed gradient when it was NOT needed.

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicoptimize(minbleicstate state, ndimensional_func func, ndimensional_rep rep, object obj)
    public static void minbleicoptimize(minbleicstate state, ndimensional_grad grad, ndimensional_rep rep, object obj)

Examples: [1] [2] [3] [4]

### `minbleicrestartfrom` function

    /*************************************************************************
    This subroutine restarts algorithm from new point.
    All optimization parameters (including constraints) are left unchanged.

    This  function  allows  to  solve multiple  optimization  problems  (which
    must have  same number of dimensions) without object reallocation penalty.

    INPUT PARAMETERS:
        State   -   structure previously allocated with MinBLEICCreate call.
        X       -   new starting point.

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicrestartfrom(minbleicstate state, double[] x)

### `minbleicresults` function

    /*************************************************************************
    BLEIC results

    INPUT PARAMETERS:
        State   -   algorithm state

    OUTPUT PARAMETERS:
        X       -   array[0..N-1], solution
        Rep     -   optimization report. You should check Rep.TerminationType
                    in  order  to  distinguish  successful  termination  from
                    unsuccessful one.
                    More information about fields of this  structure  can  be
                    found in the comments on MinBLEICReport datatype.

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicresults(
        minbleicstate state,
        out double[] x,
        out minbleicreport rep)

Examples: [1] [2] [3] [4]

### `minbleicresultsbuf` function

    /*************************************************************************
    BLEIC results

    Buffered implementation of MinBLEICResults() which uses pre-allocated buffer
    to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
    intended to be used in the inner cycles of performance critical algorithms
    where array reallocation penalty is too large to be ignored.

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicresultsbuf(
        minbleicstate state,
        ref double[] x,
        minbleicreport rep)

### `minbleicsetbc` function

    /*************************************************************************
    This function sets boundary constraints for BLEIC optimizer.

    Boundary constraints are inactive by default (after initial creation).
    They are preserved after algorithm restart with MinBLEICRestartFrom().

    INPUT PARAMETERS:
        State   -   structure stores algorithm state
        BndL    -   lower bounds, array[N].
                    If some (all) variables are unbounded, you may specify
                    very small number or -INF.
        BndU    -   upper bounds, array[N].
                    If some (all) variables are unbounded, you may specify
                    very large number or +INF.

    NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
    variable will be "frozen" at X[i]=BndL[i]=BndU[i].

    NOTE 2: this solver has following useful properties:
    * bound constraints are always satisfied exactly
    * function is evaluated only INSIDE area specified by  bound  constraints,
      even  when  numerical  differentiation is used (algorithm adjusts  nodes
      according to boundary constraints)

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetbc(
        minbleicstate state,
        double[] bndl,
        double[] bndu)

Examples: [1] [2]

### `minbleicsetinnercond` function

    /*************************************************************************
    This function sets stopping conditions for the underlying nonlinear CG
    optimizer. It controls overall accuracy of solution. These conditions
    should be strict enough in order for algorithm to converge.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        EpsG    -   >=0
                    The  subroutine  finishes  its  work   if   the  condition
                    |v|<EpsG is satisfied, where:
                    * |.| means Euclidian norm
                    * v - scaled gradient vector, v[i]=g[i]*s[i]
                    * g - gradient
                    * s - scaling coefficients set by MinBLEICSetScale()
        EpsF    -   >=0
                    The  subroutine  finishes  its work if on k+1-th iteration
                    the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                    is satisfied.
        EpsX    -   >=0
                    The subroutine finishes its work if  on  k+1-th  iteration
                    the condition |v|<=EpsX is fulfilled, where:
                    * |.| means Euclidian norm
                    * v - scaled step vector, v[i]=dx[i]/s[i]
                    * dx - ste pvector, dx=X(k+1)-X(k)
                    * s - scaling coefficients set by MinBLEICSetScale()

    Passing EpsG=0, EpsF=0 and EpsX=0 (simultaneously) will lead to
    automatic stopping criterion selection.

    These conditions are used to terminate inner iterations. However, you
    need to tune termination conditions for outer iterations too.

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetinnercond(
        minbleicstate state,
        double epsg,
        double epsf,
        double epsx)

Examples: [1] [2] [3]

### `minbleicsetlc` function

    /*************************************************************************
    This function sets linear constraints for BLEIC optimizer.

    Linear constraints are inactive by default (after initial creation).
    They are preserved after algorithm restart with MinBLEICRestartFrom().

    INPUT PARAMETERS:
        State   -   structure previously allocated with MinBLEICCreate call.
        C       -   linear constraints, array[K,N+1].
                    Each row of C represents one constraint, either equality
                    or inequality (see below):
                    * first N elements correspond to coefficients,
                    * last element corresponds to the right part.
                    All elements of C (including right part) must be finite.
        CT      -   type of constraints, array[K]:
                    * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                    * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                    * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
        K       -   number of equality/inequality constraints, K>=0:
                    * if given, only leading K elements of C/CT are used
                    * if not given, automatically determined from sizes of C/CT

    NOTE 1: linear (non-bound) constraints are satisfied only approximately:
    * there always exists some minor violation (about Epsilon in magnitude)
      due to rounding errors
    * numerical differentiation, if used, may  lead  to  function  evaluations
      outside  of the feasible  area,   because   algorithm  does  NOT  change
      numerical differentiation formula according to linear constraints.
    If you want constraints to be  satisfied  exactly, try to reformulate your
    problem  in  such  manner  that  all constraints will become boundary ones
    (this kind of constraints is always satisfied exactly, both in  the  final
    solution and in all intermediate points).

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetlc(
        minbleicstate state,
        double[,] c,
        int[] ct)
    public static void minbleicsetlc(
        minbleicstate state,
        double[,] c,
        int[] ct,
        int k)

Examples: [1]

### `minbleicsetmaxits` function

    /*************************************************************************
    This function allows to stop algorithm after specified number of inner
    iterations.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        MaxIts  -   maximum number of inner iterations.
                    If MaxIts=0, the number of iterations is unlimited.

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetmaxits(minbleicstate state, int maxits)

### `minbleicsetoutercond` function

    /*************************************************************************
    This function sets stopping conditions for outer iteration of BLEIC algo.

    These conditions control accuracy of constraint handling and amount of
    infeasibility allowed in the solution.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        EpsX    -   >0, stopping condition on outer iteration step length
        EpsI    -   >0, stopping condition on infeasibility

    Both EpsX and EpsI must be non-zero.

    MEANING OF EpsX

    EpsX  is  a  stopping  condition for outer iterations. Algorithm will stop
    when  solution  of  the  current  modified  subproblem will be within EpsX
    (using 2-norm) of the previous solution.

    MEANING OF EpsI

    EpsI controls feasibility properties -  algorithm  won't  stop  until  all
    inequality constraints will be satisfied with error (distance from current
    point to the feasible area) at most EpsI.

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetoutercond(
        minbleicstate state,
        double epsx,
        double epsi)

Examples: [1] [2] [3]

### `minbleicsetprecdefault` function

    /*************************************************************************
    Modification of the preconditioner: preconditioning is turned off.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetprecdefault(minbleicstate state)

### `minbleicsetprecdiag` function

    /*************************************************************************
    Modification  of  the  preconditioner:  diagonal of approximate Hessian is
    used.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        D       -   diagonal of the approximate Hessian, array[0..N-1],
                    (if larger, only leading N elements are used).

    NOTE 1: D[i] should be positive. Exception will be thrown otherwise.

    NOTE 2: you should pass diagonal of approximate Hessian - NOT ITS INVERSE.

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetprecdiag(minbleicstate state, double[] d)

### `minbleicsetprecscale` function

    /*************************************************************************
    Modification of the preconditioner: scale-based diagonal preconditioning.

    This preconditioning mode can be useful when you  don't  have  approximate
    diagonal of Hessian, but you know that your  variables  are  badly  scaled
    (for  example,  one  variable is in [1,10], and another in [1000,100000]),
    and most part of the ill-conditioning comes from different scales of vars.

    In this case simple  scale-based  preconditioner,  with H[i] = 1/(s[i]^2),
    can greatly improve convergence.

    IMPRTANT: you should set scale of your variables  with  MinBLEICSetScale()
    call  (before  or after MinBLEICSetPrecScale() call). Without knowledge of
    the scale of your variables scale-based preconditioner will be  just  unit
    matrix.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetprecscale(minbleicstate state)

### `minbleicsetscale` function

    /*************************************************************************
    This function sets scaling coefficients for BLEIC optimizer.

    ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
    size and gradient are scaled before comparison with tolerances).  Scale of
    the I-th variable is a translation invariant measure of:
    a) "how large" the variable is
    b) how large the step should be to make significant changes in the function

    Scaling is also used by finite difference variant of the optimizer  - step
    along I-th axis is equal to DiffStep*S[I].

    In  most  optimizers  (and  in  the  BLEIC  too)  scaling is NOT a form of
    preconditioning. It just  affects  stopping  conditions.  You  should  set
    preconditioner  by  separate  call  to  one  of  the  MinBLEICSetPrec...()
    functions.

    There is a special  preconditioning  mode, however,  which  uses   scaling
    coefficients to form diagonal preconditioning matrix. You  can  turn  this
    mode on, if you want.   But  you should understand that scaling is not the
    same thing as preconditioning - these are two different, although  related
    forms of tuning solver.

    INPUT PARAMETERS:
        State   -   structure stores algorithm state
        S       -   array[N], non-zero scaling coefficients
                    S[i] may be negative, sign doesn't matter.

      -- ALGLIB --
         Copyright 14.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetscale(minbleicstate state, double[] s)

### `minbleicsetstpmax` function

    /*************************************************************************
    This function sets maximum step length

    IMPORTANT: this feature is hard to combine with preconditioning. You can't
    set upper limit on step length, when you solve optimization  problem  with
    linear (non-boundary) constraints AND preconditioner turned on.

    When  non-boundary  constraints  are  present,  you  have to either a) use
    preconditioner, or b) use upper limit on step length.  YOU CAN'T USE BOTH!
    In this case algorithm will terminate with appropriate error code.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                    want to limit step length.

    Use this subroutine when you optimize target function which contains exp()
    or  other  fast  growing  functions,  and optimization algorithm makes too
    large  steps  which  lead   to overflow. This function allows us to reject
    steps  that  are  too  large  (and  therefore  expose  us  to the possible
    overflow) without actually calculating function value at the x+stp*d.

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetstpmax(minbleicstate state, double stpmax)

### `minbleicsetxrep` function

    /*************************************************************************
    This function turns on/off reporting.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        NeedXRep-   whether iteration reports are needed or not

    If NeedXRep is True, algorithm will call rep() callback function if  it is
    provided to MinBLEICOptimize().

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetxrep(minbleicstate state, bool needxrep)

### minbleic_d_1 example

    public static void function1_grad(double[] x, ref double func, double[] grad, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        // and its derivatives df/d0 and df/dx1
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
        grad[0] = 400*System.Math.Pow(x[0]+3,3);
        grad[1] = 4*System.Math.Pow(x[1]-3,3);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
        // subject to bound constraints -1<=x<=+1, -1<=y<=+1, using BLEIC optimizer.
        //
        double[] x = new double[]{0,0};
        double[] bndl = new double[]{-1,-1};
        double[] bndu = new double[]{+1,+1};
        alglib.minbleicstate state;
        alglib.minbleicreport rep;

        //
        // These variables define stopping conditions for the underlying CG algorithm.
        // They should be stringent enough in order to guarantee overall stability
        // of the outer iterations.
        //
        // We use very simple condition - |g|<=epsg
        //
        double epsg = 0.000001;
        double epsf = 0;
        double epsx = 0;

        //
        // These variables define stopping conditions for the outer iterations:
        // * epso controls convergence of outer iterations; algorithm will stop
        //   when difference between solutions of subsequent unconstrained problems
        //   will be less than 0.0001
        // * epsi controls amount of infeasibility allowed in the final solution
        //
        double epso = 0.00001;
        double epsi = 0.00001;

        //
        // Now we are ready to actually optimize something:
        // * first we create optimizer
        // * we add boundary constraints
        // * we tune stopping conditions
        // * and, finally, optimize and obtain results...
        //
        alglib.minbleiccreate(x, out state);
        alglib.minbleicsetbc(state, bndl, bndu);
        alglib.minbleicsetinnercond(state, epsg, epsf, epsx);
        alglib.minbleicsetoutercond(state, epso, epsi);
        alglib.minbleicoptimize(state, function1_grad, null, null);
        alglib.minbleicresults(state, out x, out rep);

        //
        // ...and evaluate these results
        //
        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-1,1]
        System.Console.ReadLine();
        return 0;
    }

### minbleic_d_2 example

    public static void function1_grad(double[] x, ref double func, double[] grad, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        // and its derivatives df/d0 and df/dx1
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
        grad[0] = 400*System.Math.Pow(x[0]+3,3);
        grad[1] = 4*System.Math.Pow(x[1]-3,3);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
        // subject to inequality constraints:
        // * x>=2 (posed as general linear constraint),
        // * x+y>=6
        // using BLEIC optimizer.
        //
        double[] x = new double[]{5,5};
        double[,] c = new double[,]{{1,0,2},{1,1,6}};
        int[] ct = new int[]{1,1};
        alglib.minbleicstate state;
        alglib.minbleicreport rep;

        //
        // These variables define stopping conditions for the underlying CG algorithm.
        // They should be stringent enough in order to guarantee overall stability
        // of the outer iterations.
        //
        // We use very simple condition - |g|<=epsg
        //
        double epsg = 0.000001;
        double epsf = 0;
        double epsx = 0;

        //
        // These variables define stopping conditions for the outer iterations:
        // * epso controls convergence of outer iterations; algorithm will stop
        //   when difference between solutions of subsequent unconstrained problems
        //   will be less than 0.0001
        // * epsi controls amount of infeasibility allowed in the final solution
        //
        double epso = 0.00001;
        double epsi = 0.00001;

        //
        // Now we are ready to actually optimize something:
        // * first we create optimizer
        // * we add linear constraints
        // * we tune stopping conditions
        // * and, finally, optimize and obtain results...
        //
        alglib.minbleiccreate(x, out state);
        alglib.minbleicsetlc(state, c, ct);
        alglib.minbleicsetinnercond(state, epsg, epsf, epsx);
        alglib.minbleicsetoutercond(state, epso, epsi);
        alglib.minbleicoptimize(state, function1_grad, null, null);
        alglib.minbleicresults(state, out x, out rep);

        //
        // ...and evaluate these results
        //
        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [2,4]
        System.Console.ReadLine();
        return 0;
    }

### minbleic_ftrim example

    public static void s1_grad(double[] x, ref double func, double[] grad, object obj)
    {
        //
        // this callback calculates f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x and its gradient.
        //
        // function is trimmed when we calculate it near the singular points or outside of the [-1,+1].
        // Note that we do NOT calculate gradient in this case.
        //
        if( (x[0]<=-0.999999999999) || (x[0]>=+0.999999999999) )
        {
            func = 1.0E+300;
            return;
        }
        func = System.Math.Pow(1+x[0],-0.2) + System.Math.Pow(1-x[0],-0.3) + 1000*x[0];
        grad[0] = -0.2*System.Math.Pow(1+x[0],-1.2) +0.3*System.Math.Pow(1-x[0],-1.3) + 1000;
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x.
        //
        // This function is undefined outside of (-1,+1) and has singularities at x=-1 and x=+1.
        // Special technique called "function trimming" allows us to solve this optimization problem
        // - without using boundary constraints!
        //
        // See http://www.alglib.net/optimization/tipsandtricks.php#ftrimming for more information
        // on this subject.
        //
        double[] x = new double[]{0};
        double epsg = 1.0e-6;
        double epsf = 0;
        double epsx = 0;
        double epso = 1.0e-6;
        double epsi = 1.0e-6;
        alglib.minbleicstate state;
        alglib.minbleicreport rep;

        alglib.minbleiccreate(x, out state);
        alglib.minbleicsetinnercond(state, epsg, epsf, epsx);
        alglib.minbleicsetoutercond(state, epso, epsi);
        alglib.minbleicoptimize(state, s1_grad, null, null);
        alglib.minbleicresults(state, out x, out rep);

        System.Console.WriteLine("{0}", alglib.ap.format(x,5)); // EXPECTED: [-0.99917305]
        System.Console.ReadLine();
        return 0;
    }

### minbleic_numdiff example

    public static void function1_func(double[] x, ref double func, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
        // subject to bound constraints -1<=x<=+1, -1<=y<=+1, using BLEIC optimizer.
        //
        double[] x = new double[]{0,0};
        double[] bndl = new double[]{-1,-1};
        double[] bndu = new double[]{+1,+1};
        alglib.minbleicstate state;
        alglib.minbleicreport rep;

        //
        // These variables define stopping conditions for the underlying CG algorithm.
        // They should be stringent enough in order to guarantee overall stability
        // of the outer iterations.
        //
        // We use very simple condition - |g|<=epsg
        //
        double epsg = 0.000001;
        double epsf = 0;
        double epsx = 0;

        //
        // These variables define stopping conditions for the outer iterations:
        // * epso controls convergence of outer iterations; algorithm will stop
        //   when difference between solutions of subsequent unconstrained problems
        //   will be less than 0.0001
        // * epsi controls amount of infeasibility allowed in the final solution
        //
        double epso = 0.00001;
        double epsi = 0.00001;

        //
        // This variable contains differentiation step
        //
        double diffstep = 1.0e-6;

        //
        // Now we are ready to actually optimize something:
        // * first we create optimizer
        // * we add boundary constraints
        // * we tune stopping conditions
        // * and, finally, optimize and obtain results...
        //
        alglib.minbleiccreatef(x, diffstep, out state);
        alglib.minbleicsetbc(state, bndl, bndu);
        alglib.minbleicsetinnercond(state, epsg, epsf, epsx);
        alglib.minbleicsetoutercond(state, epso, epsi);
        alglib.minbleicoptimize(state, function1_func, null, null);
        alglib.minbleicresults(state, out x, out rep);

        //
        // ...and evaluate these results
        //
        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-1,1]
        System.Console.ReadLine();
        return 0;
    }

## `mincg` subpackage

### Classes

mincgreport
mincgstate

### Functions

mincgcreate
mincgcreatef
mincgoptimize
mincgrestartfrom
mincgresults
mincgresultsbuf
mincgsetcgtype
mincgsetcond
mincgsetprecdefault
mincgsetprecdiag
mincgsetprecscale
mincgsetscale
mincgsetstpmax
mincgsetxrep
mincgsuggeststep

### Examples

mincg_d_1 |  | Nonlinear optimization by CG
mincg_d_2 |  | Nonlinear optimization with additional settings and restarts
mincg_ftrim |  | Nonlinear optimization by CG, function with singularities
mincg_numdiff |  | Nonlinear optimization by CG with numerical differentiation

### `mincgreport` class

    /*************************************************************************

    *************************************************************************/
    public class mincgreport
    {
        public int                  iterationscount;
        public int                  nfev;
        public int                  terminationtype;
    }

### `mincgstate` class

    /*************************************************************************
    This object stores state of the nonlinear CG optimizer.

    You should use ALGLIB functions to work with this object.
    *************************************************************************/
    public class mincgstate
    {
    }

### `mincgcreate` function

    /*************************************************************************
            NONLINEAR CONJUGATE GRADIENT METHOD

    DESCRIPTION:
    The subroutine minimizes function F(x) of N arguments by using one of  the
    nonlinear conjugate gradient methods.

    These CG methods are globally convergent (even on non-convex functions) as
    long as grad(f) is Lipschitz continuous in  a  some  neighborhood  of  the
    L = { x : f(x)<=f(x0) }.

    REQUIREMENTS:
    Algorithm will request following information during its operation:
    * function value F and its gradient G (simultaneously) at given point X

    USAGE:
    1. User initializes algorithm state with MinCGCreate() call
    2. User tunes solver parameters with MinCGSetCond(), MinCGSetStpMax() and
       other functions
    3. User calls MinCGOptimize() function which takes algorithm  state   and
       pointer (delegate, etc.) to callback function which calculates F/G.
    4. User calls MinCGResults() to get solution
    5. Optionally, user may call MinCGRestartFrom() to solve another  problem
       with same N but another starting point and/or another function.
       MinCGRestartFrom() allows to reuse already initialized structure.

    INPUT PARAMETERS:
        N       -   problem dimension, N>0:
                    * if given, only leading N elements of X are used
                    * if not given, automatically determined from size of X
        X       -   starting point, array[0..N-1].

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

      -- ALGLIB --
         Copyright 25.03.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgcreate(double[] x, out mincgstate state)
    public static void mincgcreate(int n, double[] x, out mincgstate state)

Examples: [1] [2] [3]

### `mincgcreatef` function

    /*************************************************************************
    The subroutine is finite difference variant of MinCGCreate(). It uses
    finite differences in order to differentiate target function.

    Description below contains information which is specific to this function
    only. We recommend to read comments on MinCGCreate() in order to get more
    information about creation of CG optimizer.

    INPUT PARAMETERS:
        N       -   problem dimension, N>0:
                    * if given, only leading N elements of X are used
                    * if not given, automatically determined from size of X
        X       -   starting point, array[0..N-1].
        DiffStep-   differentiation step, >0

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

    NOTES:
    1. algorithm uses 4-point central formula for differentiation.
    2. differentiation step along I-th axis is equal to DiffStep*S[I] where
       S[] is scaling vector which can be set by MinCGSetScale() call.
    3. we recommend you to use moderate values of  differentiation  step.  Too
       large step will result in too large truncation  errors, while too small
       step will result in too large numerical  errors.  1.0E-6  can  be  good
       value to start with.
    4. Numerical  differentiation  is   very   inefficient  -   one   gradient
       calculation needs 4*N function evaluations. This function will work for
       any N - either small (1...10), moderate (10...100) or  large  (100...).
       However, performance penalty will be too severe for any N's except  for
       small ones.
       We should also say that code which relies on numerical  differentiation
       is  less  robust  and  precise.  L-BFGS  needs  exact  gradient values.
       Imprecise  gradient may slow down  convergence,  especially  on  highly
       nonlinear problems.
       Thus  we  recommend to use this function for fast prototyping on small-
       dimensional problems only, and to implement analytical gradient as soon
       as possible.

      -- ALGLIB --
         Copyright 16.05.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgcreatef(
        double[] x,
        double diffstep,
        out mincgstate state)
    public static void mincgcreatef(
        int n,
        double[] x,
        double diffstep,
        out mincgstate state)

Examples: [1]

### `mincgoptimize` function

    /*************************************************************************
    This family of functions is used to launcn iterations of nonlinear optimizer

    These functions accept following parameters:
        state   -   algorithm state
        func    -   callback which calculates function (or merit function)
                    value func at given point x
        grad    -   callback which calculates function (or merit function)
                    value func and gradient grad at given point x
        rep     -   optional callback which is called after each iteration
                    can be NULL
        ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                    can be NULL

    NOTES:

    1. This function has two different implementations: one which  uses  exact
       (analytical) user-supplied  gradient, and one which uses function value
       only  and  numerically  differentiates  function  in  order  to  obtain
       gradient.

       Depending  on  the  specific  function  used to create optimizer object
       (either MinCGCreate()  for analytical gradient  or  MinCGCreateF()  for
       numerical differentiation) you should  choose  appropriate  variant  of
       MinCGOptimize() - one which accepts function AND gradient or one  which
       accepts function ONLY.

       Be careful to choose variant of MinCGOptimize()  which  corresponds  to
       your optimization scheme! Table below lists different  combinations  of
       callback (function/gradient) passed  to  MinCGOptimize()  and  specific
       function used to create optimizer.

                      |         USER PASSED TO MinCGOptimize()
       CREATED WITH   |  function only   |  function and gradient
       ------------------------------------------------------------
       MinCGCreateF() |     work                FAIL
       MinCGCreate()  |     FAIL                work

       Here "FAIL" denotes inappropriate combinations  of  optimizer  creation
       function and MinCGOptimize() version. Attemps to use  such  combination
       (for  example,  to create optimizer with  MinCGCreateF()  and  to  pass
       gradient information to MinCGOptimize()) will lead to  exception  being
       thrown. Either  you  did  not  pass  gradient when it WAS needed or you
       passed gradient when it was NOT needed.

      -- ALGLIB --
         Copyright 20.04.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgoptimize(mincgstate state, ndimensional_func func, ndimensional_rep rep, object obj)
    public static void mincgoptimize(mincgstate state, ndimensional_grad grad, ndimensional_rep rep, object obj)

Examples: [1] [2] [3] [4]

### `mincgrestartfrom` function

    /*************************************************************************
    This  subroutine  restarts  CG  algorithm from new point. All optimization
    parameters are left unchanged.

    This  function  allows  to  solve multiple  optimization  problems  (which
    must have same number of dimensions) without object reallocation penalty.

    INPUT PARAMETERS:
        State   -   structure used to store algorithm state.
        X       -   new starting point.

      -- ALGLIB --
         Copyright 30.07.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgrestartfrom(mincgstate state, double[] x)

Examples: [1]

### `mincgresults` function

    /*************************************************************************
    Conjugate gradient results

    INPUT PARAMETERS:
        State   -   algorithm state

    OUTPUT PARAMETERS:
        X       -   array[0..N-1], solution
        Rep     -   optimization report:
                    * Rep.TerminationType completetion code:
                        *  1    relative function improvement is no more than
                                EpsF.
                        *  2    relative step is no more than EpsX.
                        *  4    gradient norm is no more than EpsG
                        *  5    MaxIts steps was taken
                        *  7    stopping conditions are too stringent,
                                further improvement is impossible,
                                we return best X found so far
                        *  8    terminated by user
                    * Rep.IterationsCount contains iterations count
                    * NFEV countains number of function calculations

      -- ALGLIB --
         Copyright 20.04.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgresults(
        mincgstate state,
        out double[] x,
        out mincgreport rep)

Examples: [1] [2] [3] [4]

### `mincgresultsbuf` function

    /*************************************************************************
    Conjugate gradient results

    Buffered implementation of MinCGResults(), which uses pre-allocated buffer
    to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
    intended to be used in the inner cycles of performance critical algorithms
    where array reallocation penalty is too large to be ignored.

      -- ALGLIB --
         Copyright 20.04.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgresultsbuf(
        mincgstate state,
        ref double[] x,
        mincgreport rep)

### `mincgsetcgtype` function

    /*************************************************************************
    This function sets CG algorithm.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        CGType  -   algorithm type:
                    * -1    automatic selection of the best algorithm
                    * 0     DY (Dai and Yuan) algorithm
                    * 1     Hybrid DY-HS algorithm

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgsetcgtype(mincgstate state, int cgtype)

### `mincgsetcond` function

    /*************************************************************************
    This function sets stopping conditions for CG optimization algorithm.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        EpsG    -   >=0
                    The  subroutine  finishes  its  work   if   the  condition
                    |v|<EpsG is satisfied, where:
                    * |.| means Euclidian norm
                    * v - scaled gradient vector, v[i]=g[i]*s[i]
                    * g - gradient
                    * s - scaling coefficients set by MinCGSetScale()
        EpsF    -   >=0
                    The  subroutine  finishes  its work if on k+1-th iteration
                    the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                    is satisfied.
        EpsX    -   >=0
                    The subroutine finishes its work if  on  k+1-th  iteration
                    the condition |v|<=EpsX is fulfilled, where:
                    * |.| means Euclidian norm
                    * v - scaled step vector, v[i]=dx[i]/s[i]
                    * dx - ste pvector, dx=X(k+1)-X(k)
                    * s - scaling coefficients set by MinCGSetScale()
        MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                    iterations is unlimited.

    Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to
    automatic stopping criterion selection (small EpsX).

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgsetcond(
        mincgstate state,
        double epsg,
        double epsf,
        double epsx,
        int maxits)

Examples: [1] [2] [3]

### `mincgsetprecdefault` function

    /*************************************************************************
    Modification of the preconditioner: preconditioning is turned off.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state

    NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
    iterations.

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgsetprecdefault(mincgstate state)

### `mincgsetprecdiag` function

    /*************************************************************************
    Modification  of  the  preconditioner:  diagonal of approximate Hessian is
    used.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        D       -   diagonal of the approximate Hessian, array[0..N-1],
                    (if larger, only leading N elements are used).

    NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
    iterations.

    NOTE 2: D[i] should be positive. Exception will be thrown otherwise.

    NOTE 3: you should pass diagonal of approximate Hessian - NOT ITS INVERSE.

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgsetprecdiag(mincgstate state, double[] d)

### `mincgsetprecscale` function

    /*************************************************************************
    Modification of the preconditioner: scale-based diagonal preconditioning.

    This preconditioning mode can be useful when you  don't  have  approximate
    diagonal of Hessian, but you know that your  variables  are  badly  scaled
    (for  example,  one  variable is in [1,10], and another in [1000,100000]),
    and most part of the ill-conditioning comes from different scales of vars.

    In this case simple  scale-based  preconditioner,  with H[i] = 1/(s[i]^2),
    can greatly improve convergence.

    IMPRTANT: you should set scale of your variables with MinCGSetScale() call
    (before or after MinCGSetPrecScale() call). Without knowledge of the scale
    of your variables scale-based preconditioner will be just unit matrix.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state

    NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
    iterations.

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgsetprecscale(mincgstate state)

### `mincgsetscale` function

    /*************************************************************************
    This function sets scaling coefficients for CG optimizer.

    ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
    size and gradient are scaled before comparison with tolerances).  Scale of
    the I-th variable is a translation invariant measure of:
    a) "how large" the variable is
    b) how large the step should be to make significant changes in the function

    Scaling is also used by finite difference variant of CG optimizer  -  step
    along I-th axis is equal to DiffStep*S[I].

    In   most   optimizers  (and  in  the  CG  too)  scaling is NOT a form  of
    preconditioning. It just  affects  stopping  conditions.  You  should  set
    preconditioner by separate call to one of the MinCGSetPrec...() functions.

    There  is  special  preconditioning  mode, however,  which  uses   scaling
    coefficients to form diagonal preconditioning matrix. You  can  turn  this
    mode on, if you want.   But  you should understand that scaling is not the
    same thing as preconditioning - these are two different, although  related
    forms of tuning solver.

    INPUT PARAMETERS:
        State   -   structure stores algorithm state
        S       -   array[N], non-zero scaling coefficients
                    S[i] may be negative, sign doesn't matter.

      -- ALGLIB --
         Copyright 14.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgsetscale(mincgstate state, double[] s)

### `mincgsetstpmax` function

    /*************************************************************************
    This function sets maximum step length

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                    want to limit step length.

    Use this subroutine when you optimize target function which contains exp()
    or  other  fast  growing  functions,  and optimization algorithm makes too
    large  steps  which  leads  to overflow. This function allows us to reject
    steps  that  are  too  large  (and  therefore  expose  us  to the possible
    overflow) without actually calculating function value at the x+stp*d.

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgsetstpmax(mincgstate state, double stpmax)

### `mincgsetxrep` function

    /*************************************************************************
    This function turns on/off reporting.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        NeedXRep-   whether iteration reports are needed or not

    If NeedXRep is True, algorithm will call rep() callback function if  it is
    provided to MinCGOptimize().

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgsetxrep(mincgstate state, bool needxrep)

### `mincgsuggeststep` function

    /*************************************************************************
    This function allows to suggest initial step length to the CG algorithm.

    Suggested  step  length  is used as starting point for the line search. It
    can be useful when you have  badly  scaled  problem,  i.e.  when  ||grad||
    (which is used as initial estimate for the first step) is many  orders  of
    magnitude different from the desired step.

    Line search  may  fail  on  such problems without good estimate of initial
    step length. Imagine, for example, problem with ||grad||=10^50 and desired
    step equal to 0.1 Line  search function will use 10^50  as  initial  step,
    then  it  will  decrease step length by 2 (up to 20 attempts) and will get
    10^44, which is still too large.

    This function allows us to tell than line search should  be  started  from
    some moderate step length, like 1.0, so algorithm will be able  to  detect
    desired step length in a several searches.

    Default behavior (when no step is suggested) is to use preconditioner,  if
    it is available, to generate initial estimate of step length.

    This function influences only first iteration of algorithm. It  should  be
    called between MinCGCreate/MinCGRestartFrom() call and MinCGOptimize call.
    Suggested step is ignored if you have preconditioner.

    INPUT PARAMETERS:
        State   -   structure used to store algorithm state.
        Stp     -   initial estimate of the step length.
                    Can be zero (no estimate).

      -- ALGLIB --
         Copyright 30.07.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mincgsuggeststep(mincgstate state, double stp)

### mincg_d_1 example

    public static void function1_grad(double[] x, ref double func, double[] grad, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        // and its derivatives df/d0 and df/dx1
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
        grad[0] = 400*System.Math.Pow(x[0]+3,3);
        grad[1] = 4*System.Math.Pow(x[1]-3,3);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
        // with nonlinear conjugate gradient method.
        //
        double[] x = new double[]{0,0};
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        int maxits = 0;
        alglib.mincgstate state;
        alglib.mincgreport rep;

        alglib.mincgcreate(x, out state);
        alglib.mincgsetcond(state, epsg, epsf, epsx, maxits);
        alglib.mincgoptimize(state, function1_grad, null, null);
        alglib.mincgresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,3]
        System.Console.ReadLine();
        return 0;
    }

### mincg_d_2 example

    public static void function1_grad(double[] x, ref double func, double[] grad, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        // and its derivatives df/d0 and df/dx1
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
        grad[0] = 400*System.Math.Pow(x[0]+3,3);
        grad[1] = 4*System.Math.Pow(x[1]-3,3);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
        // with nonlinear conjugate gradient method.
        //
        // Several advanced techniques are demonstrated:
        // * upper limit on step size
        // * restart from new point
        //
        double[] x = new double[]{0,0};
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        double stpmax = 0.1;
        int maxits = 0;
        alglib.mincgstate state;
        alglib.mincgreport rep;

        // first run
        alglib.mincgcreate(x, out state);
        alglib.mincgsetcond(state, epsg, epsf, epsx, maxits);
        alglib.mincgsetstpmax(state, stpmax);
        alglib.mincgoptimize(state, function1_grad, null, null);
        alglib.mincgresults(state, out x, out rep);

        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,3]

        // second run - algorithm is restarted with mincgrestartfrom()
        x = new double[]{10,10};
        alglib.mincgrestartfrom(state, x);
        alglib.mincgoptimize(state, function1_grad, null, null);
        alglib.mincgresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,3]
        System.Console.ReadLine();
        return 0;
    }

### mincg_ftrim example

    public static void s1_grad(double[] x, ref double func, double[] grad, object obj)
    {
        //
        // this callback calculates f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x and its gradient.
        //
        // function is trimmed when we calculate it near the singular points or outside of the [-1,+1].
        // Note that we do NOT calculate gradient in this case.
        //
        if( (x[0]<=-0.999999999999) || (x[0]>=+0.999999999999) )
        {
            func = 1.0E+300;
            return;
        }
        func = System.Math.Pow(1+x[0],-0.2) + System.Math.Pow(1-x[0],-0.3) + 1000*x[0];
        grad[0] = -0.2*System.Math.Pow(1+x[0],-1.2) +0.3*System.Math.Pow(1-x[0],-1.3) + 1000;
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x.
        // This function has singularities at the boundary of the [-1,+1], but technique called
        // "function trimming" allows us to solve this optimization problem.
        //
        // See http://www.alglib.net/optimization/tipsandtricks.php#ftrimming for more information
        // on this subject.
        //
        double[] x = new double[]{0};
        double epsg = 1.0e-6;
        double epsf = 0;
        double epsx = 0;
        int maxits = 0;
        alglib.mincgstate state;
        alglib.mincgreport rep;

        alglib.mincgcreate(x, out state);
        alglib.mincgsetcond(state, epsg, epsf, epsx, maxits);
        alglib.mincgoptimize(state, s1_grad, null, null);
        alglib.mincgresults(state, out x, out rep);

        System.Console.WriteLine("{0}", alglib.ap.format(x,5)); // EXPECTED: [-0.99917305]
        System.Console.ReadLine();
        return 0;
    }

### mincg_numdiff example

    public static void function1_func(double[] x, ref double func, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
        // using numerical differentiation to calculate gradient.
        //
        double[] x = new double[]{0,0};
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        double diffstep = 1.0e-6;
        int maxits = 0;
        alglib.mincgstate state;
        alglib.mincgreport rep;

        alglib.mincgcreatef(x, diffstep, out state);
        alglib.mincgsetcond(state, epsg, epsf, epsx, maxits);
        alglib.mincgoptimize(state, function1_func, null, null);
        alglib.mincgresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,3]
        System.Console.ReadLine();
        return 0;
    }

## `mincomp` subpackage

### Classes

minasareport
minasastate

### Functions

minasacreate
minasaoptimize
minasarestartfrom
minasaresults
minasaresultsbuf
minasasetalgorithm
minasasetcond
minasasetstpmax
minasasetxrep
minbleicsetbarrierdecay
minbleicsetbarrierwidth
minlbfgssetcholeskypreconditioner
minlbfgssetdefaultpreconditioner

### Examples

### `minasareport` class

    /*************************************************************************

    *************************************************************************/
    public class minasareport
    {
        public int                  iterationscount;
        public int                  nfev;
        public int                  terminationtype;
        public int                  activeconstraints;
    }

### `minasastate` class

    /*************************************************************************

    *************************************************************************/
    public class minasastate
    {
    }

### `minasacreate` function

    /*************************************************************************
    Obsolete optimization algorithm.
    Was replaced by MinBLEIC subpackage.

      -- ALGLIB --
         Copyright 25.03.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minasacreate(
        double[] x,
        double[] bndl,
        double[] bndu,
        out minasastate state)
    public static void minasacreate(
        int n,
        double[] x,
        double[] bndl,
        double[] bndu,
        out minasastate state)

### `minasaoptimize` function

    /*************************************************************************
    This family of functions is used to launcn iterations of nonlinear optimizer

    These functions accept following parameters:
        state   -   algorithm state
        grad    -   callback which calculates function (or merit function)
                    value func and gradient grad at given point x
        rep     -   optional callback which is called after each iteration
                    can be NULL
        ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                    can be NULL

      -- ALGLIB --
         Copyright 20.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minasaoptimize(minasastate state, ndimensional_grad grad, ndimensional_rep rep, object obj)

### `minasarestartfrom` function

    /*************************************************************************
    Obsolete optimization algorithm.
    Was replaced by MinBLEIC subpackage.

      -- ALGLIB --
         Copyright 30.07.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minasarestartfrom(
        minasastate state,
        double[] x,
        double[] bndl,
        double[] bndu)

### `minasaresults` function

    /*************************************************************************
    Obsolete optimization algorithm.
    Was replaced by MinBLEIC subpackage.

      -- ALGLIB --
         Copyright 20.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minasaresults(
        minasastate state,
        out double[] x,
        out minasareport rep)

### `minasaresultsbuf` function

    /*************************************************************************
    Obsolete optimization algorithm.
    Was replaced by MinBLEIC subpackage.

      -- ALGLIB --
         Copyright 20.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minasaresultsbuf(
        minasastate state,
        ref double[] x,
        minasareport rep)

### `minasasetalgorithm` function

    /*************************************************************************
    Obsolete optimization algorithm.
    Was replaced by MinBLEIC subpackage.

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minasasetalgorithm(minasastate state, int algotype)

### `minasasetcond` function

    /*************************************************************************
    Obsolete optimization algorithm.
    Was replaced by MinBLEIC subpackage.

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minasasetcond(
        minasastate state,
        double epsg,
        double epsf,
        double epsx,
        int maxits)

### `minasasetstpmax` function

    /*************************************************************************
    Obsolete optimization algorithm.
    Was replaced by MinBLEIC subpackage.

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minasasetstpmax(minasastate state, double stpmax)

### `minasasetxrep` function

    /*************************************************************************
    Obsolete optimization algorithm.
    Was replaced by MinBLEIC subpackage.

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minasasetxrep(minasastate state, bool needxrep)

### `minbleicsetbarrierdecay` function

    /*************************************************************************
    This is obsolete function which was used by previous version of the  BLEIC
    optimizer. It does nothing in the current version of BLEIC.

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetbarrierdecay(
        minbleicstate state,
        double mudecay)

### `minbleicsetbarrierwidth` function

    /*************************************************************************
    This is obsolete function which was used by previous version of the  BLEIC
    optimizer. It does nothing in the current version of BLEIC.

      -- ALGLIB --
         Copyright 28.11.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minbleicsetbarrierwidth(minbleicstate state, double mu)

### `minlbfgssetcholeskypreconditioner` function

    /*************************************************************************
    Obsolete function, use MinLBFGSSetCholeskyPreconditioner() instead.

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgssetcholeskypreconditioner(
        minlbfgsstate state,
        double[,] p,
        bool isupper)

### `minlbfgssetdefaultpreconditioner` function

    /*************************************************************************
    Obsolete function, use MinLBFGSSetPrecDefault() instead.

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgssetdefaultpreconditioner(minlbfgsstate state)

## `minlbfgs` subpackage

### Classes

minlbfgsreport
minlbfgsstate

### Functions

minlbfgscreate
minlbfgscreatef
minlbfgsoptimize
minlbfgsrestartfrom
minlbfgsresults
minlbfgsresultsbuf
minlbfgssetcond
minlbfgssetpreccholesky
minlbfgssetprecdefault
minlbfgssetprecdiag
minlbfgssetprecscale
minlbfgssetscale
minlbfgssetstpmax
minlbfgssetxrep

### Examples

minlbfgs_d_1 |  | Nonlinear optimization by L-BFGS
minlbfgs_d_2 |  | Nonlinear optimization with additional settings and restarts
minlbfgs_ftrim |  | Nonlinear optimization by LBFGS, function with singularities
minlbfgs_numdiff |  | Nonlinear optimization by L-BFGS with numerical differentiation

### `minlbfgsreport` class

    /*************************************************************************

    *************************************************************************/
    public class minlbfgsreport
    {
        public int                  iterationscount;
        public int                  nfev;
        public int                  terminationtype;
    }

### `minlbfgsstate` class

    /*************************************************************************

    *************************************************************************/
    public class minlbfgsstate
    {
    }

### `minlbfgscreate` function

    /*************************************************************************
            LIMITED MEMORY BFGS METHOD FOR LARGE SCALE OPTIMIZATION

    DESCRIPTION:
    The subroutine minimizes function F(x) of N arguments by  using  a  quasi-
    Newton method (LBFGS scheme) which is optimized to use  a  minimum  amount
    of memory.
    The subroutine generates the approximation of an inverse Hessian matrix by
    using information about the last M steps of the algorithm  (instead of N).
    It lessens a required amount of memory from a value  of  order  N^2  to  a
    value of order 2*N*M.

    REQUIREMENTS:
    Algorithm will request following information during its operation:
    * function value F and its gradient G (simultaneously) at given point X

    USAGE:
    1. User initializes algorithm state with MinLBFGSCreate() call
    2. User tunes solver parameters with MinLBFGSSetCond() MinLBFGSSetStpMax()
       and other functions
    3. User calls MinLBFGSOptimize() function which takes algorithm  state and
       pointer (delegate, etc.) to callback function which calculates F/G.
    4. User calls MinLBFGSResults() to get solution
    5. Optionally user may call MinLBFGSRestartFrom() to solve another problem
       with same N/M but another starting point and/or another function.
       MinLBFGSRestartFrom() allows to reuse already initialized structure.

    INPUT PARAMETERS:
        N       -   problem dimension. N>0
        M       -   number of corrections in the BFGS scheme of Hessian
                    approximation update. Recommended value:  3<=M<=7. The smaller
                    value causes worse convergence, the bigger will  not  cause  a
                    considerably better convergence, but will cause a fall in  the
                    performance. M<=N.
        X       -   initial solution approximation, array[0..N-1].

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

    NOTES:
    1. you may tune stopping conditions with MinLBFGSSetCond() function
    2. if target function contains exp() or other fast growing functions,  and
       optimization algorithm makes too large steps which leads  to  overflow,
       use MinLBFGSSetStpMax() function to bound algorithm's  steps.  However,
       L-BFGS rarely needs such a tuning.

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgscreate(
        int m,
        double[] x,
        out minlbfgsstate state)
    public static void minlbfgscreate(
        int n,
        int m,
        double[] x,
        out minlbfgsstate state)

Examples: [1] [2] [3]

### `minlbfgscreatef` function

    /*************************************************************************
    The subroutine is finite difference variant of MinLBFGSCreate().  It  uses
    finite differences in order to differentiate target function.

    Description below contains information which is specific to  this function
    only. We recommend to read comments on MinLBFGSCreate() in  order  to  get
    more information about creation of LBFGS optimizer.

    INPUT PARAMETERS:
        N       -   problem dimension, N>0:
                    * if given, only leading N elements of X are used
                    * if not given, automatically determined from size of X
        M       -   number of corrections in the BFGS scheme of Hessian
                    approximation update. Recommended value:  3<=M<=7. The smaller
                    value causes worse convergence, the bigger will  not  cause  a
                    considerably better convergence, but will cause a fall in  the
                    performance. M<=N.
        X       -   starting point, array[0..N-1].
        DiffStep-   differentiation step, >0

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

    NOTES:
    1. algorithm uses 4-point central formula for differentiation.
    2. differentiation step along I-th axis is equal to DiffStep*S[I] where
       S[] is scaling vector which can be set by MinLBFGSSetScale() call.
    3. we recommend you to use moderate values of  differentiation  step.  Too
       large step will result in too large truncation  errors, while too small
       step will result in too large numerical  errors.  1.0E-6  can  be  good
       value to start with.
    4. Numerical  differentiation  is   very   inefficient  -   one   gradient
       calculation needs 4*N function evaluations. This function will work for
       any N - either small (1...10), moderate (10...100) or  large  (100...).
       However, performance penalty will be too severe for any N's except  for
       small ones.
       We should also say that code which relies on numerical  differentiation
       is   less  robust  and  precise.  LBFGS  needs  exact  gradient values.
       Imprecise gradient may slow  down  convergence,  especially  on  highly
       nonlinear problems.
       Thus  we  recommend to use this function for fast prototyping on small-
       dimensional problems only, and to implement analytical gradient as soon
       as possible.

      -- ALGLIB --
         Copyright 16.05.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgscreatef(
        int m,
        double[] x,
        double diffstep,
        out minlbfgsstate state)
    public static void minlbfgscreatef(
        int n,
        int m,
        double[] x,
        double diffstep,
        out minlbfgsstate state)

Examples: [1]

### `minlbfgsoptimize` function

    /*************************************************************************
    This family of functions is used to launcn iterations of nonlinear optimizer

    These functions accept following parameters:
        state   -   algorithm state
        func    -   callback which calculates function (or merit function)
                    value func at given point x
        grad    -   callback which calculates function (or merit function)
                    value func and gradient grad at given point x
        rep     -   optional callback which is called after each iteration
                    can be NULL
        ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                    can be NULL

    NOTES:

    1. This function has two different implementations: one which  uses  exact
       (analytical) user-supplied gradient,  and one which uses function value
       only  and  numerically  differentiates  function  in  order  to  obtain
       gradient.

       Depending  on  the  specific  function  used to create optimizer object
       (either MinLBFGSCreate() for analytical gradient  or  MinLBFGSCreateF()
       for numerical differentiation) you should choose appropriate variant of
       MinLBFGSOptimize() - one  which  accepts  function  AND gradient or one
       which accepts function ONLY.

       Be careful to choose variant of MinLBFGSOptimize() which corresponds to
       your optimization scheme! Table below lists different  combinations  of
       callback (function/gradient) passed to MinLBFGSOptimize()  and specific
       function used to create optimizer.

                         |         USER PASSED TO MinLBFGSOptimize()
       CREATED WITH      |  function only   |  function and gradient
       ------------------------------------------------------------
       MinLBFGSCreateF() |     work                FAIL
       MinLBFGSCreate()  |     FAIL                work

       Here "FAIL" denotes inappropriate combinations  of  optimizer  creation
       function  and  MinLBFGSOptimize()  version.   Attemps   to   use   such
       combination (for example, to create optimizer with MinLBFGSCreateF() and
       to pass gradient information to MinCGOptimize()) will lead to exception
       being thrown. Either  you  did  not pass gradient when it WAS needed or
       you passed gradient when it was NOT needed.

      -- ALGLIB --
         Copyright 20.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgsoptimize(minlbfgsstate state, ndimensional_func func, ndimensional_rep rep, object obj)
    public static void minlbfgsoptimize(minlbfgsstate state, ndimensional_grad grad, ndimensional_rep rep, object obj)

Examples: [1] [2] [3] [4]

### `minlbfgsrestartfrom` function

    /*************************************************************************
    This  subroutine restarts LBFGS algorithm from new point. All optimization
    parameters are left unchanged.

    This  function  allows  to  solve multiple  optimization  problems  (which
    must have same number of dimensions) without object reallocation penalty.

    INPUT PARAMETERS:
        State   -   structure used to store algorithm state
        X       -   new starting point.

      -- ALGLIB --
         Copyright 30.07.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgsrestartfrom(minlbfgsstate state, double[] x)

Examples: [1]

### `minlbfgsresults` function

    /*************************************************************************
    L-BFGS algorithm results

    INPUT PARAMETERS:
        State   -   algorithm state

    OUTPUT PARAMETERS:
        X       -   array[0..N-1], solution
        Rep     -   optimization report:
                    * Rep.TerminationType completetion code:
                        * -2    rounding errors prevent further improvement.
                                X contains best point found.
                        * -1    incorrect parameters were specified
                        *  1    relative function improvement is no more than
                                EpsF.
                        *  2    relative step is no more than EpsX.
                        *  4    gradient norm is no more than EpsG
                        *  5    MaxIts steps was taken
                        *  7    stopping conditions are too stringent,
                                further improvement is impossible
                    * Rep.IterationsCount contains iterations count
                    * NFEV countains number of function calculations

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgsresults(
        minlbfgsstate state,
        out double[] x,
        out minlbfgsreport rep)

Examples: [1] [2] [3] [4]

### `minlbfgsresultsbuf` function

    /*************************************************************************
    L-BFGS algorithm results

    Buffered implementation of MinLBFGSResults which uses pre-allocated buffer
    to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
    intended to be used in the inner cycles of performance critical algorithms
    where array reallocation penalty is too large to be ignored.

      -- ALGLIB --
         Copyright 20.08.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgsresultsbuf(
        minlbfgsstate state,
        ref double[] x,
        minlbfgsreport rep)

### `minlbfgssetcond` function

    /*************************************************************************
    This function sets stopping conditions for L-BFGS optimization algorithm.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        EpsG    -   >=0
                    The  subroutine  finishes  its  work   if   the  condition
                    |v|<EpsG is satisfied, where:
                    * |.| means Euclidian norm
                    * v - scaled gradient vector, v[i]=g[i]*s[i]
                    * g - gradient
                    * s - scaling coefficients set by MinLBFGSSetScale()
        EpsF    -   >=0
                    The  subroutine  finishes  its work if on k+1-th iteration
                    the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                    is satisfied.
        EpsX    -   >=0
                    The subroutine finishes its work if  on  k+1-th  iteration
                    the condition |v|<=EpsX is fulfilled, where:
                    * |.| means Euclidian norm
                    * v - scaled step vector, v[i]=dx[i]/s[i]
                    * dx - ste pvector, dx=X(k+1)-X(k)
                    * s - scaling coefficients set by MinLBFGSSetScale()
        MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                    iterations is unlimited.

    Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to
    automatic stopping criterion selection (small EpsX).

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgssetcond(
        minlbfgsstate state,
        double epsg,
        double epsf,
        double epsx,
        int maxits)

Examples: [1] [2] [3]

### `minlbfgssetpreccholesky` function

    /*************************************************************************
    Modification of the preconditioner: Cholesky factorization of  approximate
    Hessian is used.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        P       -   triangular preconditioner, Cholesky factorization of
                    the approximate Hessian. array[0..N-1,0..N-1],
                    (if larger, only leading N elements are used).
        IsUpper -   whether upper or lower triangle of P is given
                    (other triangle is not referenced)

    After call to this function preconditioner is changed to P  (P  is  copied
    into the internal buffer).

    NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
    iterations.

    NOTE 2:  P  should  be nonsingular. Exception will be thrown otherwise.

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgssetpreccholesky(
        minlbfgsstate state,
        double[,] p,
        bool isupper)

### `minlbfgssetprecdefault` function

    /*************************************************************************
    Modification  of  the  preconditioner:  default  preconditioner    (simple
    scaling, same for all elements of X) is used.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state

    NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
    iterations.

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgssetprecdefault(minlbfgsstate state)

### `minlbfgssetprecdiag` function

    /*************************************************************************
    Modification  of  the  preconditioner:  diagonal of approximate Hessian is
    used.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        D       -   diagonal of the approximate Hessian, array[0..N-1],
                    (if larger, only leading N elements are used).

    NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
    iterations.

    NOTE 2: D[i] should be positive. Exception will be thrown otherwise.

    NOTE 3: you should pass diagonal of approximate Hessian - NOT ITS INVERSE.

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgssetprecdiag(minlbfgsstate state, double[] d)

### `minlbfgssetprecscale` function

    /*************************************************************************
    Modification of the preconditioner: scale-based diagonal preconditioning.

    This preconditioning mode can be useful when you  don't  have  approximate
    diagonal of Hessian, but you know that your  variables  are  badly  scaled
    (for  example,  one  variable is in [1,10], and another in [1000,100000]),
    and most part of the ill-conditioning comes from different scales of vars.

    In this case simple  scale-based  preconditioner,  with H[i] = 1/(s[i]^2),
    can greatly improve convergence.

    IMPRTANT: you should set scale of your variables  with  MinLBFGSSetScale()
    call  (before  or after MinLBFGSSetPrecScale() call). Without knowledge of
    the scale of your variables scale-based preconditioner will be  just  unit
    matrix.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state

      -- ALGLIB --
         Copyright 13.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgssetprecscale(minlbfgsstate state)

### `minlbfgssetscale` function

    /*************************************************************************
    This function sets scaling coefficients for LBFGS optimizer.

    ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
    size and gradient are scaled before comparison with tolerances).  Scale of
    the I-th variable is a translation invariant measure of:
    a) "how large" the variable is
    b) how large the step should be to make significant changes in the function

    Scaling is also used by finite difference variant of the optimizer  - step
    along I-th axis is equal to DiffStep*S[I].

    In  most  optimizers  (and  in  the  LBFGS  too)  scaling is NOT a form of
    preconditioning. It just  affects  stopping  conditions.  You  should  set
    preconditioner  by  separate  call  to  one  of  the  MinLBFGSSetPrec...()
    functions.

    There  is  special  preconditioning  mode, however,  which  uses   scaling
    coefficients to form diagonal preconditioning matrix. You  can  turn  this
    mode on, if you want.   But  you should understand that scaling is not the
    same thing as preconditioning - these are two different, although  related
    forms of tuning solver.

    INPUT PARAMETERS:
        State   -   structure stores algorithm state
        S       -   array[N], non-zero scaling coefficients
                    S[i] may be negative, sign doesn't matter.

      -- ALGLIB --
         Copyright 14.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgssetscale(minlbfgsstate state, double[] s)

### `minlbfgssetstpmax` function

    /*************************************************************************
    This function sets maximum step length

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        StpMax  -   maximum step length, >=0. Set StpMax to 0.0 (default),  if
                    you don't want to limit step length.

    Use this subroutine when you optimize target function which contains exp()
    or  other  fast  growing  functions,  and optimization algorithm makes too
    large  steps  which  leads  to overflow. This function allows us to reject
    steps  that  are  too  large  (and  therefore  expose  us  to the possible
    overflow) without actually calculating function value at the x+stp*d.

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgssetstpmax(minlbfgsstate state, double stpmax)

### `minlbfgssetxrep` function

    /*************************************************************************
    This function turns on/off reporting.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        NeedXRep-   whether iteration reports are needed or not

    If NeedXRep is True, algorithm will call rep() callback function if  it is
    provided to MinLBFGSOptimize().

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlbfgssetxrep(minlbfgsstate state, bool needxrep)

### minlbfgs_d_1 example

    public static void function1_grad(double[] x, ref double func, double[] grad, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        // and its derivatives df/d0 and df/dx1
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
        grad[0] = 400*System.Math.Pow(x[0]+3,3);
        grad[1] = 4*System.Math.Pow(x[1]-3,3);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
        // using LBFGS method.
        //
        double[] x = new double[]{0,0};
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        int maxits = 0;
        alglib.minlbfgsstate state;
        alglib.minlbfgsreport rep;

        alglib.minlbfgscreate(1, x, out state);
        alglib.minlbfgssetcond(state, epsg, epsf, epsx, maxits);
        alglib.minlbfgsoptimize(state, function1_grad, null, null);
        alglib.minlbfgsresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,3]
        System.Console.ReadLine();
        return 0;
    }

### minlbfgs_d_2 example

    public static void function1_grad(double[] x, ref double func, double[] grad, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        // and its derivatives df/d0 and df/dx1
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
        grad[0] = 400*System.Math.Pow(x[0]+3,3);
        grad[1] = 4*System.Math.Pow(x[1]-3,3);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
        // using LBFGS method.
        //
        // Several advanced techniques are demonstrated:
        // * upper limit on step size
        // * restart from new point
        //
        double[] x = new double[]{0,0};
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        double stpmax = 0.1;
        int maxits = 0;
        alglib.minlbfgsstate state;
        alglib.minlbfgsreport rep;

        // first run
        alglib.minlbfgscreate(1, x, out state);
        alglib.minlbfgssetcond(state, epsg, epsf, epsx, maxits);
        alglib.minlbfgssetstpmax(state, stpmax);
        alglib.minlbfgsoptimize(state, function1_grad, null, null);
        alglib.minlbfgsresults(state, out x, out rep);

        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,3]

        // second run - algorithm is restarted
        x = new double[]{10,10};
        alglib.minlbfgsrestartfrom(state, x);
        alglib.minlbfgsoptimize(state, function1_grad, null, null);
        alglib.minlbfgsresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,3]
        System.Console.ReadLine();
        return 0;
    }

### minlbfgs_ftrim example

    public static void s1_grad(double[] x, ref double func, double[] grad, object obj)
    {
        //
        // this callback calculates f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x and its gradient.
        //
        // function is trimmed when we calculate it near the singular points or outside of the [-1,+1].
        // Note that we do NOT calculate gradient in this case.
        //
        if( (x[0]<=-0.999999999999) || (x[0]>=+0.999999999999) )
        {
            func = 1.0E+300;
            return;
        }
        func = System.Math.Pow(1+x[0],-0.2) + System.Math.Pow(1-x[0],-0.3) + 1000*x[0];
        grad[0] = -0.2*System.Math.Pow(1+x[0],-1.2) +0.3*System.Math.Pow(1-x[0],-1.3) + 1000;
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x.
        // This function has singularities at the boundary of the [-1,+1], but technique called
        // "function trimming" allows us to solve this optimization problem.
        //
        // See http://www.alglib.net/optimization/tipsandtricks.php#ftrimming for more information
        // on this subject.
        //
        double[] x = new double[]{0};
        double epsg = 1.0e-6;
        double epsf = 0;
        double epsx = 0;
        int maxits = 0;
        alglib.minlbfgsstate state;
        alglib.minlbfgsreport rep;

        alglib.minlbfgscreate(1, x, out state);
        alglib.minlbfgssetcond(state, epsg, epsf, epsx, maxits);
        alglib.minlbfgsoptimize(state, s1_grad, null, null);
        alglib.minlbfgsresults(state, out x, out rep);

        System.Console.WriteLine("{0}", alglib.ap.format(x,5)); // EXPECTED: [-0.99917305]
        System.Console.ReadLine();
        return 0;
    }

### minlbfgs_numdiff example

    public static void function1_func(double[] x, ref double func, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
        // using numerical differentiation to calculate gradient.
        //
        double[] x = new double[]{0,0};
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        double diffstep = 1.0e-6;
        int maxits = 0;
        alglib.minlbfgsstate state;
        alglib.minlbfgsreport rep;

        alglib.minlbfgscreatef(1, x, diffstep, out state);
        alglib.minlbfgssetcond(state, epsg, epsf, epsx, maxits);
        alglib.minlbfgsoptimize(state, function1_func, null, null);
        alglib.minlbfgsresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,3]
        System.Console.ReadLine();
        return 0;
    }

## `minlm` subpackage

### Classes

minlmreport
minlmstate

### Functions

minlmcreatefgh
minlmcreatefgj
minlmcreatefj
minlmcreatev
minlmcreatevgj
minlmcreatevj
minlmoptimize
minlmrestartfrom
minlmresults
minlmresultsbuf
minlmsetacctype
minlmsetbc
minlmsetcond
minlmsetscale
minlmsetstpmax
minlmsetxrep

### Examples

minlm_d_fgh |  | Nonlinear Hessian-based optimization for general functions
minlm_d_restarts |  | Efficient restarts of LM optimizer
minlm_d_v |  | Nonlinear least squares optimization using function vector only
minlm_d_vb |  | Bound constrained nonlinear least squares optimization
minlm_d_vj |  | Nonlinear least squares optimization using function vector and Jacobian

### `minlmreport` class

    /*************************************************************************
    Optimization report, filled by MinLMResults() function

    FIELDS:
    * TerminationType, completetion code:
        * -9    derivative correctness check failed;
                see Rep.WrongNum, Rep.WrongI, Rep.WrongJ for
                more information.
        *  1    relative function improvement is no more than
                EpsF.
        *  2    relative step is no more than EpsX.
        *  4    gradient is no more than EpsG.
        *  5    MaxIts steps was taken
        *  7    stopping conditions are too stringent,
                further improvement is impossible
    * IterationsCount, contains iterations count
    * NFunc, number of function calculations
    * NJac, number of Jacobi matrix calculations
    * NGrad, number of gradient calculations
    * NHess, number of Hessian calculations
    * NCholesky, number of Cholesky decomposition calculations
    *************************************************************************/
    public class minlmreport
    {
        public int                  iterationscount;
        public int                  terminationtype;
        public int                  nfunc;
        public int                  njac;
        public int                  ngrad;
        public int                  nhess;
        public int                  ncholesky;
    }

### `minlmstate` class

    /*************************************************************************
    Levenberg-Marquardt optimizer.

    This structure should be created using one of the MinLMCreate???()
    functions. You should not access its fields directly; use ALGLIB functions
    to work with it.
    *************************************************************************/
    public class minlmstate
    {
    }

### `minlmcreatefgh` function

    /*************************************************************************
        LEVENBERG-MARQUARDT-LIKE METHOD FOR NON-LINEAR OPTIMIZATION

    DESCRIPTION:
    This  function  is  used  to  find  minimum  of general form (not "sum-of-
    -squares") function
        F = F(x[0], ..., x[n-1])
    using  its  gradient  and  Hessian.  Levenberg-Marquardt modification with
    L-BFGS pre-optimization and internal pre-conditioned  L-BFGS  optimization
    after each Levenberg-Marquardt step is used.

    REQUIREMENTS:
    This algorithm will request following information during its operation:

    * function value F at given point X
    * F and gradient G (simultaneously) at given point X
    * F, G and Hessian H (simultaneously) at given point X

    There are several overloaded versions of  MinLMOptimize()  function  which
    correspond  to  different LM-like optimization algorithms provided by this
    unit. You should choose version which accepts func(),  grad()  and  hess()
    function pointers. First pointer is used to calculate F  at  given  point,
    second  one  calculates  F(x)  and  grad F(x),  third one calculates F(x),
    grad F(x), hess F(x).

    You can try to initialize MinLMState structure with FGH-function and  then
    use incorrect version of MinLMOptimize() (for example, version which  does
    not provide Hessian matrix), but it will lead to  exception  being  thrown
    after first attempt to calculate Hessian.

    USAGE:
    1. User initializes algorithm state with MinLMCreateFGH() call
    2. User tunes solver parameters with MinLMSetCond(),  MinLMSetStpMax() and
       other functions
    3. User calls MinLMOptimize() function which  takes algorithm  state   and
       pointers (delegates, etc.) to callback functions.
    4. User calls MinLMResults() to get solution
    5. Optionally, user may call MinLMRestartFrom() to solve  another  problem
       with same N but another starting point and/or another function.
       MinLMRestartFrom() allows to reuse already initialized structure.

    INPUT PARAMETERS:
        N       -   dimension, N>1
                    * if given, only leading N elements of X are used
                    * if not given, automatically determined from size of X
        X       -   initial solution, array[0..N-1]

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

    NOTES:
    1. you may tune stopping conditions with MinLMSetCond() function
    2. if target function contains exp() or other fast growing functions,  and
       optimization algorithm makes too large steps which leads  to  overflow,
       use MinLMSetStpMax() function to bound algorithm's steps.

      -- ALGLIB --
         Copyright 30.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmcreatefgh(double[] x, out minlmstate state)
    public static void minlmcreatefgh(int n, double[] x, out minlmstate state)

Examples: [1]

### `minlmcreatefgj` function

    /*************************************************************************
    This is obsolete function.

    Since ALGLIB 3.3 it is equivalent to MinLMCreateFJ().

      -- ALGLIB --
         Copyright 30.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmcreatefgj(int m, double[] x, out minlmstate state)
    public static void minlmcreatefgj(
        int n,
        int m,
        double[] x,
        out minlmstate state)

### `minlmcreatefj` function

    /*************************************************************************
    This function is considered obsolete since ALGLIB 3.1.0 and is present for
    backward  compatibility  only.  We  recommend  to use MinLMCreateVJ, which
    provides similar, but more consistent and feature-rich interface.

      -- ALGLIB --
         Copyright 30.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmcreatefj(int m, double[] x, out minlmstate state)
    public static void minlmcreatefj(
        int n,
        int m,
        double[] x,
        out minlmstate state)

### `minlmcreatev` function

    /*************************************************************************
                    IMPROVED LEVENBERG-MARQUARDT METHOD FOR
                     NON-LINEAR LEAST SQUARES OPTIMIZATION

    DESCRIPTION:
    This function is used to find minimum of function which is represented  as
    sum of squares:
        F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1])
    using value of function vector f[] only. Finite differences  are  used  to
    calculate Jacobian.

    REQUIREMENTS:
    This algorithm will request following information during its operation:
    * function vector f[] at given point X

    There are several overloaded versions of  MinLMOptimize()  function  which
    correspond  to  different LM-like optimization algorithms provided by this
    unit. You should choose version which accepts fvec() callback.

    You can try to initialize MinLMState structure with VJ  function and  then
    use incorrect version  of  MinLMOptimize()  (for  example,  version  which
    works with general form function and does not accept function vector), but
    it will  lead  to  exception being thrown after first attempt to calculate
    Jacobian.

    USAGE:
    1. User initializes algorithm state with MinLMCreateV() call
    2. User tunes solver parameters with MinLMSetCond(),  MinLMSetStpMax() and
       other functions
    3. User calls MinLMOptimize() function which  takes algorithm  state   and
       callback functions.
    4. User calls MinLMResults() to get solution
    5. Optionally, user may call MinLMRestartFrom() to solve  another  problem
       with same N/M but another starting point and/or another function.
       MinLMRestartFrom() allows to reuse already initialized structure.

    INPUT PARAMETERS:
        N       -   dimension, N>1
                    * if given, only leading N elements of X are used
                    * if not given, automatically determined from size of X
        M       -   number of functions f[i]
        X       -   initial solution, array[0..N-1]
        DiffStep-   differentiation step, >0

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

    See also MinLMIteration, MinLMResults.

    NOTES:
    1. you may tune stopping conditions with MinLMSetCond() function
    2. if target function contains exp() or other fast growing functions,  and
       optimization algorithm makes too large steps which leads  to  overflow,
       use MinLMSetStpMax() function to bound algorithm's steps.

      -- ALGLIB --
         Copyright 30.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmcreatev(
        int m,
        double[] x,
        double diffstep,
        out minlmstate state)
    public static void minlmcreatev(
        int n,
        int m,
        double[] x,
        double diffstep,
        out minlmstate state)

Examples: [1] [2] [3]

### `minlmcreatevgj` function

    /*************************************************************************
    This is obsolete function.

    Since ALGLIB 3.3 it is equivalent to MinLMCreateVJ().

      -- ALGLIB --
         Copyright 30.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmcreatevgj(int m, double[] x, out minlmstate state)
    public static void minlmcreatevgj(
        int n,
        int m,
        double[] x,
        out minlmstate state)

### `minlmcreatevj` function

    /*************************************************************************
                    IMPROVED LEVENBERG-MARQUARDT METHOD FOR
                     NON-LINEAR LEAST SQUARES OPTIMIZATION

    DESCRIPTION:
    This function is used to find minimum of function which is represented  as
    sum of squares:
        F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1])
    using value of function vector f[] and Jacobian of f[].

    REQUIREMENTS:
    This algorithm will request following information during its operation:

    * function vector f[] at given point X
    * function vector f[] and Jacobian of f[] (simultaneously) at given point

    There are several overloaded versions of  MinLMOptimize()  function  which
    correspond  to  different LM-like optimization algorithms provided by this
    unit. You should choose version which accepts fvec()  and jac() callbacks.
    First  one  is used to calculate f[] at given point, second one calculates
    f[] and Jacobian df[i]/dx[j].

    You can try to initialize MinLMState structure with VJ  function and  then
    use incorrect version  of  MinLMOptimize()  (for  example,  version  which
    works  with  general  form function and does not provide Jacobian), but it
    will  lead  to  exception  being  thrown  after first attempt to calculate
    Jacobian.

    USAGE:
    1. User initializes algorithm state with MinLMCreateVJ() call
    2. User tunes solver parameters with MinLMSetCond(),  MinLMSetStpMax() and
       other functions
    3. User calls MinLMOptimize() function which  takes algorithm  state   and
       callback functions.
    4. User calls MinLMResults() to get solution
    5. Optionally, user may call MinLMRestartFrom() to solve  another  problem
       with same N/M but another starting point and/or another function.
       MinLMRestartFrom() allows to reuse already initialized structure.

    INPUT PARAMETERS:
        N       -   dimension, N>1
                    * if given, only leading N elements of X are used
                    * if not given, automatically determined from size of X
        M       -   number of functions f[i]
        X       -   initial solution, array[0..N-1]

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

    NOTES:
    1. you may tune stopping conditions with MinLMSetCond() function
    2. if target function contains exp() or other fast growing functions,  and
       optimization algorithm makes too large steps which leads  to  overflow,
       use MinLMSetStpMax() function to bound algorithm's steps.

      -- ALGLIB --
         Copyright 30.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmcreatevj(int m, double[] x, out minlmstate state)
    public static void minlmcreatevj(
        int n,
        int m,
        double[] x,
        out minlmstate state)

Examples: [1]

### `minlmoptimize` function

    /*************************************************************************
    This family of functions is used to launcn iterations of nonlinear optimizer

    These functions accept following parameters:
        state   -   algorithm state
        func    -   callback which calculates function (or merit function)
                    value func at given point x
        grad    -   callback which calculates function (or merit function)
                    value func and gradient grad at given point x
        hess    -   callback which calculates function (or merit function)
                    value func, gradient grad and Hessian hess at given point x
        fvec    -   callback which calculates function vector fi[]
                    at given point x
        jac     -   callback which calculates function vector fi[]
                    and Jacobian jac at given point x
        rep     -   optional callback which is called after each iteration
                    can be NULL
        ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                    can be NULL

    NOTES:

    1. Depending on function used to create state  structure,  this  algorithm
       may accept Jacobian and/or Hessian and/or gradient.  According  to  the
       said above, there ase several versions of this function,  which  accept
       different sets of callbacks.

       This flexibility opens way to subtle errors - you may create state with
       MinLMCreateFGH() (optimization using Hessian), but call function  which
       does not accept Hessian. So when algorithm will request Hessian,  there
       will be no callback to call. In this case exception will be thrown.

       Be careful to avoid such errors because there is no way to find them at
       compile time - you can see them at runtime only.

      -- ALGLIB --
         Copyright 10.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmoptimize(minlmstate state, ndimensional_fvec  fvec, ndimensional_rep rep, object obj)
    public static void minlmoptimize(minlmstate state, ndimensional_fvec  fvec, ndimensional_jac  jac, ndimensional_rep rep, object obj)
    public static void minlmoptimize(minlmstate state, ndimensional_func func, ndimensional_grad grad, ndimensional_hess hess, ndimensional_rep rep, object obj)
    public static void minlmoptimize(minlmstate state, ndimensional_func func, ndimensional_jac  jac, ndimensional_rep rep, object obj)
    public static void minlmoptimize(minlmstate state, ndimensional_func func, ndimensional_grad grad, ndimensional_jac  jac, ndimensional_rep rep, object obj)

Examples: [1] [2] [3] [4] [5]

### `minlmrestartfrom` function

    /*************************************************************************
    This  subroutine  restarts  LM  algorithm from new point. All optimization
    parameters are left unchanged.

    This  function  allows  to  solve multiple  optimization  problems  (which
    must have same number of dimensions) without object reallocation penalty.

    INPUT PARAMETERS:
        State   -   structure used for reverse communication previously
                    allocated with MinLMCreateXXX call.
        X       -   new starting point.

      -- ALGLIB --
         Copyright 30.07.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmrestartfrom(minlmstate state, double[] x)

Examples: [1]

### `minlmresults` function

    /*************************************************************************
    Levenberg-Marquardt algorithm results

    INPUT PARAMETERS:
        State   -   algorithm state

    OUTPUT PARAMETERS:
        X       -   array[0..N-1], solution
        Rep     -   optimization report;
                    see comments for this structure for more info.

      -- ALGLIB --
         Copyright 10.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmresults(
        minlmstate state,
        out double[] x,
        out minlmreport rep)

Examples: [1] [2] [3] [4] [5]

### `minlmresultsbuf` function

    /*************************************************************************
    Levenberg-Marquardt algorithm results

    Buffered implementation of MinLMResults(), which uses pre-allocated buffer
    to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
    intended to be used in the inner cycles of performance critical algorithms
    where array reallocation penalty is too large to be ignored.

      -- ALGLIB --
         Copyright 10.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmresultsbuf(
        minlmstate state,
        ref double[] x,
        minlmreport rep)

### `minlmsetacctype` function

    /*************************************************************************
    This function is used to change acceleration settings

    You can choose between three acceleration strategies:
    * AccType=0, no acceleration.
    * AccType=1, secant updates are used to update quadratic model after  each
      iteration. After fixed number of iterations (or after  model  breakdown)
      we  recalculate  quadratic  model  using  analytic  Jacobian  or  finite
      differences. Number of secant-based iterations depends  on  optimization
      settings: about 3 iterations - when we have analytic Jacobian, up to 2*N
      iterations - when we use finite differences to calculate Jacobian.

    AccType=1 is recommended when Jacobian  calculation  cost  is  prohibitive
    high (several Mx1 function vector calculations  followed  by  several  NxN
    Cholesky factorizations are faster than calculation of one M*N  Jacobian).
    It should also be used when we have no Jacobian, because finite difference
    approximation takes too much time to compute.

    Table below list  optimization  protocols  (XYZ  protocol  corresponds  to
    MinLMCreateXYZ) and acceleration types they support (and use by  default).

    ACCELERATION TYPES SUPPORTED BY OPTIMIZATION PROTOCOLS:

    protocol    0   1   comment
    V           +   +
    VJ          +   +
    FGH         +

    DAFAULT VALUES:

    protocol    0   1   comment
    V               x   without acceleration it is so slooooooooow
    VJ          x
    FGH         x

    NOTE: this  function should be called before optimization. Attempt to call
    it during algorithm iterations may result in unexpected behavior.

    NOTE: attempt to call this function with unsupported protocol/acceleration
    combination will result in exception being thrown.

      -- ALGLIB --
         Copyright 14.10.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmsetacctype(minlmstate state, int acctype)

### `minlmsetbc` function

    /*************************************************************************
    This function sets boundary constraints for LM optimizer

    Boundary constraints are inactive by default (after initial creation).
    They are preserved until explicitly turned off with another SetBC() call.

    INPUT PARAMETERS:
        State   -   structure stores algorithm state
        BndL    -   lower bounds, array[N].
                    If some (all) variables are unbounded, you may specify
                    very small number or -INF (latter is recommended because
                    it will allow solver to use better algorithm).
        BndU    -   upper bounds, array[N].
                    If some (all) variables are unbounded, you may specify
                    very large number or +INF (latter is recommended because
                    it will allow solver to use better algorithm).

    NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
    variable will be "frozen" at X[i]=BndL[i]=BndU[i].

    NOTE 2: this solver has following useful properties:
    * bound constraints are always satisfied exactly
    * function is evaluated only INSIDE area specified by bound constraints
      or at its boundary

      -- ALGLIB --
         Copyright 14.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmsetbc(
        minlmstate state,
        double[] bndl,
        double[] bndu)

### `minlmsetcond` function

    /*************************************************************************
    This function sets stopping conditions for Levenberg-Marquardt optimization
    algorithm.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        EpsG    -   >=0
                    The  subroutine  finishes  its  work   if   the  condition
                    |v|<EpsG is satisfied, where:
                    * |.| means Euclidian norm
                    * v - scaled gradient vector, v[i]=g[i]*s[i]
                    * g - gradient
                    * s - scaling coefficients set by MinLMSetScale()
        EpsF    -   >=0
                    The  subroutine  finishes  its work if on k+1-th iteration
                    the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                    is satisfied.
        EpsX    -   >=0
                    The subroutine finishes its work if  on  k+1-th  iteration
                    the condition |v|<=EpsX is fulfilled, where:
                    * |.| means Euclidian norm
                    * v - scaled step vector, v[i]=dx[i]/s[i]
                    * dx - ste pvector, dx=X(k+1)-X(k)
                    * s - scaling coefficients set by MinLMSetScale()
        MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                    iterations   is    unlimited.   Only   Levenberg-Marquardt
                    iterations  are  counted  (L-BFGS/CG  iterations  are  NOT
                    counted because their cost is very low compared to that of
                    LM).

    Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to
    automatic stopping criterion selection (small EpsX).

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmsetcond(
        minlmstate state,
        double epsg,
        double epsf,
        double epsx,
        int maxits)

Examples: [1] [2] [3] [4] [5]

### `minlmsetscale` function

    /*************************************************************************
    This function sets scaling coefficients for LM optimizer.

    ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
    size and gradient are scaled before comparison with tolerances).  Scale of
    the I-th variable is a translation invariant measure of:
    a) "how large" the variable is
    b) how large the step should be to make significant changes in the function

    Generally, scale is NOT considered to be a form of preconditioner.  But LM
    optimizer is unique in that it uses scaling matrix both  in  the  stopping
    condition tests and as Marquardt damping factor.

    Proper scaling is very important for the algorithm performance. It is less
    important for the quality of results, but still has some influence (it  is
    easier  to  converge  when  variables  are  properly  scaled, so premature
    stopping is possible when very badly scalled variables are  combined  with
    relaxed stopping conditions).

    INPUT PARAMETERS:
        State   -   structure stores algorithm state
        S       -   array[N], non-zero scaling coefficients
                    S[i] may be negative, sign doesn't matter.

      -- ALGLIB --
         Copyright 14.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmsetscale(minlmstate state, double[] s)

### `minlmsetstpmax` function

    /*************************************************************************
    This function sets maximum step length

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                    want to limit step length.

    Use this subroutine when you optimize target function which contains exp()
    or  other  fast  growing  functions,  and optimization algorithm makes too
    large  steps  which  leads  to overflow. This function allows us to reject
    steps  that  are  too  large  (and  therefore  expose  us  to the possible
    overflow) without actually calculating function value at the x+stp*d.

    NOTE: non-zero StpMax leads to moderate  performance  degradation  because
    intermediate  step  of  preconditioned L-BFGS optimization is incompatible
    with limits on step size.

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmsetstpmax(minlmstate state, double stpmax)

### `minlmsetxrep` function

    /*************************************************************************
    This function turns on/off reporting.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        NeedXRep-   whether iteration reports are needed or not

    If NeedXRep is True, algorithm will call rep() callback function if  it is
    provided to MinLMOptimize(). Both Levenberg-Marquardt and internal  L-BFGS
    iterations are reported.

      -- ALGLIB --
         Copyright 02.04.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void minlmsetxrep(minlmstate state, bool needxrep)

### minlm_d_fgh example

    public static void function1_func(double[] x, ref double func, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
    }
    public static void function1_grad(double[] x, ref double func, double[] grad, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        // and its derivatives df/d0 and df/dx1
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
        grad[0] = 400*System.Math.Pow(x[0]+3,3);
        grad[1] = 4*System.Math.Pow(x[1]-3,3);
    }
    public static void function1_hess(double[] x, ref double func, double[] grad, double[,] hess, object obj)
    {
        // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
        // its derivatives df/d0 and df/dx1
        // and its Hessian.
        func = 100*System.Math.Pow(x[0]+3,4) + System.Math.Pow(x[1]-3,4);
        grad[0] = 400*System.Math.Pow(x[0]+3,3);
        grad[1] = 4*System.Math.Pow(x[1]-3,3);
        hess[0,0] = 1200*System.Math.Pow(x[0]+3,2);
        hess[0,1] = 0;
        hess[1,0] = 0;
        hess[1,1] = 12*System.Math.Pow(x[1]-3,2);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of F(x0,x1) = 100*(x0+3)^4+(x1-3)^4
        // using "FGH" mode of the Levenberg-Marquardt optimizer.
        //
        // F is treated like a monolitic function without internal structure,
        // i.e. we do NOT represent it as a sum of squares.
        //
        // Optimization algorithm uses:
        // * function value F(x0,x1)
        // * gradient G={dF/dxi}
        // * Hessian H={d2F/(dxi*dxj)}
        //
        double[] x = new double[]{0,0};
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        int maxits = 0;
        alglib.minlmstate state;
        alglib.minlmreport rep;

        alglib.minlmcreatefgh(x, out state);
        alglib.minlmsetcond(state, epsg, epsf, epsx, maxits);
        alglib.minlmoptimize(state, function1_func, function1_grad, function1_hess, null, null);
        alglib.minlmresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,+3]
        System.Console.ReadLine();
        return 0;
    }

### minlm_d_restarts example

    public static void  function1_fvec(double[] x, double[] fi, object obj)
    {
        //
        // this callback calculates
        // f0(x0,x1) = 100*(x0+3)^4,
        // f1(x0,x1) = (x1-3)^4
        //
        fi[0] = 10*System.Math.Pow(x[0]+3,2);
        fi[1] = System.Math.Pow(x[1]-3,2);
    }
    public static void  function2_fvec(double[] x, double[] fi, object obj)
    {
        //
        // this callback calculates
        // f0(x0,x1) = 100*(x0+3)^4,
        // f1(x0,x1) = (x1-3)^4
        //
        fi[0] = x[0]*x[0]+1;
        fi[1] = x[1]-1;
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where
        //
        //     f0(x0,x1) = 10*(x0+3)^2
        //     f1(x0,x1) = (x1-3)^2
        //
        // using several starting points and efficient restarts.
        //
        double[] x;
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        int maxits = 0;
        alglib.minlmstate state;
        alglib.minlmreport rep;

        //
        // create optimizer using minlmcreatev()
        //
        x = new double[]{10,10};
        alglib.minlmcreatev(2, x, 0.0001, out state);
        alglib.minlmsetcond(state, epsg, epsf, epsx, maxits);
        alglib.minlmoptimize(state, function1_fvec, null, null);
        alglib.minlmresults(state, out x, out rep);
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,+3]

        //
        // restart optimizer using minlmrestartfrom()
        //
        // we can use different starting point, different function,
        // different stopping conditions, but problem size
        // must remain unchanged.
        //
        x = new double[]{4,4};
        alglib.minlmrestartfrom(state, x);
        alglib.minlmoptimize(state, function2_fvec, null, null);
        alglib.minlmresults(state, out x, out rep);
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [0,1]
        System.Console.ReadLine();
        return 0;
    }

### minlm_d_v example

    public static void  function1_fvec(double[] x, double[] fi, object obj)
    {
        //
        // this callback calculates
        // f0(x0,x1) = 100*(x0+3)^4,
        // f1(x0,x1) = (x1-3)^4
        //
        fi[0] = 10*System.Math.Pow(x[0]+3,2);
        fi[1] = System.Math.Pow(x[1]-3,2);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where
        //
        //     f0(x0,x1) = 10*(x0+3)^2
        //     f1(x0,x1) = (x1-3)^2
        //
        // using "V" mode of the Levenberg-Marquardt optimizer.
        //
        // Optimization algorithm uses:
        // * function vector f[] = {f1,f2}
        //
        // No other information (Jacobian, gradient, etc.) is needed.
        //
        double[] x = new double[]{0,0};
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        int maxits = 0;
        alglib.minlmstate state;
        alglib.minlmreport rep;

        alglib.minlmcreatev(2, x, 0.0001, out state);
        alglib.minlmsetcond(state, epsg, epsf, epsx, maxits);
        alglib.minlmoptimize(state, function1_fvec, null, null);
        alglib.minlmresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,+3]
        System.Console.ReadLine();
        return 0;
    }

### minlm_d_vb example

    public static void  function1_fvec(double[] x, double[] fi, object obj)
    {
        //
        // this callback calculates
        // f0(x0,x1) = 100*(x0+3)^4,
        // f1(x0,x1) = (x1-3)^4
        //
        fi[0] = 10*System.Math.Pow(x[0]+3,2);
        fi[1] = System.Math.Pow(x[1]-3,2);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where
        //
        //     f0(x0,x1) = 10*(x0+3)^2
        //     f1(x0,x1) = (x1-3)^2
        //
        // with boundary constraints
        //
        //     -1 <= x0 <= +1
        //     -1 <= x1 <= +1
        //
        // using "V" mode of the Levenberg-Marquardt optimizer.
        //
        // Optimization algorithm uses:
        // * function vector f[] = {f1,f2}
        //
        // No other information (Jacobian, gradient, etc.) is needed.
        //
        double[] x = new double[]{0,0};
        double[] bndl = new double[]{-1,-1};
        double[] bndu = new double[]{+1,+1};
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        int maxits = 0;
        alglib.minlmstate state;
        alglib.minlmreport rep;

        alglib.minlmcreatev(2, x, 0.0001, out state);
        alglib.minlmsetbc(state, bndl, bndu);
        alglib.minlmsetcond(state, epsg, epsf, epsx, maxits);
        alglib.minlmoptimize(state, function1_fvec, null, null);
        alglib.minlmresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-1,+1]
        System.Console.ReadLine();
        return 0;
    }

### minlm_d_vj example

    public static void  function1_fvec(double[] x, double[] fi, object obj)
    {
        //
        // this callback calculates
        // f0(x0,x1) = 100*(x0+3)^4,
        // f1(x0,x1) = (x1-3)^4
        //
        fi[0] = 10*System.Math.Pow(x[0]+3,2);
        fi[1] = System.Math.Pow(x[1]-3,2);
    }
    public static void  function1_jac(double[] x, double[] fi, double[,] jac, object obj)
    {
        // this callback calculates
        // f0(x0,x1) = 100*(x0+3)^4,
        // f1(x0,x1) = (x1-3)^4
        // and Jacobian matrix J = [dfi/dxj]
        fi[0] = 10*System.Math.Pow(x[0]+3,2);
        fi[1] = System.Math.Pow(x[1]-3,2);
        jac[0,0] = 20*(x[0]+3);
        jac[0,1] = 0;
        jac[1,0] = 0;
        jac[1,1] = 2*(x[1]-3);
    }
    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where
        //
        //     f0(x0,x1) = 10*(x0+3)^2
        //     f1(x0,x1) = (x1-3)^2
        //
        // using "VJ" mode of the Levenberg-Marquardt optimizer.
        //
        // Optimization algorithm uses:
        // * function vector f[] = {f1,f2}
        // * Jacobian matrix J = {dfi/dxj}.
        //
        double[] x = new double[]{0,0};
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        int maxits = 0;
        alglib.minlmstate state;
        alglib.minlmreport rep;

        alglib.minlmcreatevj(2, x, out state);
        alglib.minlmsetcond(state, epsg, epsf, epsx, maxits);
        alglib.minlmoptimize(state, function1_fvec, function1_jac, null, null);
        alglib.minlmresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [-3,+3]
        System.Console.ReadLine();
        return 0;
    }

## `minqp` subpackage

### Classes

minqpreport
minqpstate

### Functions

minqpcreate
minqpoptimize
minqpresults
minqpresultsbuf
minqpsetalgocholesky
minqpsetbc
minqpsetlinearterm
minqpsetorigin
minqpsetquadraticterm
minqpsetstartingpoint

### Examples

minqp_d_bc1 |  | Constrained dense quadratic programming
minqp_d_u1 |  | Unconstrained dense quadratic programming

### `minqpreport` class

    /*************************************************************************
    This structure stores optimization report:
    * InnerIterationsCount      number of inner iterations
    * OuterIterationsCount      number of outer iterations
    * NCholesky                 number of Cholesky decomposition
    * NMV                       number of matrix-vector products
                                (only products calculated as part of iterative
                                process are counted)
    * TerminationType           completion code (see below)

    Completion codes:
    * -5    inappropriate solver was used:
            * Cholesky solver for semidefinite or indefinite problems
            * Cholesky solver for problems with non-boundary constraints
    * -3    inconsistent constraints (or, maybe, feasible point is
            too hard to find). If you are sure that constraints are feasible,
            try to restart optimizer with better initial approximation.
    *  4    successful completion
    *  5    MaxIts steps was taken
    *  7    stopping conditions are too stringent,
            further improvement is impossible,
            X contains best point found so far.
    *************************************************************************/
    public class minqpreport
    {
        public int                  inneriterationscount;
        public int                  outeriterationscount;
        public int                  nmv;
        public int                  ncholesky;
        public int                  terminationtype;
    }

### `minqpstate` class

    /*************************************************************************
    This object stores nonlinear optimizer state.
    You should use functions provided by MinQP subpackage to work with this
    object
    *************************************************************************/
    public class minqpstate
    {
    }

### `minqpcreate` function

    /*************************************************************************
                        CONSTRAINED QUADRATIC PROGRAMMING

    The subroutine creates QP optimizer. After initial creation,  it  contains
    default optimization problem with zero quadratic and linear terms  and  no
    constraints. You should set quadratic/linear terms with calls to functions
    provided by MinQP subpackage.

    INPUT PARAMETERS:
        N       -   problem size

    OUTPUT PARAMETERS:
        State   -   optimizer with zero quadratic/linear terms
                    and no constraints

      -- ALGLIB --
         Copyright 11.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minqpcreate(int n, out minqpstate state)

Examples: [1] [2]

### `minqpoptimize` function

    /*************************************************************************
    This function solves quadratic programming problem.
    You should call it after setting solver options with MinQPSet...() calls.

    INPUT PARAMETERS:
        State   -   algorithm state

    You should use MinQPResults() function to access results after calls
    to this function.

      -- ALGLIB --
         Copyright 11.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minqpoptimize(minqpstate state)

Examples: [1] [2]

### `minqpresults` function

    /*************************************************************************
    QP solver results

    INPUT PARAMETERS:
        State   -   algorithm state

    OUTPUT PARAMETERS:
        X       -   array[0..N-1], solution
        Rep     -   optimization report. You should check Rep.TerminationType,
                    which contains completion code, and you may check  another
                    fields which contain another information  about  algorithm
                    functioning.

      -- ALGLIB --
         Copyright 11.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minqpresults(
        minqpstate state,
        out double[] x,
        out minqpreport rep)

Examples: [1] [2]

### `minqpresultsbuf` function

    /*************************************************************************
    QP results

    Buffered implementation of MinQPResults() which uses pre-allocated  buffer
    to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
    intended to be used in the inner cycles of performance critical algorithms
    where array reallocation penalty is too large to be ignored.

      -- ALGLIB --
         Copyright 11.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minqpresultsbuf(
        minqpstate state,
        ref double[] x,
        minqpreport rep)

### `minqpsetalgocholesky` function

    /*************************************************************************
    This function tells solver to use Cholesky-based algorithm.

    Cholesky-based algorithm can be used when:
    * problem is convex
    * there is no constraints or only boundary constraints are present

    This algorithm has O(N^3) complexity for unconstrained problem and  is  up
    to several times slower on bound constrained  problems  (these  additional
    iterations are needed to identify active constraints).

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state

      -- ALGLIB --
         Copyright 11.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minqpsetalgocholesky(minqpstate state)

### `minqpsetbc` function

    /*************************************************************************
    This function sets boundary constraints for QP solver

    Boundary constraints are inactive by default (after initial creation).
    After  being  set,  they  are  preserved  until explicitly turned off with
    another SetBC() call.

    INPUT PARAMETERS:
        State   -   structure stores algorithm state
        BndL    -   lower bounds, array[N].
                    If some (all) variables are unbounded, you may specify
                    very small number or -INF (latter is recommended because
                    it will allow solver to use better algorithm).
        BndU    -   upper bounds, array[N].
                    If some (all) variables are unbounded, you may specify
                    very large number or +INF (latter is recommended because
                    it will allow solver to use better algorithm).

    NOTE: it is possible to specify BndL[i]=BndU[i]. In this case I-th
    variable will be "frozen" at X[i]=BndL[i]=BndU[i].

      -- ALGLIB --
         Copyright 11.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minqpsetbc(
        minqpstate state,
        double[] bndl,
        double[] bndu)

### `minqpsetlinearterm` function

    /*************************************************************************
    This function sets linear term for QP solver.

    By default, linear term is zero.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        B       -   linear term, array[N].

      -- ALGLIB --
         Copyright 11.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minqpsetlinearterm(minqpstate state, double[] b)

Examples: [1] [2]

### `minqpsetorigin` function

    /*************************************************************************
    This  function sets origin for QP solver. By default, following QP program
    is solved:

        min(0.5*x'*A*x+b'*x)

    This function allows to solve different problem:

        min(0.5*(x-x_origin)'*A*(x-x_origin)+b'*(x-x_origin))

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        XOrigin -   origin, array[N].

      -- ALGLIB --
         Copyright 11.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minqpsetorigin(minqpstate state, double[] xorigin)

### `minqpsetquadraticterm` function

    /*************************************************************************
    This function sets quadratic term for QP solver.

    By default quadratic term is zero.

    IMPORTANT: this solver minimizes following  function:
        f(x) = 0.5*x'*A*x + b'*x.
    Note that quadratic term has 0.5 before it. So if  you  want  to  minimize
        f(x) = x^2 + x
    you should rewrite your problem as follows:
        f(x) = 0.5*(2*x^2) + x
    and your matrix A will be equal to [[2.0]], not to [[1.0]]

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        A       -   matrix, array[N,N]
        IsUpper -   (optional) storage type:
                    * if True, symmetric matrix  A  is  given  by  its  upper
                      triangle, and the lower triangle isn’t used
                    * if False, symmetric matrix  A  is  given  by  its lower
                      triangle, and the upper triangle isn’t used
                    * if not given, both lower and upper  triangles  must  be
                      filled.

      -- ALGLIB --
         Copyright 11.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minqpsetquadraticterm(minqpstate state, double[,] a)
    public static void minqpsetquadraticterm(
        minqpstate state,
        double[,] a,
        bool isupper)

Examples: [1] [2]

### `minqpsetstartingpoint` function

    /*************************************************************************
    This function sets starting point for QP solver. It is useful to have
    good initial approximation to the solution, because it will increase
    speed of convergence and identification of active constraints.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        X       -   starting point, array[N].

      -- ALGLIB --
         Copyright 11.01.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void minqpsetstartingpoint(minqpstate state, double[] x)

Examples: [1] [2]

### minqp_d_bc1 example

    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1
        // subject to bound constraints 0<=x0<=2.5, 0<=x1<=2.5
        //
        // Exact solution is [x0,x1] = [2.5,2]
        //
        // We provide algorithm with starting point. With such small problem good starting
        // point is not really necessary, but with high-dimensional problem it can save us
        // a lot of time.
        //
        // IMPORTANT: this solver minimizes  following  function:
        //     f(x) = 0.5*x'*A*x + b'*x.
        // Note that quadratic term has 0.5 before it. So if you want to minimize
        // quadratic function, you should rewrite it in such way that quadratic term
        // is multiplied by 0.5 too.
        // For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as
        //     f(x) = 0.5*(2*x0^2+2*x1^2) + ....
        // and pass diag(2,2) as quadratic term - NOT diag(1,1)!
        //
        double[,] a = new double[,]{{2,0},{0,2}};
        double[] b = new double[]{-6,-4};
        double[] x0 = new double[]{0,1};
        double[] bndl = new double[]{0.0,0.0};
        double[] bndu = new double[]{2.5,2.5};
        double[] x;
        alglib.minqpstate state;
        alglib.minqpreport rep;

        alglib.minqpcreate(2, out state);
        alglib.minqpsetquadraticterm(state, a);
        alglib.minqpsetlinearterm(state, b);
        alglib.minqpsetstartingpoint(state, x0);
        alglib.minqpsetbc(state, bndl, bndu);
        alglib.minqpoptimize(state);
        alglib.minqpresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [2.5,2]
        System.Console.ReadLine();
        return 0;
    }

### minqp_d_u1 example

    public static int Main(string[] args)
    {
        //
        // This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1
        //
        // Exact solution is [x0,x1] = [3,2]
        //
        // We provide algorithm with starting point, although in this case
        // (dense matrix, no constraints) it can work without such information.
        //
        // IMPORTANT: this solver minimizes  following  function:
        //     f(x) = 0.5*x'*A*x + b'*x.
        // Note that quadratic term has 0.5 before it. So if you want to minimize
        // quadratic function, you should rewrite it in such way that quadratic term
        // is multiplied by 0.5 too.
        // For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as
        //     f(x) = 0.5*(2*x0^2+2*x1^2) + ....
        // and pass diag(2,2) as quadratic term - NOT diag(1,1)!
        //
        double[,] a = new double[,]{{2,0},{0,2}};
        double[] b = new double[]{-6,-4};
        double[] x0 = new double[]{0,1};
        double[] x;
        alglib.minqpstate state;
        alglib.minqpreport rep;

        alglib.minqpcreate(2, out state);
        alglib.minqpsetquadraticterm(state, a);
        alglib.minqpsetlinearterm(state, b);
        alglib.minqpsetstartingpoint(state, x0);
        alglib.minqpoptimize(state);
        alglib.minqpresults(state, out x, out rep);

        System.Console.WriteLine("{0}", rep.terminationtype); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(x,2)); // EXPECTED: [3,2]
        System.Console.ReadLine();
        return 0;
    }

## `mlpbase` subpackage

### Classes

multilayerperceptron

### Functions

mlpactivationfunction
mlpavgce
mlpavgerror
mlpavgrelerror
mlpclserror
mlpcreate0
mlpcreate1
mlpcreate2
mlpcreateb0
mlpcreateb1
mlpcreateb2
mlpcreatec0
mlpcreatec1
mlpcreatec2
mlpcreater0
mlpcreater1
mlpcreater2
mlperror
mlperrorn
mlpgetinputscaling
mlpgetlayerscount
mlpgetlayersize
mlpgetneuroninfo
mlpgetoutputscaling
mlpgetweight
mlpgrad
mlpgradbatch
mlpgradn
mlpgradnbatch
mlphessianbatch
mlphessiannbatch
mlpissoftmax
mlpprocess
mlpprocessi
mlpproperties
mlprandomize
mlprandomizefull
mlprelclserror
mlprmserror
mlpserialize
mlpsetinputscaling
mlpsetneuroninfo
mlpsetoutputscaling
mlpsetweight
mlpunserialize

### Examples

### `multilayerperceptron` class

    /*************************************************************************

    *************************************************************************/
    public class multilayerperceptron
    {
    }

### `mlpactivationfunction` function

    /*************************************************************************
    Neural network activation function

    INPUT PARAMETERS:
        NET         -   neuron input
        K           -   function index (zero for linear function)

    OUTPUT PARAMETERS:
        F           -   function
        DF          -   its derivative
        D2F         -   its second derivative

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpactivationfunction(
        double net,
        int k,
        out double f,
        out double df,
        out double d2f)

### `mlpavgce` function

    /*************************************************************************
    Average cross-entropy (in bits per element) on the test set

    INPUT PARAMETERS:
        Network -   neural network
        XY      -   test set
        NPoints -   test set size

    RESULT:
        CrossEntropy/(NPoints*LN(2)).
        Zero if network solves regression task.

      -- ALGLIB --
         Copyright 08.01.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double mlpavgce(
        multilayerperceptron network,
        double[,] xy,
        int npoints)

### `mlpavgerror` function

    /*************************************************************************
    Average error on the test set

    INPUT PARAMETERS:
        Network -   neural network
        XY      -   test set
        NPoints -   test set size

    RESULT:
        Its meaning for regression task is obvious. As for
        classification task, it means average error when estimating posterior
        probabilities.

      -- ALGLIB --
         Copyright 11.03.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double mlpavgerror(
        multilayerperceptron network,
        double[,] xy,
        int npoints)

### `mlpavgrelerror` function

    /*************************************************************************
    Average relative error on the test set

    INPUT PARAMETERS:
        Network -   neural network
        XY      -   test set
        NPoints -   test set size

    RESULT:
        Its meaning for regression task is obvious. As for
        classification task, it means average relative error when estimating
        posterior probability of belonging to the correct class.

      -- ALGLIB --
         Copyright 11.03.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double mlpavgrelerror(
        multilayerperceptron network,
        double[,] xy,
        int npoints)

### `mlpclserror` function

    /*************************************************************************
    Classification error

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static int mlpclserror(
        multilayerperceptron network,
        double[,] xy,
        int ssize)

### `mlpcreate0` function

    /*************************************************************************
    Creates  neural  network  with  NIn  inputs,  NOut outputs, without hidden
    layers, with linear output layer. Network weights are  filled  with  small
    random values.

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreate0(
        int nin,
        int nout,
        out multilayerperceptron network)

### `mlpcreate1` function

    /*************************************************************************
    Same  as  MLPCreate0,  but  with  one  hidden  layer  (NHid  neurons) with
    non-linear activation function. Output layer is linear.

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreate1(
        int nin,
        int nhid,
        int nout,
        out multilayerperceptron network)

### `mlpcreate2` function

    /*************************************************************************
    Same as MLPCreate0, but with two hidden layers (NHid1 and  NHid2  neurons)
    with non-linear activation function. Output layer is linear.
     $ALL

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreate2(
        int nin,
        int nhid1,
        int nhid2,
        int nout,
        out multilayerperceptron network)

### `mlpcreateb0` function

    /*************************************************************************
    Creates  neural  network  with  NIn  inputs,  NOut outputs, without hidden
    layers with non-linear output layer. Network weights are filled with small
    random values.

    Activation function of the output layer takes values:

        (B, +INF), if D>=0

    or

        (-INF, B), if D<0.

      -- ALGLIB --
         Copyright 30.03.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreateb0(
        int nin,
        int nout,
        double b,
        double d,
        out multilayerperceptron network)

### `mlpcreateb1` function

    /*************************************************************************
    Same as MLPCreateB0 but with non-linear hidden layer.

      -- ALGLIB --
         Copyright 30.03.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreateb1(
        int nin,
        int nhid,
        int nout,
        double b,
        double d,
        out multilayerperceptron network)

### `mlpcreateb2` function

    /*************************************************************************
    Same as MLPCreateB0 but with two non-linear hidden layers.

      -- ALGLIB --
         Copyright 30.03.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreateb2(
        int nin,
        int nhid1,
        int nhid2,
        int nout,
        double b,
        double d,
        out multilayerperceptron network)

### `mlpcreatec0` function

    /*************************************************************************
    Creates classifier network with NIn  inputs  and  NOut  possible  classes.
    Network contains no hidden layers and linear output  layer  with  SOFTMAX-
    normalization  (so  outputs  sums  up  to  1.0  and  converge to posterior
    probabilities).

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreatec0(
        int nin,
        int nout,
        out multilayerperceptron network)

### `mlpcreatec1` function

    /*************************************************************************
    Same as MLPCreateC0, but with one non-linear hidden layer.

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreatec1(
        int nin,
        int nhid,
        int nout,
        out multilayerperceptron network)

### `mlpcreatec2` function

    /*************************************************************************
    Same as MLPCreateC0, but with two non-linear hidden layers.

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreatec2(
        int nin,
        int nhid1,
        int nhid2,
        int nout,
        out multilayerperceptron network)

### `mlpcreater0` function

    /*************************************************************************
    Creates  neural  network  with  NIn  inputs,  NOut outputs, without hidden
    layers with non-linear output layer. Network weights are filled with small
    random values. Activation function of the output layer takes values [A,B].

      -- ALGLIB --
         Copyright 30.03.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreater0(
        int nin,
        int nout,
        double a,
        double b,
        out multilayerperceptron network)

### `mlpcreater1` function

    /*************************************************************************
    Same as MLPCreateR0, but with non-linear hidden layer.

      -- ALGLIB --
         Copyright 30.03.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreater1(
        int nin,
        int nhid,
        int nout,
        double a,
        double b,
        out multilayerperceptron network)

### `mlpcreater2` function

    /*************************************************************************
    Same as MLPCreateR0, but with two non-linear hidden layers.

      -- ALGLIB --
         Copyright 30.03.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpcreater2(
        int nin,
        int nhid1,
        int nhid2,
        int nout,
        double a,
        double b,
        out multilayerperceptron network)

### `mlperror` function

    /*************************************************************************
    Error function for neural network, internal subroutine.

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static double mlperror(
        multilayerperceptron network,
        double[,] xy,
        int ssize)

### `mlperrorn` function

    /*************************************************************************
    Natural error function for neural network, internal subroutine.

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static double mlperrorn(
        multilayerperceptron network,
        double[,] xy,
        int ssize)

### `mlpgetinputscaling` function

    /*************************************************************************
    This function returns offset/scaling coefficients for I-th input of the
    network.

    INPUT PARAMETERS:
        Network     -   network
        I           -   input index

    OUTPUT PARAMETERS:
        Mean        -   mean term
        Sigma       -   sigma term, guaranteed to be nonzero.

    I-th input is passed through linear transformation
        IN[i] = (IN[i]-Mean)/Sigma
    before feeding to the network

      -- ALGLIB --
         Copyright 25.03.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpgetinputscaling(
        multilayerperceptron network,
        int i,
        out double mean,
        out double sigma)

### `mlpgetlayerscount` function

    /*************************************************************************
    This function returns total number of layers (including input, hidden and
    output layers).

      -- ALGLIB --
         Copyright 25.03.2011 by Bochkanov Sergey
    *************************************************************************/
    public static int mlpgetlayerscount(multilayerperceptron network)

### `mlpgetlayersize` function

    /*************************************************************************
    This function returns size of K-th layer.

    K=0 corresponds to input layer, K=CNT-1 corresponds to output layer.

    Size of the output layer is always equal to the number of outputs, although
    when we have softmax-normalized network, last neuron doesn't have any
    connections - it is just zero.

      -- ALGLIB --
         Copyright 25.03.2011 by Bochkanov Sergey
    *************************************************************************/
    public static int mlpgetlayersize(multilayerperceptron network, int k)

### `mlpgetneuroninfo` function

    /*************************************************************************
    This function returns information about Ith neuron of Kth layer

    INPUT PARAMETERS:
        Network     -   network
        K           -   layer index
        I           -   neuron index (within layer)

    OUTPUT PARAMETERS:
        FKind       -   activation function type (used by MLPActivationFunction())
                        this value is zero for input or linear neurons
        Threshold   -   also called offset, bias
                        zero for input neurons

    NOTE: this function throws exception if layer or neuron with  given  index
    do not exists.

      -- ALGLIB --
         Copyright 25.03.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpgetneuroninfo(
        multilayerperceptron network,
        int k,
        int i,
        out int fkind,
        out double threshold)

### `mlpgetoutputscaling` function

    /*************************************************************************
    This function returns offset/scaling coefficients for I-th output of the
    network.

    INPUT PARAMETERS:
        Network     -   network
        I           -   input index

    OUTPUT PARAMETERS:
        Mean        -   mean term
        Sigma       -   sigma term, guaranteed to be nonzero.

    I-th output is passed through linear transformation
        OUT[i] = OUT[i]*Sigma+Mean
    before returning it to user. In case we have SOFTMAX-normalized network,
    we return (Mean,Sigma)=(0.0,1.0).

      -- ALGLIB --
         Copyright 25.03.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpgetoutputscaling(
        multilayerperceptron network,
        int i,
        out double mean,
        out double sigma)

### `mlpgetweight` function

    /*************************************************************************
    This function returns information about connection from I0-th neuron of
    K0-th layer to I1-th neuron of K1-th layer.

    INPUT PARAMETERS:
        Network     -   network
        K0          -   layer index
        I0          -   neuron index (within layer)
        K1          -   layer index
        I1          -   neuron index (within layer)

    RESULT:
        connection weight (zero for non-existent connections)

    This function:
    1. throws exception if layer or neuron with given index do not exists.
    2. returns zero if neurons exist, but there is no connection between them

      -- ALGLIB --
         Copyright 25.03.2011 by Bochkanov Sergey
    *************************************************************************/
    public static double mlpgetweight(
        multilayerperceptron network,
        int k0,
        int i0,
        int k1,
        int i1)

### `mlpgrad` function

    /*************************************************************************
    Gradient calculation

    INPUT PARAMETERS:
        Network -   network initialized with one of the network creation funcs
        X       -   input vector, length of array must be at least NIn
        DesiredY-   desired outputs, length of array must be at least NOut
        Grad    -   possibly preallocated array. If size of array is smaller
                    than WCount, it will be reallocated. It is recommended to
                    reuse previously allocated array to reduce allocation
                    overhead.

    OUTPUT PARAMETERS:
        E       -   error function, SUM(sqr(y[i]-desiredy[i])/2,i)
        Grad    -   gradient of E with respect to weights of network, array[WCount]

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpgrad(
        multilayerperceptron network,
        double[] x,
        double[] desiredy,
        out double e,
        ref double[] grad)

### `mlpgradbatch` function

    /*************************************************************************
    Batch gradient calculation for a set of inputs/outputs

    INPUT PARAMETERS:
        Network -   network initialized with one of the network creation funcs
        XY      -   set of inputs/outputs; one sample = one row;
                    first NIn columns contain inputs,
                    next NOut columns - desired outputs.
        SSize   -   number of elements in XY
        Grad    -   possibly preallocated array. If size of array is smaller
                    than WCount, it will be reallocated. It is recommended to
                    reuse previously allocated array to reduce allocation
                    overhead.

    OUTPUT PARAMETERS:
        E       -   error function, SUM(sqr(y[i]-desiredy[i])/2,i)
        Grad    -   gradient of E with respect to weights of network, array[WCount]

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpgradbatch(
        multilayerperceptron network,
        double[,] xy,
        int ssize,
        out double e,
        ref double[] grad)

### `mlpgradn` function

    /*************************************************************************
    Gradient calculation (natural error function is used)

    INPUT PARAMETERS:
        Network -   network initialized with one of the network creation funcs
        X       -   input vector, length of array must be at least NIn
        DesiredY-   desired outputs, length of array must be at least NOut
        Grad    -   possibly preallocated array. If size of array is smaller
                    than WCount, it will be reallocated. It is recommended to
                    reuse previously allocated array to reduce allocation
                    overhead.

    OUTPUT PARAMETERS:
        E       -   error function, sum-of-squares for regression networks,
                    cross-entropy for classification networks.
        Grad    -   gradient of E with respect to weights of network, array[WCount]

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpgradn(
        multilayerperceptron network,
        double[] x,
        double[] desiredy,
        out double e,
        ref double[] grad)

### `mlpgradnbatch` function

    /*************************************************************************
    Batch gradient calculation for a set of inputs/outputs
    (natural error function is used)

    INPUT PARAMETERS:
        Network -   network initialized with one of the network creation funcs
        XY      -   set of inputs/outputs; one sample = one row;
                    first NIn columns contain inputs,
                    next NOut columns - desired outputs.
        SSize   -   number of elements in XY
        Grad    -   possibly preallocated array. If size of array is smaller
                    than WCount, it will be reallocated. It is recommended to
                    reuse previously allocated array to reduce allocation
                    overhead.

    OUTPUT PARAMETERS:
        E       -   error function, sum-of-squares for regression networks,
                    cross-entropy for classification networks.
        Grad    -   gradient of E with respect to weights of network, array[WCount]

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpgradnbatch(
        multilayerperceptron network,
        double[,] xy,
        int ssize,
        out double e,
        ref double[] grad)

### `mlphessianbatch` function

    /*************************************************************************
    Batch Hessian calculation using R-algorithm.
    Internal subroutine.

      -- ALGLIB --
         Copyright 26.01.2008 by Bochkanov Sergey.

         Hessian calculation based on R-algorithm described in
         "Fast Exact Multiplication by the Hessian",
         B. A. Pearlmutter,
         Neural Computation, 1994.
    *************************************************************************/
    public static void mlphessianbatch(
        multilayerperceptron network,
        double[,] xy,
        int ssize,
        out double e,
        ref double[] grad,
        ref double[,] h)

### `mlphessiannbatch` function

    /*************************************************************************
    Batch Hessian calculation (natural error function) using R-algorithm.
    Internal subroutine.

      -- ALGLIB --
         Copyright 26.01.2008 by Bochkanov Sergey.

         Hessian calculation based on R-algorithm described in
         "Fast Exact Multiplication by the Hessian",
         B. A. Pearlmutter,
         Neural Computation, 1994.
    *************************************************************************/
    public static void mlphessiannbatch(
        multilayerperceptron network,
        double[,] xy,
        int ssize,
        out double e,
        ref double[] grad,
        ref double[,] h)

### `mlpissoftmax` function

    /*************************************************************************
    Tells whether network is SOFTMAX-normalized (i.e. classifier) or not.

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static bool mlpissoftmax(multilayerperceptron network)

### `mlpprocess` function

    /*************************************************************************
    Procesing

    INPUT PARAMETERS:
        Network -   neural network
        X       -   input vector,  array[0..NIn-1].

    OUTPUT PARAMETERS:
        Y       -   result. Regression estimate when solving regression  task,
                    vector of posterior probabilities for classification task.

    See also MLPProcessI

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpprocess(
        multilayerperceptron network,
        double[] x,
        ref double[] y)

### `mlpprocessi` function

    /*************************************************************************
    'interactive'  variant  of  MLPProcess  for  languages  like  Python which
    support constructs like "Y = MLPProcess(NN,X)" and interactive mode of the
    interpreter

    This function allocates new array on each call,  so  it  is  significantly
    slower than its 'non-interactive' counterpart, but it is  more  convenient
    when you call it from command line.

      -- ALGLIB --
         Copyright 21.09.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpprocessi(
        multilayerperceptron network,
        double[] x,
        out double[] y)

### `mlpproperties` function

    /*************************************************************************
    Returns information about initialized network: number of inputs, outputs,
    weights.

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpproperties(
        multilayerperceptron network,
        out int nin,
        out int nout,
        out int wcount)

### `mlprandomize` function

    /*************************************************************************
    Randomization of neural network weights

      -- ALGLIB --
         Copyright 06.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlprandomize(multilayerperceptron network)

### `mlprandomizefull` function

    /*************************************************************************
    Randomization of neural network weights and standartisator

      -- ALGLIB --
         Copyright 10.03.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void mlprandomizefull(multilayerperceptron network)

### `mlprelclserror` function

    /*************************************************************************
    Relative classification error on the test set

    INPUT PARAMETERS:
        Network -   network
        XY      -   test set
        NPoints -   test set size

    RESULT:
        percent of incorrectly classified cases. Works both for
        classifier networks and general purpose networks used as
        classifiers.

      -- ALGLIB --
         Copyright 25.12.2008 by Bochkanov Sergey
    *************************************************************************/
    public static double mlprelclserror(
        multilayerperceptron network,
        double[,] xy,
        int npoints)

### `mlprmserror` function

    /*************************************************************************
    RMS error on the test set

    INPUT PARAMETERS:
        Network -   neural network
        XY      -   test set
        NPoints -   test set size

    RESULT:
        root mean square error.
        Its meaning for regression task is obvious. As for
        classification task, RMS error means error when estimating posterior
        probabilities.

      -- ALGLIB --
         Copyright 04.11.2007 by Bochkanov Sergey
    *************************************************************************/
    public static double mlprmserror(
        multilayerperceptron network,
        double[,] xy,
        int npoints)

### `mlpserialize` function

    /*************************************************************************
    This function serializes data structure to string.

    Important properties of s_out:
    * it contains alphanumeric characters, dots, underscores, minus signs
    * these symbols are grouped into words, which are separated by spaces
      and Windows-style (CR+LF) newlines
    * although  serializer  uses  spaces and CR+LF as separators, you can
      replace any separator character by arbitrary combination of spaces,
      tabs, Windows or Unix newlines. It allows flexible reformatting  of
      the  string  in  case you want to include it into text or XML file.
      But you should not insert separators into the middle of the "words"
      nor you should change case of letters.
    * s_out can be freely moved between 32-bit and 64-bit systems, little
      and big endian machines, and so on. You can serialize structure  on
      32-bit machine and unserialize it on 64-bit one (or vice versa), or
      serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also
      serialize  it  in  C++ version of ALGLIB and unserialize in C# one,
      and vice versa.
    *************************************************************************/
    public static void mlpserialize(multilayerperceptron obj, out string s_out)

### `mlpsetinputscaling` function

    /*************************************************************************
    This function sets offset/scaling coefficients for I-th input of the
    network.

    INPUT PARAMETERS:
        Network     -   network
        I           -   input index
        Mean        -   mean term
        Sigma       -   sigma term (if zero, will be replaced by 1.0)

    NTE: I-th input is passed through linear transformation
        IN[i] = (IN[i]-Mean)/Sigma
    before feeding to the network. This function sets Mean and Sigma.

      -- ALGLIB --
         Copyright 25.03.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpsetinputscaling(
        multilayerperceptron network,
        int i,
        double mean,
        double sigma)

### `mlpsetneuroninfo` function

    /*************************************************************************
    This function modifies information about Ith neuron of Kth layer

    INPUT PARAMETERS:
        Network     -   network
        K           -   layer index
        I           -   neuron index (within layer)
        FKind       -   activation function type (used by MLPActivationFunction())
                        this value must be zero for input neurons
                        (you can not set activation function for input neurons)
        Threshold   -   also called offset, bias
                        this value must be zero for input neurons
                        (you can not set threshold for input neurons)

    NOTES:
    1. this function throws exception if layer or neuron with given index do
       not exists.
    2. this function also throws exception when you try to set non-linear
       activation function for input neurons (any kind of network) or for output
       neurons of classifier network.
    3. this function throws exception when you try to set non-zero threshold for
       input neurons (any kind of network).

      -- ALGLIB --
         Copyright 25.03.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpsetneuroninfo(
        multilayerperceptron network,
        int k,
        int i,
        int fkind,
        double threshold)

### `mlpsetoutputscaling` function

    /*************************************************************************
    This function sets offset/scaling coefficients for I-th output of the
    network.

    INPUT PARAMETERS:
        Network     -   network
        I           -   input index
        Mean        -   mean term
        Sigma       -   sigma term (if zero, will be replaced by 1.0)

    OUTPUT PARAMETERS:

    NOTE: I-th output is passed through linear transformation
        OUT[i] = OUT[i]*Sigma+Mean
    before returning it to user. This function sets Sigma/Mean. In case we
    have SOFTMAX-normalized network, you can not set (Sigma,Mean) to anything
    other than(0.0,1.0) - this function will throw exception.

      -- ALGLIB --
         Copyright 25.03.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpsetoutputscaling(
        multilayerperceptron network,
        int i,
        double mean,
        double sigma)

### `mlpsetweight` function

    /*************************************************************************
    This function modifies information about connection from I0-th neuron of
    K0-th layer to I1-th neuron of K1-th layer.

    INPUT PARAMETERS:
        Network     -   network
        K0          -   layer index
        I0          -   neuron index (within layer)
        K1          -   layer index
        I1          -   neuron index (within layer)
        W           -   connection weight (must be zero for non-existent
                        connections)

    This function:
    1. throws exception if layer or neuron with given index do not exists.
    2. throws exception if you try to set non-zero weight for non-existent
       connection

      -- ALGLIB --
         Copyright 25.03.2011 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpsetweight(
        multilayerperceptron network,
        int k0,
        int i0,
        int k1,
        int i1,
        double w)

### `mlpunserialize` function

    /*************************************************************************
    This function unserializes data structure from string.
    *************************************************************************/
    public static void mlpunserialize(string s_in, out multilayerperceptron obj)

## `mlpe` subpackage

### Classes

mlpensemble

### Functions

mlpeavgce
mlpeavgerror
mlpeavgrelerror
mlpebagginglbfgs
mlpebagginglm
mlpecreate0
mlpecreate1
mlpecreate2
mlpecreateb0
mlpecreateb1
mlpecreateb2
mlpecreatec0
mlpecreatec1
mlpecreatec2
mlpecreatefromnetwork
mlpecreater0
mlpecreater1
mlpecreater2
mlpeissoftmax
mlpeprocess
mlpeprocessi
mlpeproperties
mlperandomize
mlperelclserror
mlpermserror
mlpetraines

### Examples

### `mlpensemble` class

    /*************************************************************************
    Neural networks ensemble
    *************************************************************************/
    public class mlpensemble
    {
    }

### `mlpeavgce` function

    /*************************************************************************
    Average cross-entropy (in bits per element) on the test set

    INPUT PARAMETERS:
        Ensemble-   ensemble
        XY      -   test set
        NPoints -   test set size

    RESULT:
        CrossEntropy/(NPoints*LN(2)).
        Zero if ensemble solves regression task.

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double mlpeavgce(
        mlpensemble ensemble,
        double[,] xy,
        int npoints)

### `mlpeavgerror` function

    /*************************************************************************
    Average error on the test set

    INPUT PARAMETERS:
        Ensemble-   ensemble
        XY      -   test set
        NPoints -   test set size

    RESULT:
        Its meaning for regression task is obvious. As for classification task
    it means average error when estimating posterior probabilities.

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double mlpeavgerror(
        mlpensemble ensemble,
        double[,] xy,
        int npoints)

### `mlpeavgrelerror` function

    /*************************************************************************
    Average relative error on the test set

    INPUT PARAMETERS:
        Ensemble-   ensemble
        XY      -   test set
        NPoints -   test set size

    RESULT:
        Its meaning for regression task is obvious. As for classification task
    it means average relative error when estimating posterior probabilities.

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double mlpeavgrelerror(
        mlpensemble ensemble,
        double[,] xy,
        int npoints)

### `mlpebagginglbfgs` function

    /*************************************************************************
    Training neural networks ensemble using  bootstrap  aggregating (bagging).
    L-BFGS algorithm is used as base training method.

    INPUT PARAMETERS:
        Ensemble    -   model with initialized geometry
        XY          -   training set
        NPoints     -   training set size
        Decay       -   weight decay coefficient, >=0.001
        Restarts    -   restarts, >0.
        WStep       -   stopping criterion, same as in MLPTrainLBFGS
        MaxIts      -   stopping criterion, same as in MLPTrainLBFGS

    OUTPUT PARAMETERS:
        Ensemble    -   trained model
        Info        -   return code:
                        * -8, if both WStep=0 and MaxIts=0
                        * -2, if there is a point with class number
                              outside of [0..NClasses-1].
                        * -1, if incorrect parameters was passed
                              (NPoints<0, Restarts<1).
                        *  2, if task has been solved.
        Rep         -   training report.
        OOBErrors   -   out-of-bag generalization error estimate

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpebagginglbfgs(
        mlpensemble ensemble,
        double[,] xy,
        int npoints,
        double decay,
        int restarts,
        double wstep,
        int maxits,
        out int info,
        out mlpreport rep,
        out mlpcvreport ooberrors)

### `mlpebagginglm` function

    /*************************************************************************
    Training neural networks ensemble using  bootstrap  aggregating (bagging).
    Modified Levenberg-Marquardt algorithm is used as base training method.

    INPUT PARAMETERS:
        Ensemble    -   model with initialized geometry
        XY          -   training set
        NPoints     -   training set size
        Decay       -   weight decay coefficient, >=0.001
        Restarts    -   restarts, >0.

    OUTPUT PARAMETERS:
        Ensemble    -   trained model
        Info        -   return code:
                        * -2, if there is a point with class number
                              outside of [0..NClasses-1].
                        * -1, if incorrect parameters was passed
                              (NPoints<0, Restarts<1).
                        *  2, if task has been solved.
        Rep         -   training report.
        OOBErrors   -   out-of-bag generalization error estimate

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpebagginglm(
        mlpensemble ensemble,
        double[,] xy,
        int npoints,
        double decay,
        int restarts,
        out int info,
        out mlpreport rep,
        out mlpcvreport ooberrors)

### `mlpecreate0` function

    /*************************************************************************
    Like MLPCreate0, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreate0(
        int nin,
        int nout,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreate1` function

    /*************************************************************************
    Like MLPCreate1, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreate1(
        int nin,
        int nhid,
        int nout,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreate2` function

    /*************************************************************************
    Like MLPCreate2, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreate2(
        int nin,
        int nhid1,
        int nhid2,
        int nout,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreateb0` function

    /*************************************************************************
    Like MLPCreateB0, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreateb0(
        int nin,
        int nout,
        double b,
        double d,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreateb1` function

    /*************************************************************************
    Like MLPCreateB1, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreateb1(
        int nin,
        int nhid,
        int nout,
        double b,
        double d,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreateb2` function

    /*************************************************************************
    Like MLPCreateB2, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreateb2(
        int nin,
        int nhid1,
        int nhid2,
        int nout,
        double b,
        double d,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreatec0` function

    /*************************************************************************
    Like MLPCreateC0, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreatec0(
        int nin,
        int nout,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreatec1` function

    /*************************************************************************
    Like MLPCreateC1, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreatec1(
        int nin,
        int nhid,
        int nout,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreatec2` function

    /*************************************************************************
    Like MLPCreateC2, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreatec2(
        int nin,
        int nhid1,
        int nhid2,
        int nout,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreatefromnetwork` function

    /*************************************************************************
    Creates ensemble from network. Only network geometry is copied.

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreatefromnetwork(
        multilayerperceptron network,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreater0` function

    /*************************************************************************
    Like MLPCreateR0, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreater0(
        int nin,
        int nout,
        double a,
        double b,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreater1` function

    /*************************************************************************
    Like MLPCreateR1, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreater1(
        int nin,
        int nhid,
        int nout,
        double a,
        double b,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpecreater2` function

    /*************************************************************************
    Like MLPCreateR2, but for ensembles.

      -- ALGLIB --
         Copyright 18.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpecreater2(
        int nin,
        int nhid1,
        int nhid2,
        int nout,
        double a,
        double b,
        int ensemblesize,
        out mlpensemble ensemble)

### `mlpeissoftmax` function

    /*************************************************************************
    Return normalization type (whether ensemble is SOFTMAX-normalized or not).

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static bool mlpeissoftmax(mlpensemble ensemble)

### `mlpeprocess` function

    /*************************************************************************
    Procesing

    INPUT PARAMETERS:
        Ensemble-   neural networks ensemble
        X       -   input vector,  array[0..NIn-1].
        Y       -   (possibly) preallocated buffer; if size of Y is less than
                    NOut, it will be reallocated. If it is large enough, it
                    is NOT reallocated, so we can save some time on reallocation.

    OUTPUT PARAMETERS:
        Y       -   result. Regression estimate when solving regression  task,
                    vector of posterior probabilities for classification task.

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpeprocess(
        mlpensemble ensemble,
        double[] x,
        ref double[] y)

### `mlpeprocessi` function

    /*************************************************************************
    'interactive'  variant  of  MLPEProcess  for  languages  like Python which
    support constructs like "Y = MLPEProcess(LM,X)" and interactive mode of the
    interpreter

    This function allocates new array on each call,  so  it  is  significantly
    slower than its 'non-interactive' counterpart, but it is  more  convenient
    when you call it from command line.

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpeprocessi(
        mlpensemble ensemble,
        double[] x,
        out double[] y)

### `mlpeproperties` function

    /*************************************************************************
    Return ensemble properties (number of inputs and outputs).

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpeproperties(
        mlpensemble ensemble,
        out int nin,
        out int nout)

### `mlperandomize` function

    /*************************************************************************
    Randomization of MLP ensemble

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlperandomize(mlpensemble ensemble)

### `mlperelclserror` function

    /*************************************************************************
    Relative classification error on the test set

    INPUT PARAMETERS:
        Ensemble-   ensemble
        XY      -   test set
        NPoints -   test set size

    RESULT:
        percent of incorrectly classified cases.
        Works both for classifier betwork and for regression networks which
    are used as classifiers.

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double mlperelclserror(
        mlpensemble ensemble,
        double[,] xy,
        int npoints)

### `mlpermserror` function

    /*************************************************************************
    RMS error on the test set

    INPUT PARAMETERS:
        Ensemble-   ensemble
        XY      -   test set
        NPoints -   test set size

    RESULT:
        root mean square error.
        Its meaning for regression task is obvious. As for classification task
    RMS error means error when estimating posterior probabilities.

      -- ALGLIB --
         Copyright 17.02.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double mlpermserror(
        mlpensemble ensemble,
        double[,] xy,
        int npoints)

### `mlpetraines` function

    /*************************************************************************
    Training neural networks ensemble using early stopping.

    INPUT PARAMETERS:
        Ensemble    -   model with initialized geometry
        XY          -   training set
        NPoints     -   training set size
        Decay       -   weight decay coefficient, >=0.001
        Restarts    -   restarts, >0.

    OUTPUT PARAMETERS:
        Ensemble    -   trained model
        Info        -   return code:
                        * -2, if there is a point with class number
                              outside of [0..NClasses-1].
                        * -1, if incorrect parameters was passed
                              (NPoints<0, Restarts<1).
                        *  6, if task has been solved.
        Rep         -   training report.
        OOBErrors   -   out-of-bag generalization error estimate

      -- ALGLIB --
         Copyright 10.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpetraines(
        mlpensemble ensemble,
        double[,] xy,
        int npoints,
        double decay,
        int restarts,
        out int info,
        out mlpreport rep)

## `mlptrain` subpackage

### Classes

mlpcvreport
mlpreport

### Functions

mlpkfoldcvlbfgs
mlpkfoldcvlm
mlptraines
mlptrainlbfgs
mlptrainlm

### Examples

### `mlpcvreport` class

    /*************************************************************************
    Cross-validation estimates of generalization error
    *************************************************************************/
    public class mlpcvreport
    {
        public double               relclserror;
        public double               avgce;
        public double               rmserror;
        public double               avgerror;
        public double               avgrelerror;
    }

### `mlpreport` class

    /*************************************************************************
    Training report:
        * NGrad     - number of gradient calculations
        * NHess     - number of Hessian calculations
        * NCholesky - number of Cholesky decompositions
    *************************************************************************/
    public class mlpreport
    {
        public int                  ngrad;
        public int                  nhess;
        public int                  ncholesky;
    }

### `mlpkfoldcvlbfgs` function

    /*************************************************************************
    Cross-validation estimate of generalization error.

    Base algorithm - L-BFGS.

    INPUT PARAMETERS:
        Network     -   neural network with initialized geometry.   Network is
                        not changed during cross-validation -  it is used only
                        as a representative of its architecture.
        XY          -   training set.
        SSize       -   training set size
        Decay       -   weight  decay, same as in MLPTrainLBFGS
        Restarts    -   number of restarts, >0.
                        restarts are counted for each partition separately, so
                        total number of restarts will be Restarts*FoldsCount.
        WStep       -   stopping criterion, same as in MLPTrainLBFGS
        MaxIts      -   stopping criterion, same as in MLPTrainLBFGS
        FoldsCount  -   number of folds in k-fold cross-validation,
                        2<=FoldsCount<=SSize.
                        recommended value: 10.

    OUTPUT PARAMETERS:
        Info        -   return code, same as in MLPTrainLBFGS
        Rep         -   report, same as in MLPTrainLM/MLPTrainLBFGS
        CVRep       -   generalization error estimates

      -- ALGLIB --
         Copyright 09.12.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpkfoldcvlbfgs(
        multilayerperceptron network,
        double[,] xy,
        int npoints,
        double decay,
        int restarts,
        double wstep,
        int maxits,
        int foldscount,
        out int info,
        out mlpreport rep,
        out mlpcvreport cvrep)

### `mlpkfoldcvlm` function

    /*************************************************************************
    Cross-validation estimate of generalization error.

    Base algorithm - Levenberg-Marquardt.

    INPUT PARAMETERS:
        Network     -   neural network with initialized geometry.   Network is
                        not changed during cross-validation -  it is used only
                        as a representative of its architecture.
        XY          -   training set.
        SSize       -   training set size
        Decay       -   weight  decay, same as in MLPTrainLBFGS
        Restarts    -   number of restarts, >0.
                        restarts are counted for each partition separately, so
                        total number of restarts will be Restarts*FoldsCount.
        FoldsCount  -   number of folds in k-fold cross-validation,
                        2<=FoldsCount<=SSize.
                        recommended value: 10.

    OUTPUT PARAMETERS:
        Info        -   return code, same as in MLPTrainLBFGS
        Rep         -   report, same as in MLPTrainLM/MLPTrainLBFGS
        CVRep       -   generalization error estimates

      -- ALGLIB --
         Copyright 09.12.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlpkfoldcvlm(
        multilayerperceptron network,
        double[,] xy,
        int npoints,
        double decay,
        int restarts,
        int foldscount,
        out int info,
        out mlpreport rep,
        out mlpcvreport cvrep)

### `mlptraines` function

    /*************************************************************************
    Neural network training using early stopping (base algorithm - L-BFGS with
    regularization).

    INPUT PARAMETERS:
        Network     -   neural network with initialized geometry
        TrnXY       -   training set
        TrnSize     -   training set size
        ValXY       -   validation set
        ValSize     -   validation set size
        Decay       -   weight decay constant, >=0.001
                        Decay term 'Decay*||Weights||^2' is added to error
                        function.
                        If you don't know what Decay to choose, use 0.001.
        Restarts    -   number of restarts from random position, >0.
                        If you don't know what Restarts to choose, use 2.

    OUTPUT PARAMETERS:
        Network     -   trained neural network.
        Info        -   return code:
                        * -2, if there is a point with class number
                              outside of [0..NOut-1].
                        * -1, if wrong parameters specified
                              (NPoints<0, Restarts<1, ...).
                        *  2, task has been solved, stopping  criterion  met -
                              sufficiently small step size.  Not expected  (we
                              use  EARLY  stopping)  but  possible  and not an
                              error.
                        *  6, task has been solved, stopping  criterion  met -
                              increasing of validation set error.
        Rep         -   training report

    NOTE:

    Algorithm stops if validation set error increases for  a  long  enough  or
    step size is small enought  (there  are  task  where  validation  set  may
    decrease for eternity). In any case solution returned corresponds  to  the
    minimum of validation set error.

      -- ALGLIB --
         Copyright 10.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlptraines(
        multilayerperceptron network,
        double[,] trnxy,
        int trnsize,
        double[,] valxy,
        int valsize,
        double decay,
        int restarts,
        out int info,
        out mlpreport rep)

### `mlptrainlbfgs` function

    /*************************************************************************
    Neural  network  training  using  L-BFGS  algorithm  with  regularization.
    Subroutine  trains  neural  network  with  restarts from random positions.
    Algorithm  is  well  suited  for  problems  of  any dimensionality (memory
    requirements and step complexity are linear by weights number).

    INPUT PARAMETERS:
        Network     -   neural network with initialized geometry
        XY          -   training set
        NPoints     -   training set size
        Decay       -   weight decay constant, >=0.001
                        Decay term 'Decay*||Weights||^2' is added to error
                        function.
                        If you don't know what Decay to choose, use 0.001.
        Restarts    -   number of restarts from random position, >0.
                        If you don't know what Restarts to choose, use 2.
        WStep       -   stopping criterion. Algorithm stops if  step  size  is
                        less than WStep. Recommended value - 0.01.  Zero  step
                        size means stopping after MaxIts iterations.
        MaxIts      -   stopping   criterion.  Algorithm  stops  after  MaxIts
                        iterations (NOT gradient  calculations).  Zero  MaxIts
                        means stopping when step is sufficiently small.

    OUTPUT PARAMETERS:
        Network     -   trained neural network.
        Info        -   return code:
                        * -8, if both WStep=0 and MaxIts=0
                        * -2, if there is a point with class number
                              outside of [0..NOut-1].
                        * -1, if wrong parameters specified
                              (NPoints<0, Restarts<1).
                        *  2, if task has been solved.
        Rep         -   training report

      -- ALGLIB --
         Copyright 09.12.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void mlptrainlbfgs(
        multilayerperceptron network,
        double[,] xy,
        int npoints,
        double decay,
        int restarts,
        double wstep,
        int maxits,
        out int info,
        out mlpreport rep)

### `mlptrainlm` function

    /*************************************************************************
    Neural network training  using  modified  Levenberg-Marquardt  with  exact
    Hessian calculation and regularization. Subroutine trains  neural  network
    with restarts from random positions. Algorithm is well  suited  for  small
    and medium scale problems (hundreds of weights).

    INPUT PARAMETERS:
        Network     -   neural network with initialized geometry
        XY          -   training set
        NPoints     -   training set size
        Decay       -   weight decay constant, >=0.001
                        Decay term 'Decay*||Weights||^2' is added to error
                        function.
                        If you don't know what Decay to choose, use 0.001.
        Restarts    -   number of restarts from random position, >0.
                        If you don't know what Restarts to choose, use 2.

    OUTPUT PARAMETERS:
        Network     -   trained neural network.
        Info        -   return code:
                        * -9, if internal matrix inverse subroutine failed
                        * -2, if there is a point with class number
                              outside of [0..NOut-1].
                        * -1, if wrong parameters specified
                              (NPoints<0, Restarts<1).
                        *  2, if task has been solved.
        Rep         -   training report

      -- ALGLIB --
         Copyright 10.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void mlptrainlm(
        multilayerperceptron network,
        double[,] xy,
        int npoints,
        double decay,
        int restarts,
        out int info,
        out mlpreport rep)

## `nearestneighbor` subpackage

### Classes

kdtree

### Functions

kdtreebuild
kdtreebuildtagged
kdtreequeryaknn
kdtreequeryknn
kdtreequeryresultsdistances
kdtreequeryresultsdistancesi
kdtreequeryresultstags
kdtreequeryresultstagsi
kdtreequeryresultsx
kdtreequeryresultsxi
kdtreequeryresultsxy
kdtreequeryresultsxyi
kdtreequeryrnn
kdtreeserialize
kdtreeunserialize

### Examples

nneighbor_d_1 |  | Nearest neighbor search, KNN queries
nneighbor_d_2 |  | Serialization of KD-trees

### `kdtree` class

    /*************************************************************************

    *************************************************************************/
    public class kdtree
    {
    }

### `kdtreebuild` function

    /*************************************************************************
    KD-tree creation

    This subroutine creates KD-tree from set of X-values and optional Y-values

    INPUT PARAMETERS
        XY      -   dataset, array[0..N-1,0..NX+NY-1].
                    one row corresponds to one point.
                    first NX columns contain X-values, next NY (NY may be zero)
                    columns may contain associated Y-values
        N       -   number of points, N>=1
        NX      -   space dimension, NX>=1.
        NY      -   number of optional Y-values, NY>=0.
        NormType-   norm type:
                    * 0 denotes infinity-norm
                    * 1 denotes 1-norm
                    * 2 denotes 2-norm (Euclidean norm)

    OUTPUT PARAMETERS
        KDT     -   KD-tree

    NOTES

    1. KD-tree  creation  have O(N*logN) complexity and O(N*(2*NX+NY))  memory
       requirements.
    2. Although KD-trees may be used with any combination of N  and  NX,  they
       are more efficient than brute-force search only when N >> 4^NX. So they
       are most useful in low-dimensional tasks (NX=2, NX=3). NX=1  is another
       inefficient case, because  simple  binary  search  (without  additional
       structures) is much more efficient in such tasks than KD-trees.

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void kdtreebuild(
        double[,] xy,
        int nx,
        int ny,
        int normtype,
        out kdtree kdt)
    public static void kdtreebuild(
        double[,] xy,
        int n,
        int nx,
        int ny,
        int normtype,
        out kdtree kdt)

Examples: [1] [2]

### `kdtreebuildtagged` function

    /*************************************************************************
    KD-tree creation

    This  subroutine  creates  KD-tree  from set of X-values, integer tags and
    optional Y-values

    INPUT PARAMETERS
        XY      -   dataset, array[0..N-1,0..NX+NY-1].
                    one row corresponds to one point.
                    first NX columns contain X-values, next NY (NY may be zero)
                    columns may contain associated Y-values
        Tags    -   tags, array[0..N-1], contains integer tags associated
                    with points.
        N       -   number of points, N>=1
        NX      -   space dimension, NX>=1.
        NY      -   number of optional Y-values, NY>=0.
        NormType-   norm type:
                    * 0 denotes infinity-norm
                    * 1 denotes 1-norm
                    * 2 denotes 2-norm (Euclidean norm)

    OUTPUT PARAMETERS
        KDT     -   KD-tree

    NOTES

    1. KD-tree  creation  have O(N*logN) complexity and O(N*(2*NX+NY))  memory
       requirements.
    2. Although KD-trees may be used with any combination of N  and  NX,  they
       are more efficient than brute-force search only when N >> 4^NX. So they
       are most useful in low-dimensional tasks (NX=2, NX=3). NX=1  is another
       inefficient case, because  simple  binary  search  (without  additional
       structures) is much more efficient in such tasks than KD-trees.

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void kdtreebuildtagged(
        double[,] xy,
        int[] tags,
        int nx,
        int ny,
        int normtype,
        out kdtree kdt)
    public static void kdtreebuildtagged(
        double[,] xy,
        int[] tags,
        int n,
        int nx,
        int ny,
        int normtype,
        out kdtree kdt)

Examples: [1]

### `kdtreequeryaknn` function

    /*************************************************************************
    K-NN query: approximate K nearest neighbors

    INPUT PARAMETERS
        KDT         -   KD-tree
        X           -   point, array[0..NX-1].
        K           -   number of neighbors to return, K>=1
        SelfMatch   -   whether self-matches are allowed:
                        * if True, nearest neighbor may be the point itself
                          (if it exists in original dataset)
                        * if False, then only points with non-zero distance
                          are returned
                        * if not given, considered True
        Eps         -   approximation factor, Eps>=0. eps-approximate  nearest
                        neighbor  is  a  neighbor  whose distance from X is at
                        most (1+eps) times distance of true nearest neighbor.

    RESULT
        number of actual neighbors found (either K or N, if K>N).

    NOTES
        significant performance gain may be achieved only when Eps  is  is  on
        the order of magnitude of 1 or larger.

    This  subroutine  performs  query  and  stores  its result in the internal
    structures of the KD-tree. You can use  following  subroutines  to  obtain
    these results:
    * KDTreeQueryResultsX() to get X-values
    * KDTreeQueryResultsXY() to get X- and Y-values
    * KDTreeQueryResultsTags() to get tag values
    * KDTreeQueryResultsDistances() to get distances

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static int kdtreequeryaknn(
        kdtree kdt,
        double[] x,
        int k,
        double eps)
    public static int kdtreequeryaknn(
        kdtree kdt,
        double[] x,
        int k,
        bool selfmatch,
        double eps)

Examples: [1]

### `kdtreequeryknn` function

    /*************************************************************************
    K-NN query: K nearest neighbors

    INPUT PARAMETERS
        KDT         -   KD-tree
        X           -   point, array[0..NX-1].
        K           -   number of neighbors to return, K>=1
        SelfMatch   -   whether self-matches are allowed:
                        * if True, nearest neighbor may be the point itself
                          (if it exists in original dataset)
                        * if False, then only points with non-zero distance
                          are returned
                        * if not given, considered True

    RESULT
        number of actual neighbors found (either K or N, if K>N).

    This  subroutine  performs  query  and  stores  its result in the internal
    structures of the KD-tree. You can use  following  subroutines  to  obtain
    these results:
    * KDTreeQueryResultsX() to get X-values
    * KDTreeQueryResultsXY() to get X- and Y-values
    * KDTreeQueryResultsTags() to get tag values
    * KDTreeQueryResultsDistances() to get distances

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static int kdtreequeryknn(kdtree kdt, double[] x, int k)
    public static int kdtreequeryknn(
        kdtree kdt,
        double[] x,
        int k,
        bool selfmatch)

Examples: [1]

### `kdtreequeryresultsdistances` function

    /*************************************************************************
    Distances from last query

    INPUT PARAMETERS
        KDT     -   KD-tree
        R       -   possibly pre-allocated buffer. If X is too small to store
                    result, it is resized. If size(X) is enough to store
                    result, it is left unchanged.

    OUTPUT PARAMETERS
        R       -   filled with distances (in corresponding norm)

    NOTES
    1. points are ordered by distance from the query point (first = closest)
    2. if  XY is larger than required to store result, only leading part  will
       be overwritten; trailing part will be left unchanged. So  if  on  input
       XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
       XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
       you want function  to  resize  array  according  to  result  size,  use
       function with same name and suffix 'I'.

    SEE ALSO
    * KDTreeQueryResultsX()             X-values
    * KDTreeQueryResultsXY()            X- and Y-values
    * KDTreeQueryResultsTags()          tag values

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void kdtreequeryresultsdistances(kdtree kdt, ref double[] r)

Examples: [1]

### `kdtreequeryresultsdistancesi` function

    /*************************************************************************
    Distances from last query; 'interactive' variant for languages like Python
    which  support  constructs   like  "R = KDTreeQueryResultsDistancesI(KDT)"
    and interactive mode of interpreter.

    This function allocates new array on each call,  so  it  is  significantly
    slower than its 'non-interactive' counterpart, but it is  more  convenient
    when you call it from command line.

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void kdtreequeryresultsdistancesi(
        kdtree kdt,
        out double[] r)

### `kdtreequeryresultstags` function

    /*************************************************************************
    Tags from last query

    INPUT PARAMETERS
        KDT     -   KD-tree
        Tags    -   possibly pre-allocated buffer. If X is too small to store
                    result, it is resized. If size(X) is enough to store
                    result, it is left unchanged.

    OUTPUT PARAMETERS
        Tags    -   filled with tags associated with points,
                    or, when no tags were supplied, with zeros

    NOTES
    1. points are ordered by distance from the query point (first = closest)
    2. if  XY is larger than required to store result, only leading part  will
       be overwritten; trailing part will be left unchanged. So  if  on  input
       XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
       XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
       you want function  to  resize  array  according  to  result  size,  use
       function with same name and suffix 'I'.

    SEE ALSO
    * KDTreeQueryResultsX()             X-values
    * KDTreeQueryResultsXY()            X- and Y-values
    * KDTreeQueryResultsDistances()     distances

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void kdtreequeryresultstags(kdtree kdt, ref int[] tags)

Examples: [1]

### `kdtreequeryresultstagsi` function

    /*************************************************************************
    Tags  from  last  query;  'interactive' variant for languages like  Python
    which  support  constructs  like "Tags = KDTreeQueryResultsTagsI(KDT)" and
    interactive mode of interpreter.

    This function allocates new array on each call,  so  it  is  significantly
    slower than its 'non-interactive' counterpart, but it is  more  convenient
    when you call it from command line.

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void kdtreequeryresultstagsi(kdtree kdt, out int[] tags)

### `kdtreequeryresultsx` function

    /*************************************************************************
    X-values from last query

    INPUT PARAMETERS
        KDT     -   KD-tree
        X       -   possibly pre-allocated buffer. If X is too small to store
                    result, it is resized. If size(X) is enough to store
                    result, it is left unchanged.

    OUTPUT PARAMETERS
        X       -   rows are filled with X-values

    NOTES
    1. points are ordered by distance from the query point (first = closest)
    2. if  XY is larger than required to store result, only leading part  will
       be overwritten; trailing part will be left unchanged. So  if  on  input
       XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
       XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
       you want function  to  resize  array  according  to  result  size,  use
       function with same name and suffix 'I'.

    SEE ALSO
    * KDTreeQueryResultsXY()            X- and Y-values
    * KDTreeQueryResultsTags()          tag values
    * KDTreeQueryResultsDistances()     distances

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void kdtreequeryresultsx(kdtree kdt, ref double[,] x)

Examples: [1]

### `kdtreequeryresultsxi` function

    /*************************************************************************
    X-values from last query; 'interactive' variant for languages like  Python
    which   support    constructs   like  "X = KDTreeQueryResultsXI(KDT)"  and
    interactive mode of interpreter.

    This function allocates new array on each call,  so  it  is  significantly
    slower than its 'non-interactive' counterpart, but it is  more  convenient
    when you call it from command line.

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void kdtreequeryresultsxi(kdtree kdt, out double[,] x)

### `kdtreequeryresultsxy` function

    /*************************************************************************
    X- and Y-values from last query

    INPUT PARAMETERS
        KDT     -   KD-tree
        XY      -   possibly pre-allocated buffer. If XY is too small to store
                    result, it is resized. If size(XY) is enough to store
                    result, it is left unchanged.

    OUTPUT PARAMETERS
        XY      -   rows are filled with points: first NX columns with
                    X-values, next NY columns - with Y-values.

    NOTES
    1. points are ordered by distance from the query point (first = closest)
    2. if  XY is larger than required to store result, only leading part  will
       be overwritten; trailing part will be left unchanged. So  if  on  input
       XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
       XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
       you want function  to  resize  array  according  to  result  size,  use
       function with same name and suffix 'I'.

    SEE ALSO
    * KDTreeQueryResultsX()             X-values
    * KDTreeQueryResultsTags()          tag values
    * KDTreeQueryResultsDistances()     distances

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void kdtreequeryresultsxy(kdtree kdt, ref double[,] xy)

Examples: [1]

### `kdtreequeryresultsxyi` function

    /*************************************************************************
    XY-values from last query; 'interactive' variant for languages like Python
    which   support    constructs   like "XY = KDTreeQueryResultsXYI(KDT)" and
    interactive mode of interpreter.

    This function allocates new array on each call,  so  it  is  significantly
    slower than its 'non-interactive' counterpart, but it is  more  convenient
    when you call it from command line.

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void kdtreequeryresultsxyi(kdtree kdt, out double[,] xy)

### `kdtreequeryrnn` function

    /*************************************************************************
    R-NN query: all points within R-sphere centered at X

    INPUT PARAMETERS
        KDT         -   KD-tree
        X           -   point, array[0..NX-1].
        R           -   radius of sphere (in corresponding norm), R>0
        SelfMatch   -   whether self-matches are allowed:
                        * if True, nearest neighbor may be the point itself
                          (if it exists in original dataset)
                        * if False, then only points with non-zero distance
                          are returned
                        * if not given, considered True

    RESULT
        number of neighbors found, >=0

    This  subroutine  performs  query  and  stores  its result in the internal
    structures of the KD-tree. You can use  following  subroutines  to  obtain
    actual results:
    * KDTreeQueryResultsX() to get X-values
    * KDTreeQueryResultsXY() to get X- and Y-values
    * KDTreeQueryResultsTags() to get tag values
    * KDTreeQueryResultsDistances() to get distances

      -- ALGLIB --
         Copyright 28.02.2010 by Bochkanov Sergey
    *************************************************************************/
    public static int kdtreequeryrnn(kdtree kdt, double[] x, double r)
    public static int kdtreequeryrnn(
        kdtree kdt,
        double[] x,
        double r,
        bool selfmatch)

Examples: [1]

### `kdtreeserialize` function

    /*************************************************************************
    This function serializes data structure to string.

    Important properties of s_out:
    * it contains alphanumeric characters, dots, underscores, minus signs
    * these symbols are grouped into words, which are separated by spaces
      and Windows-style (CR+LF) newlines
    * although  serializer  uses  spaces and CR+LF as separators, you can
      replace any separator character by arbitrary combination of spaces,
      tabs, Windows or Unix newlines. It allows flexible reformatting  of
      the  string  in  case you want to include it into text or XML file.
      But you should not insert separators into the middle of the "words"
      nor you should change case of letters.
    * s_out can be freely moved between 32-bit and 64-bit systems, little
      and big endian machines, and so on. You can serialize structure  on
      32-bit machine and unserialize it on 64-bit one (or vice versa), or
      serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also
      serialize  it  in  C++ version of ALGLIB and unserialize in C# one,
      and vice versa.
    *************************************************************************/
    public static void kdtreeserialize(kdtree obj, out string s_out)

### `kdtreeunserialize` function

    /*************************************************************************
    This function unserializes data structure from string.
    *************************************************************************/
    public static void kdtreeunserialize(string s_in, out kdtree obj)

### nneighbor_d_1 example

    public static int Main(string[] args)
    {
        double[,] a = new double[,]{{0,0},{0,1},{1,0},{1,1}};
        int nx = 2;
        int ny = 0;
        int normtype = 2;
        alglib.kdtree kdt;
        double[] x;
        double[,] r = new double[0,0];
        int k;
        alglib.kdtreebuild(a, nx, ny, normtype, out kdt);
        x = new double[]{-1,0};
        k = alglib.kdtreequeryknn(kdt, x, 1);
        System.Console.WriteLine("{0}", k); // EXPECTED: 1
        alglib.kdtreequeryresultsx(kdt, ref r);
        System.Console.WriteLine("{0}", alglib.ap.format(r,1)); // EXPECTED: [[0,0]]
        System.Console.ReadLine();
        return 0;
    }

### nneighbor_d_2 example

    public static int Main(string[] args)
    {
        double[,] a = new double[,]{{0,0},{0,1},{1,0},{1,1}};
        int nx = 2;
        int ny = 0;
        int normtype = 2;
        alglib.kdtree kdt0;
        alglib.kdtree kdt1;
        string s;
        double[] x;
        double[,] r0 = new double[0,0];
        double[,] r1 = new double[0,0];

        //
        // Build tree and serialize it
        //
        alglib.kdtreebuild(a, nx, ny, normtype, out kdt0);
        alglib.kdtreeserialize(kdt0, out s);
        alglib.kdtreeunserialize(s, out kdt1);

        //
        // Compare results from KNN queries
        //
        x = new double[]{-1,0};
        alglib.kdtreequeryknn(kdt0, x, 1);
        alglib.kdtreequeryresultsx(kdt0, ref r0);
        alglib.kdtreequeryknn(kdt1, x, 1);
        alglib.kdtreequeryresultsx(kdt1, ref r1);
        System.Console.WriteLine("{0}", alglib.ap.format(r0,1)); // EXPECTED: [[0,0]]
        System.Console.WriteLine("{0}", alglib.ap.format(r1,1)); // EXPECTED: [[0,0]]
        System.Console.ReadLine();
        return 0;
    }

## `nleq` subpackage

### Classes

nleqreport
nleqstate

### Functions

nleqcreatelm
nleqrestartfrom
nleqresults
nleqresultsbuf
nleqsetcond
nleqsetstpmax
nleqsetxrep
nleqsolve

### Examples

### `nleqreport` class

    /*************************************************************************

    *************************************************************************/
    public class nleqreport
    {
        public int                  iterationscount;
        public int                  nfunc;
        public int                  njac;
        public int                  terminationtype;
    }

### `nleqstate` class

    /*************************************************************************

    *************************************************************************/
    public class nleqstate
    {
    }

### `nleqcreatelm` function

    /*************************************************************************
                    LEVENBERG-MARQUARDT-LIKE NONLINEAR SOLVER

    DESCRIPTION:
    This algorithm solves system of nonlinear equations
        F[0](x[0], ..., x[n-1])   = 0
        F[1](x[0], ..., x[n-1])   = 0
        ...
        F[M-1](x[0], ..., x[n-1]) = 0
    with M/N do not necessarily coincide.  Algorithm  converges  quadratically
    under following conditions:
        * the solution set XS is nonempty
        * for some xs in XS there exist such neighbourhood N(xs) that:
          * vector function F(x) and its Jacobian J(x) are continuously
            differentiable on N
          * ||F(x)|| provides local error bound on N, i.e. there  exists  such
            c1, that ||F(x)||>c1*distance(x,XS)
    Note that these conditions are much more weaker than usual non-singularity
    conditions. For example, algorithm will converge for any  affine  function
    F (whether its Jacobian singular or not).

    REQUIREMENTS:
    Algorithm will request following information during its operation:
    * function vector F[] and Jacobian matrix at given point X
    * value of merit function f(x)=F[0]^2(x)+...+F[M-1]^2(x) at given point X

    USAGE:
    1. User initializes algorithm state with NLEQCreateLM() call
    2. User tunes solver parameters with  NLEQSetCond(),  NLEQSetStpMax()  and
       other functions
    3. User  calls  NLEQSolve()  function  which  takes  algorithm  state  and
       pointers (delegates, etc.) to callback functions which calculate  merit
       function value and Jacobian.
    4. User calls NLEQResults() to get solution
    5. Optionally, user may call NLEQRestartFrom() to  solve  another  problem
       with same parameters (N/M) but another starting  point  and/or  another
       function vector. NLEQRestartFrom() allows to reuse already  initialized
       structure.

    INPUT PARAMETERS:
        N       -   space dimension, N>1:
                    * if provided, only leading N elements of X are used
                    * if not provided, determined automatically from size of X
        M       -   system size
        X       -   starting point

    OUTPUT PARAMETERS:
        State   -   structure which stores algorithm state

    NOTES:
    1. you may tune stopping conditions with NLEQSetCond() function
    2. if target function contains exp() or other fast growing functions,  and
       optimization algorithm makes too large steps which leads  to  overflow,
       use NLEQSetStpMax() function to bound algorithm's steps.
    3. this  algorithm  is  a  slightly  modified implementation of the method
       described  in  'Levenberg-Marquardt  method  for constrained  nonlinear
       equations with strong local convergence properties' by Christian Kanzow
       Nobuo Yamashita and Masao Fukushima and further  developed  in  'On the
       convergence of a New Levenberg-Marquardt Method'  by  Jin-yan  Fan  and
       Ya-Xiang Yuan.

      -- ALGLIB --
         Copyright 20.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void nleqcreatelm(int m, double[] x, out nleqstate state)
    public static void nleqcreatelm(
        int n,
        int m,
        double[] x,
        out nleqstate state)

### `nleqrestartfrom` function

    /*************************************************************************
    This  subroutine  restarts  CG  algorithm from new point. All optimization
    parameters are left unchanged.

    This  function  allows  to  solve multiple  optimization  problems  (which
    must have same number of dimensions) without object reallocation penalty.

    INPUT PARAMETERS:
        State   -   structure used for reverse communication previously
                    allocated with MinCGCreate call.
        X       -   new starting point.
        BndL    -   new lower bounds
        BndU    -   new upper bounds

      -- ALGLIB --
         Copyright 30.07.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void nleqrestartfrom(nleqstate state, double[] x)

### `nleqresults` function

    /*************************************************************************
    NLEQ solver results

    INPUT PARAMETERS:
        State   -   algorithm state.

    OUTPUT PARAMETERS:
        X       -   array[0..N-1], solution
        Rep     -   optimization report:
                    * Rep.TerminationType completetion code:
                        * -4    ERROR:  algorithm   has   converged   to   the
                                stationary point Xf which is local minimum  of
                                f=F[0]^2+...+F[m-1]^2, but is not solution  of
                                nonlinear system.
                        *  1    sqrt(f)<=EpsF.
                        *  5    MaxIts steps was taken
                        *  7    stopping conditions are too stringent,
                                further improvement is impossible
                    * Rep.IterationsCount contains iterations count
                    * NFEV countains number of function calculations
                    * ActiveConstraints contains number of active constraints

      -- ALGLIB --
         Copyright 20.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void nleqresults(
        nleqstate state,
        out double[] x,
        out nleqreport rep)

### `nleqresultsbuf` function

    /*************************************************************************
    NLEQ solver results

    Buffered implementation of NLEQResults(), which uses pre-allocated  buffer
    to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
    intended to be used in the inner cycles of performance critical algorithms
    where array reallocation penalty is too large to be ignored.

      -- ALGLIB --
         Copyright 20.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void nleqresultsbuf(
        nleqstate state,
        ref double[] x,
        nleqreport rep)

### `nleqsetcond` function

    /*************************************************************************
    This function sets stopping conditions for the nonlinear solver

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        EpsF    -   >=0
                    The subroutine finishes  its work if on k+1-th iteration
                    the condition ||F||<=EpsF is satisfied
        MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                    iterations is unlimited.

    Passing EpsF=0 and MaxIts=0 simultaneously will lead to  automatic
    stopping criterion selection (small EpsF).

    NOTES:

      -- ALGLIB --
         Copyright 20.08.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void nleqsetcond(nleqstate state, double epsf, int maxits)

### `nleqsetstpmax` function

    /*************************************************************************
    This function sets maximum step length

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                    want to limit step length.

    Use this subroutine when target function  contains  exp()  or  other  fast
    growing functions, and algorithm makes  too  large  steps  which  lead  to
    overflow. This function allows us to reject steps that are too large  (and
    therefore expose us to the possible overflow) without actually calculating
    function value at the x+stp*d.

      -- ALGLIB --
         Copyright 20.08.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void nleqsetstpmax(nleqstate state, double stpmax)

### `nleqsetxrep` function

    /*************************************************************************
    This function turns on/off reporting.

    INPUT PARAMETERS:
        State   -   structure which stores algorithm state
        NeedXRep-   whether iteration reports are needed or not

    If NeedXRep is True, algorithm will call rep() callback function if  it is
    provided to NLEQSolve().

      -- ALGLIB --
         Copyright 20.08.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void nleqsetxrep(nleqstate state, bool needxrep)

### `nleqsolve` function

    /*************************************************************************
    This family of functions is used to launcn iterations of nonlinear solver

    These functions accept following parameters:
        state   -   algorithm state
        func    -   callback which calculates function (or merit function)
                    value func at given point x
        jac     -   callback which calculates function vector fi[]
                    and Jacobian jac at given point x
        rep     -   optional callback which is called after each iteration
                    can be NULL
        ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                    can be NULL

      -- ALGLIB --
         Copyright 20.03.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void nleqsolve(nleqstate state, ndimensional_func func, ndimensional_jac  jac, ndimensional_rep rep, object obj)

## `normaldistr` subpackage

### Functions

errorfunction
errorfunctionc
inverf
invnormaldistribution
normaldistribution

### Examples

### `errorfunction` function

    /*************************************************************************
    Error function

    The integral is

                              x
                               -
                    2         | |          2
      erf(x)  =  --------     |    exp( - t  ) dt.
                 sqrt(pi)   | |
                             -
                              0

    For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
    erf(x) = 1 - erfc(x).

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE      0,1         30000       3.7e-16     1.0e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double errorfunction(double x)

### `errorfunctionc` function

    /*************************************************************************
    Complementary error function

     1 - erf(x) =

                              inf.
                                -
                     2         | |          2
      erfc(x)  =  --------     |    exp( - t  ) dt
                  sqrt(pi)   | |
                              -
                               x

    For small x, erfc(x) = 1 - erf(x); otherwise rational
    approximations are computed.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE      0,26.6417   30000       5.7e-14     1.5e-14

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double errorfunctionc(double x)

### `inverf` function

    /*************************************************************************
    Inverse of the error function

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double inverf(double e)

### `invnormaldistribution` function

    /*************************************************************************
    Inverse of Normal distribution function

    Returns the argument, x, for which the area under the
    Gaussian probability density function (integrated from
    minus infinity to x) is equal to y.

    For small arguments 0 < y < exp(-2), the program computes
    z = sqrt( -2.0 * log(y) );  then the approximation is
    x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z).
    There are two rational functions P/Q, one for 0 < y < exp(-32)
    and the other for y up to exp(-2).  For larger arguments,
    w = y - 0.5, and  x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).

    ACCURACY:

                         Relative error:
    arithmetic   domain        # trials      peak         rms
       IEEE     0.125, 1        20000       7.2e-16     1.3e-16
       IEEE     3e-308, 0.135   50000       4.6e-16     9.8e-17

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double invnormaldistribution(double y0)

### `normaldistribution` function

    /*************************************************************************
    Normal distribution function

    Returns the area under the Gaussian probability density
    function, integrated from minus infinity to x:

                               x
                                -
                      1        | |          2
       ndtr(x)  = ---------    |    exp( - t /2 ) dt
                  sqrt(2pi)  | |
                              -
                             -inf.

                =  ( 1 + erf(z) ) / 2
                =  erfc(z) / 2

    where z = x/sqrt(2). Computation is via the functions
    erf and erfc.

    ACCURACY:

                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE     -13,0        30000       3.4e-14     6.7e-15

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double normaldistribution(double x)

## `odesolver` subpackage

### Classes

odesolverreport
odesolverstate

### Functions

odesolverresults
odesolverrkck
odesolversolve

### Examples

odesolver_d1 |  | Solving y'=-y with ODE solver

### `odesolverreport` class

    /*************************************************************************

    *************************************************************************/
    public class odesolverreport
    {
        public int                  nfev;
        public int                  terminationtype;
    }

### `odesolverstate` class

    /*************************************************************************

    *************************************************************************/
    public class odesolverstate
    {
    }

### `odesolverresults` function

    /*************************************************************************
    ODE solver results

    Called after OdeSolverIteration returned False.

    INPUT PARAMETERS:
        State   -   algorithm state (used by OdeSolverIteration).

    OUTPUT PARAMETERS:
        M       -   number of tabulated values, M>=1
        XTbl    -   array[0..M-1], values of X
        YTbl    -   array[0..M-1,0..N-1], values of Y in X[i]
        Rep     -   solver report:
                    * Rep.TerminationType completetion code:
                        * -2    X is not ordered  by  ascending/descending  or
                                there are non-distinct X[],  i.e.  X[i]=X[i+1]
                        * -1    incorrect parameters were specified
                        *  1    task has been solved
                    * Rep.NFEV contains number of function calculations

      -- ALGLIB --
         Copyright 01.09.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void odesolverresults(
        odesolverstate state,
        out int m,
        out double[] xtbl,
        out double[,] ytbl,
        out odesolverreport rep)

Examples: [1]

### `odesolverrkck` function

    /*************************************************************************
    Cash-Karp adaptive ODE solver.

    This subroutine solves ODE  Y'=f(Y,x)  with  initial  conditions  Y(xs)=Ys
    (here Y may be single variable or vector of N variables).

    INPUT PARAMETERS:
        Y       -   initial conditions, array[0..N-1].
                    contains values of Y[] at X[0]
        N       -   system size
        X       -   points at which Y should be tabulated, array[0..M-1]
                    integrations starts at X[0], ends at X[M-1],  intermediate
                    values at X[i] are returned too.
                    SHOULD BE ORDERED BY ASCENDING OR BY DESCENDING!!!!
        M       -   number of intermediate points + first point + last point:
                    * M>2 means that you need both Y(X[M-1]) and M-2 values at
                      intermediate points
                    * M=2 means that you want just to integrate from  X[0]  to
                      X[1] and don't interested in intermediate values.
                    * M=1 means that you don't want to integrate :)
                      it is degenerate case, but it will be handled correctly.
                    * M<1 means error
        Eps     -   tolerance (absolute/relative error on each  step  will  be
                    less than Eps). When passing:
                    * Eps>0, it means desired ABSOLUTE error
                    * Eps<0, it means desired RELATIVE error.  Relative errors
                      are calculated with respect to maximum values of  Y seen
                      so far. Be careful to use this criterion  when  starting
                      from Y[] that are close to zero.
        H       -   initial  step  lenth,  it  will  be adjusted automatically
                    after the first  step.  If  H=0,  step  will  be  selected
                    automatically  (usualy  it  will  be  equal  to  0.001  of
                    min(x[i]-x[j])).

    OUTPUT PARAMETERS
        State   -   structure which stores algorithm state between  subsequent
                    calls of OdeSolverIteration. Used for reverse communication.
                    This structure should be passed  to the OdeSolverIteration
                    subroutine.

    SEE ALSO
        AutoGKSmoothW, AutoGKSingular, AutoGKIteration, AutoGKResults.

      -- ALGLIB --
         Copyright 01.09.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void odesolverrkck(
        double[] y,
        double[] x,
        double eps,
        double h,
        out odesolverstate state)
    public static void odesolverrkck(
        double[] y,
        int n,
        double[] x,
        int m,
        double eps,
        double h,
        out odesolverstate state)

Examples: [1]

### `odesolversolve` function

    /*************************************************************************
    This function is used to launcn iterations of ODE solver

    It accepts following parameters:
        diff    -   callback which calculates dy/dx for given y and x
        ptr     -   optional pointer which is passed to diff; can be NULL

      -- ALGLIB --
         Copyright 01.09.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void odesolversolve(odesolverstate state, ndimensional_ode_rp diff, object obj)
    {
        if( diff==null )
            throw new alglibexception("ALGLIB: error in 'odesolversolve()' (diff is null)");
        while( alglib.odesolveriteration(state) )
        {
            if( state.needdy )
            {
                diff(state.innerobj.y, state.innerobj.x, state.innerobj.dy, obj);
                continue;
            }
            throw new alglibexception("ALGLIB: unexpected error in 'odesolversolve'");
        }
    }

Examples: [1]

### odesolver_d1 example

    public static void ode_function_1_diff(double[] y, double x, double[] dy, object obj)
    {
        // this callback calculates f(y[],x)=-y[0]
        dy[0] = -y[0];
    }
    public static int Main(string[] args)
    {
        double[] y = new double[]{1};
        double[] x = new double[]{0,1,2,3};
        double eps = 0.00001;
        double h = 0;
        alglib.odesolverstate s;
        int m;
        double[] xtbl;
        double[,] ytbl;
        alglib.odesolverreport rep;
        alglib.odesolverrkck(y, x, eps, h, out s);
        alglib.odesolversolve(s, ode_function_1_diff, null);
        alglib.odesolverresults(s, out m, out xtbl, out ytbl, out rep);
        System.Console.WriteLine("{0}", m); // EXPECTED: 4
        System.Console.WriteLine("{0}", alglib.ap.format(xtbl,2)); // EXPECTED: [0, 1, 2, 3]
        System.Console.WriteLine("{0}", alglib.ap.format(ytbl,2)); // EXPECTED: [[1], [0.367], [0.135], [0.050]]
        System.Console.ReadLine();
        return 0;
    }

## `ortfac` subpackage

### Functions

cmatrixlq
cmatrixlqunpackl
cmatrixlqunpackq
cmatrixqr
cmatrixqrunpackq
cmatrixqrunpackr
hmatrixtd
hmatrixtdunpackq
rmatrixbd
rmatrixbdmultiplybyp
rmatrixbdmultiplybyq
rmatrixbdunpackdiagonals
rmatrixbdunpackpt
rmatrixbdunpackq
rmatrixhessenberg
rmatrixhessenbergunpackh
rmatrixhessenbergunpackq
rmatrixlq
rmatrixlqunpackl
rmatrixlqunpackq
rmatrixqr
rmatrixqrunpackq
rmatrixqrunpackr
smatrixtd
smatrixtdunpackq

### Examples

### `cmatrixlq` function

    /*************************************************************************
    LQ decomposition of a rectangular complex matrix of size MxN

    Input parameters:
        A   -   matrix A whose indexes range within [0..M-1, 0..N-1]
        M   -   number of rows in matrix A.
        N   -   number of columns in matrix A.

    Output parameters:
        A   -   matrices Q and L in compact form
        Tau -   array of scalar factors which are used to form matrix Q. Array
                whose indexes range within [0.. Min(M,N)-1]

    Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size
    MxM, L - lower triangular (or lower trapezoid) matrix of size MxN.

      -- LAPACK routine (version 3.0) --
         Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
         Courant Institute, Argonne National Lab, and Rice University
         September 30, 1994
    *************************************************************************/
    public static void cmatrixlq(
        ref complex[,] a,
        int m,
        int n,
        out complex[] tau)

### `cmatrixlqunpackl` function

    /*************************************************************************
    Unpacking of matrix L from the LQ decomposition of a matrix A

    Input parameters:
        A       -   matrices Q and L in compact form.
                    Output of CMatrixLQ subroutine.
        M       -   number of rows in given matrix A. M>=0.
        N       -   number of columns in given matrix A. N>=0.

    Output parameters:
        L       -   matrix L, array[0..M-1, 0..N-1].

      -- ALGLIB routine --
         17.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixlqunpackl(
        complex[,] a,
        int m,
        int n,
        out complex[,] l)

### `cmatrixlqunpackq` function

    /*************************************************************************
    Partial unpacking of matrix Q from LQ decomposition of a complex matrix A.

    Input parameters:
        A           -   matrices Q and R in compact form.
                        Output of CMatrixLQ subroutine .
        M           -   number of rows in matrix A. M>=0.
        N           -   number of columns in matrix A. N>=0.
        Tau         -   scalar factors which are used to form Q.
                        Output of CMatrixLQ subroutine .
        QRows       -   required number of rows in matrix Q. N>=QColumns>=0.

    Output parameters:
        Q           -   first QRows rows of matrix Q.
                        Array whose index ranges within [0..QRows-1, 0..N-1].
                        If QRows=0, array isn't changed.

      -- ALGLIB routine --
         17.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixlqunpackq(
        complex[,] a,
        int m,
        int n,
        complex[] tau,
        int qrows,
        out complex[,] q)

### `cmatrixqr` function

    /*************************************************************************
    QR decomposition of a rectangular complex matrix of size MxN

    Input parameters:
        A   -   matrix A whose indexes range within [0..M-1, 0..N-1]
        M   -   number of rows in matrix A.
        N   -   number of columns in matrix A.

    Output parameters:
        A   -   matrices Q and R in compact form
        Tau -   array of scalar factors which are used to form matrix Q. Array
                whose indexes range within [0.. Min(M,N)-1]

    Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
    MxM, R - upper triangular (or upper trapezoid) matrix of size MxN.

      -- LAPACK routine (version 3.0) --
         Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
         Courant Institute, Argonne National Lab, and Rice University
         September 30, 1994
    *************************************************************************/
    public static void cmatrixqr(
        ref complex[,] a,
        int m,
        int n,
        out complex[] tau)

### `cmatrixqrunpackq` function

    /*************************************************************************
    Partial unpacking of matrix Q from QR decomposition of a complex matrix A.

    Input parameters:
        A           -   matrices Q and R in compact form.
                        Output of CMatrixQR subroutine .
        M           -   number of rows in matrix A. M>=0.
        N           -   number of columns in matrix A. N>=0.
        Tau         -   scalar factors which are used to form Q.
                        Output of CMatrixQR subroutine .
        QColumns    -   required number of columns in matrix Q. M>=QColumns>=0.

    Output parameters:
        Q           -   first QColumns columns of matrix Q.
                        Array whose index ranges within [0..M-1, 0..QColumns-1].
                        If QColumns=0, array isn't changed.

      -- ALGLIB routine --
         17.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixqrunpackq(
        complex[,] a,
        int m,
        int n,
        complex[] tau,
        int qcolumns,
        out complex[,] q)

### `cmatrixqrunpackr` function

    /*************************************************************************
    Unpacking of matrix R from the QR decomposition of a matrix A

    Input parameters:
        A       -   matrices Q and R in compact form.
                    Output of CMatrixQR subroutine.
        M       -   number of rows in given matrix A. M>=0.
        N       -   number of columns in given matrix A. N>=0.

    Output parameters:
        R       -   matrix R, array[0..M-1, 0..N-1].

      -- ALGLIB routine --
         17.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixqrunpackr(
        complex[,] a,
        int m,
        int n,
        out complex[,] r)

### `hmatrixtd` function

    /*************************************************************************
    Reduction of a Hermitian matrix which is given  by  its  higher  or  lower
    triangular part to a real  tridiagonal  matrix  using  unitary  similarity
    transformation: Q'*A*Q = T.

    Input parameters:
        A       -   matrix to be transformed
                    array with elements [0..N-1, 0..N-1].
        N       -   size of matrix A.
        IsUpper -   storage format. If IsUpper = True, then matrix A is  given
                    by its upper triangle, and the lower triangle is not  used
                    and not modified by the algorithm, and vice versa
                    if IsUpper = False.

    Output parameters:
        A       -   matrices T and Q in  compact form (see lower)
        Tau     -   array of factors which are forming matrices H(i)
                    array with elements [0..N-2].
        D       -   main diagonal of real symmetric matrix T.
                    array with elements [0..N-1].
        E       -   secondary diagonal of real symmetric matrix T.
                    array with elements [0..N-2].

      If IsUpper=True, the matrix Q is represented as a product of elementary
      reflectors

         Q = H(n-2) . . . H(2) H(0).

      Each H(i) has the form

         H(i) = I - tau * v * v'

      where tau is a complex scalar, and v is a complex vector with
      v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in
      A(0:i-1,i+1), and tau in TAU(i).

      If IsUpper=False, the matrix Q is represented as a product of elementary
      reflectors

         Q = H(0) H(2) . . . H(n-2).

      Each H(i) has the form

         H(i) = I - tau * v * v'

      where tau is a complex scalar, and v is a complex vector with
      v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i),
      and tau in TAU(i).

      The contents of A on exit are illustrated by the following examples
      with n = 5:

      if UPLO = 'U':                       if UPLO = 'L':

        (  d   e   v1  v2  v3 )              (  d                  )
        (      d   e   v2  v3 )              (  e   d              )
        (          d   e   v3 )              (  v0  e   d          )
        (              d   e  )              (  v0  v1  e   d      )
        (                  d  )              (  v0  v1  v2  e   d  )

    where d and e denote diagonal and off-diagonal elements of T, and vi
    denotes an element of the vector defining H(i).

      -- LAPACK routine (version 3.0) --
         Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
         Courant Institute, Argonne National Lab, and Rice University
         October 31, 1992
    *************************************************************************/
    public static void hmatrixtd(
        ref complex[,] a,
        int n,
        bool isupper,
        out complex[] tau,
        out double[] d,
        out double[] e)

### `hmatrixtdunpackq` function

    /*************************************************************************
    Unpacking matrix Q which reduces a Hermitian matrix to a real  tridiagonal
    form.

    Input parameters:
        A       -   the result of a HMatrixTD subroutine
        N       -   size of matrix A.
        IsUpper -   storage format (a parameter of HMatrixTD subroutine)
        Tau     -   the result of a HMatrixTD subroutine

    Output parameters:
        Q       -   transformation matrix.
                    array with elements [0..N-1, 0..N-1].

      -- ALGLIB --
         Copyright 2005-2010 by Bochkanov Sergey
    *************************************************************************/
    public static void hmatrixtdunpackq(
        complex[,] a,
        int n,
        bool isupper,
        complex[] tau,
        out complex[,] q)

### `rmatrixbd` function

    /*************************************************************************
    Reduction of a rectangular matrix to  bidiagonal form

    The algorithm reduces the rectangular matrix A to  bidiagonal form by
    orthogonal transformations P and Q: A = Q*B*P.

    Input parameters:
        A       -   source matrix. array[0..M-1, 0..N-1]
        M       -   number of rows in matrix A.
        N       -   number of columns in matrix A.

    Output parameters:
        A       -   matrices Q, B, P in compact form (see below).
        TauQ    -   scalar factors which are used to form matrix Q.
        TauP    -   scalar factors which are used to form matrix P.

    The main diagonal and one of the  secondary  diagonals  of  matrix  A  are
    replaced with bidiagonal  matrix  B.  Other  elements  contain  elementary
    reflections which form MxM matrix Q and NxN matrix P, respectively.

    If M>=N, B is the upper  bidiagonal  MxN  matrix  and  is  stored  in  the
    corresponding  elements  of  matrix  A.  Matrix  Q  is  represented  as  a
    product   of   elementary   reflections   Q = H(0)*H(1)*...*H(n-1),  where
    H(i) = 1-tau*v*v'. Here tau is a scalar which is stored  in  TauQ[i],  and
    vector v has the following  structure:  v(0:i-1)=0, v(i)=1, v(i+1:m-1)  is
    stored   in   elements   A(i+1:m-1,i).   Matrix   P  is  as  follows:  P =
    G(0)*G(1)*...*G(n-2), where G(i) = 1 - tau*u*u'. Tau is stored in TauP[i],
    u(0:i)=0, u(i+1)=1, u(i+2:n-1) is stored in elements A(i,i+2:n-1).

    If M<N, B is the  lower  bidiagonal  MxN  matrix  and  is  stored  in  the
    corresponding   elements  of  matrix  A.  Q = H(0)*H(1)*...*H(m-2),  where
    H(i) = 1 - tau*v*v', tau is stored in TauQ, v(0:i)=0, v(i+1)=1, v(i+2:m-1)
    is    stored    in   elements   A(i+2:m-1,i).    P = G(0)*G(1)*...*G(m-1),
    G(i) = 1-tau*u*u', tau is stored in  TauP,  u(0:i-1)=0, u(i)=1, u(i+1:n-1)
    is stored in A(i,i+1:n-1).

    EXAMPLE:

    m=6, n=5 (m > n):               m=5, n=6 (m < n):

    (  d   e   u1  u1  u1 )         (  d   u1  u1  u1  u1  u1 )
    (  v1  d   e   u2  u2 )         (  e   d   u2  u2  u2  u2 )
    (  v1  v2  d   e   u3 )         (  v1  e   d   u3  u3  u3 )
    (  v1  v2  v3  d   e  )         (  v1  v2  e   d   u4  u4 )
    (  v1  v2  v3  v4  d  )         (  v1  v2  v3  e   d   u5 )
    (  v1  v2  v3  v4  v5 )

    Here vi and ui are vectors which form H(i) and G(i), and d and e -
    are the diagonal and off-diagonal elements of matrix B.

      -- LAPACK routine (version 3.0) --
         Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
         Courant Institute, Argonne National Lab, and Rice University
         September 30, 1994.
         Sergey Bochkanov, ALGLIB project, translation from FORTRAN to
         pseudocode, 2007-2010.
    *************************************************************************/
    public static void rmatrixbd(
        ref double[,] a,
        int m,
        int n,
        out double[] tauq,
        out double[] taup)

### `rmatrixbdmultiplybyp` function

    /*************************************************************************
    Multiplication by matrix P which reduces matrix A to  bidiagonal form.

    The algorithm allows pre- or post-multiply by P or P'.

    Input parameters:
        QP          -   matrices Q and P in compact form.
                        Output of RMatrixBD subroutine.
        M           -   number of rows in matrix A.
        N           -   number of columns in matrix A.
        TAUP        -   scalar factors which are used to form P.
                        Output of RMatrixBD subroutine.
        Z           -   multiplied matrix.
                        Array whose indexes range within [0..ZRows-1,0..ZColumns-1].
        ZRows       -   number of rows in matrix Z. If FromTheRight=False,
                        ZRows=N, otherwise ZRows can be arbitrary.
        ZColumns    -   number of columns in matrix Z. If FromTheRight=True,
                        ZColumns=N, otherwise ZColumns can be arbitrary.
        FromTheRight -  pre- or post-multiply.
        DoTranspose -   multiply by P or P'.

    Output parameters:
        Z - product of Z and P.
                    Array whose indexes range within [0..ZRows-1,0..ZColumns-1].
                    If ZRows=0 or ZColumns=0, the array is not modified.

      -- ALGLIB --
         2005-2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixbdmultiplybyp(
        double[,] qp,
        int m,
        int n,
        double[] taup,
        ref double[,] z,
        int zrows,
        int zcolumns,
        bool fromtheright,
        bool dotranspose)

### `rmatrixbdmultiplybyq` function

    /*************************************************************************
    Multiplication by matrix Q which reduces matrix A to  bidiagonal form.

    The algorithm allows pre- or post-multiply by Q or Q'.

    Input parameters:
        QP          -   matrices Q and P in compact form.
                        Output of ToBidiagonal subroutine.
        M           -   number of rows in matrix A.
        N           -   number of columns in matrix A.
        TAUQ        -   scalar factors which are used to form Q.
                        Output of ToBidiagonal subroutine.
        Z           -   multiplied matrix.
                        array[0..ZRows-1,0..ZColumns-1]
        ZRows       -   number of rows in matrix Z. If FromTheRight=False,
                        ZRows=M, otherwise ZRows can be arbitrary.
        ZColumns    -   number of columns in matrix Z. If FromTheRight=True,
                        ZColumns=M, otherwise ZColumns can be arbitrary.
        FromTheRight -  pre- or post-multiply.
        DoTranspose -   multiply by Q or Q'.

    Output parameters:
        Z           -   product of Z and Q.
                        Array[0..ZRows-1,0..ZColumns-1]
                        If ZRows=0 or ZColumns=0, the array is not modified.

      -- ALGLIB --
         2005-2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixbdmultiplybyq(
        double[,] qp,
        int m,
        int n,
        double[] tauq,
        ref double[,] z,
        int zrows,
        int zcolumns,
        bool fromtheright,
        bool dotranspose)

### `rmatrixbdunpackdiagonals` function

    /*************************************************************************
    Unpacking of the main and secondary diagonals of bidiagonal decomposition
    of matrix A.

    Input parameters:
        B   -   output of RMatrixBD subroutine.
        M   -   number of rows in matrix B.
        N   -   number of columns in matrix B.

    Output parameters:
        IsUpper -   True, if the matrix is upper bidiagonal.
                    otherwise IsUpper is False.
        D       -   the main diagonal.
                    Array whose index ranges within [0..Min(M,N)-1].
        E       -   the secondary diagonal (upper or lower, depending on
                    the value of IsUpper).
                    Array index ranges within [0..Min(M,N)-1], the last
                    element is not used.

      -- ALGLIB --
         2005-2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixbdunpackdiagonals(
        double[,] b,
        int m,
        int n,
        out bool isupper,
        out double[] d,
        out double[] e)

### `rmatrixbdunpackpt` function

    /*************************************************************************
    Unpacking matrix P which reduces matrix A to bidiagonal form.
    The subroutine returns transposed matrix P.

    Input parameters:
        QP      -   matrices Q and P in compact form.
                    Output of ToBidiagonal subroutine.
        M       -   number of rows in matrix A.
        N       -   number of columns in matrix A.
        TAUP    -   scalar factors which are used to form P.
                    Output of ToBidiagonal subroutine.
        PTRows  -   required number of rows of matrix P^T. N >= PTRows >= 0.

    Output parameters:
        PT      -   first PTRows columns of matrix P^T
                    Array[0..PTRows-1, 0..N-1]
                    If PTRows=0, the array is not modified.

      -- ALGLIB --
         2005-2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixbdunpackpt(
        double[,] qp,
        int m,
        int n,
        double[] taup,
        int ptrows,
        out double[,] pt)

### `rmatrixbdunpackq` function

    /*************************************************************************
    Unpacking matrix Q which reduces a matrix to bidiagonal form.

    Input parameters:
        QP          -   matrices Q and P in compact form.
                        Output of ToBidiagonal subroutine.
        M           -   number of rows in matrix A.
        N           -   number of columns in matrix A.
        TAUQ        -   scalar factors which are used to form Q.
                        Output of ToBidiagonal subroutine.
        QColumns    -   required number of columns in matrix Q.
                        M>=QColumns>=0.

    Output parameters:
        Q           -   first QColumns columns of matrix Q.
                        Array[0..M-1, 0..QColumns-1]
                        If QColumns=0, the array is not modified.

      -- ALGLIB --
         2005-2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixbdunpackq(
        double[,] qp,
        int m,
        int n,
        double[] tauq,
        int qcolumns,
        out double[,] q)

### `rmatrixhessenberg` function

    /*************************************************************************
    Reduction of a square matrix to  upper Hessenberg form: Q'*A*Q = H,
    where Q is an orthogonal matrix, H - Hessenberg matrix.

    Input parameters:
        A       -   matrix A with elements [0..N-1, 0..N-1]
        N       -   size of matrix A.

    Output parameters:
        A       -   matrices Q and P in  compact form (see below).
        Tau     -   array of scalar factors which are used to form matrix Q.
                    Array whose index ranges within [0..N-2]

    Matrix H is located on the main diagonal, on the lower secondary  diagonal
    and above the main diagonal of matrix A. The elements which are used to
    form matrix Q are situated in array Tau and below the lower secondary
    diagonal of matrix A as follows:

    Matrix Q is represented as a product of elementary reflections

    Q = H(0)*H(2)*...*H(n-2),

    where each H(i) is given by

    H(i) = 1 - tau * v * (v^T)

    where tau is a scalar stored in Tau[I]; v - is a real vector,
    so that v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) stored in A(i+2:n-1,i).

      -- LAPACK routine (version 3.0) --
         Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
         Courant Institute, Argonne National Lab, and Rice University
         October 31, 1992
    *************************************************************************/
    public static void rmatrixhessenberg(
        ref double[,] a,
        int n,
        out double[] tau)

### `rmatrixhessenbergunpackh` function

    /*************************************************************************
    Unpacking matrix H (the result of matrix A reduction to upper Hessenberg form)

    Input parameters:
        A   -   output of RMatrixHessenberg subroutine.
        N   -   size of matrix A.

    Output parameters:
        H   -   matrix H. Array whose indexes range within [0..N-1, 0..N-1].

      -- ALGLIB --
         2005-2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixhessenbergunpackh(
        double[,] a,
        int n,
        out double[,] h)

### `rmatrixhessenbergunpackq` function

    /*************************************************************************
    Unpacking matrix Q which reduces matrix A to upper Hessenberg form

    Input parameters:
        A   -   output of RMatrixHessenberg subroutine.
        N   -   size of matrix A.
        Tau -   scalar factors which are used to form Q.
                Output of RMatrixHessenberg subroutine.

    Output parameters:
        Q   -   matrix Q.
                Array whose indexes range within [0..N-1, 0..N-1].

      -- ALGLIB --
         2005-2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixhessenbergunpackq(
        double[,] a,
        int n,
        double[] tau,
        out double[,] q)

### `rmatrixlq` function

    /*************************************************************************
    LQ decomposition of a rectangular matrix of size MxN

    Input parameters:
        A   -   matrix A whose indexes range within [0..M-1, 0..N-1].
        M   -   number of rows in matrix A.
        N   -   number of columns in matrix A.

    Output parameters:
        A   -   matrices L and Q in compact form (see below)
        Tau -   array of scalar factors which are used to form
                matrix Q. Array whose index ranges within [0..Min(M,N)-1].

    Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size
    MxM, L - lower triangular (or lower trapezoid) matrix of size M x N.

    The elements of matrix L are located on and below  the  main  diagonal  of
    matrix A. The elements which are located in Tau array and above  the  main
    diagonal of matrix A are used to form matrix Q as follows:

    Matrix Q is represented as a product of elementary reflections

    Q = H(k-1)*H(k-2)*...*H(1)*H(0),

    where k = min(m,n), and each H(i) is of the form

    H(i) = 1 - tau * v * (v^T)

    where tau is a scalar stored in Tau[I]; v - real vector, so that v(0:i-1)=0,
    v(i) = 1, v(i+1:n-1) stored in A(i,i+1:n-1).

      -- ALGLIB routine --
         17.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixlq(
        ref double[,] a,
        int m,
        int n,
        out double[] tau)

### `rmatrixlqunpackl` function

    /*************************************************************************
    Unpacking of matrix L from the LQ decomposition of a matrix A

    Input parameters:
        A       -   matrices Q and L in compact form.
                    Output of RMatrixLQ subroutine.
        M       -   number of rows in given matrix A. M>=0.
        N       -   number of columns in given matrix A. N>=0.

    Output parameters:
        L       -   matrix L, array[0..M-1, 0..N-1].

      -- ALGLIB routine --
         17.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixlqunpackl(
        double[,] a,
        int m,
        int n,
        out double[,] l)

### `rmatrixlqunpackq` function

    /*************************************************************************
    Partial unpacking of matrix Q from the LQ decomposition of a matrix A

    Input parameters:
        A       -   matrices L and Q in compact form.
                    Output of RMatrixLQ subroutine.
        M       -   number of rows in given matrix A. M>=0.
        N       -   number of columns in given matrix A. N>=0.
        Tau     -   scalar factors which are used to form Q.
                    Output of the RMatrixLQ subroutine.
        QRows   -   required number of rows in matrix Q. N>=QRows>=0.

    Output parameters:
        Q       -   first QRows rows of matrix Q. Array whose indexes range
                    within [0..QRows-1, 0..N-1]. If QRows=0, the array remains
                    unchanged.

      -- ALGLIB routine --
         17.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixlqunpackq(
        double[,] a,
        int m,
        int n,
        double[] tau,
        int qrows,
        out double[,] q)

### `rmatrixqr` function

    /*************************************************************************
    QR decomposition of a rectangular matrix of size MxN

    Input parameters:
        A   -   matrix A whose indexes range within [0..M-1, 0..N-1].
        M   -   number of rows in matrix A.
        N   -   number of columns in matrix A.

    Output parameters:
        A   -   matrices Q and R in compact form (see below).
        Tau -   array of scalar factors which are used to form
                matrix Q. Array whose index ranges within [0.. Min(M-1,N-1)].

    Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
    MxM, R - upper triangular (or upper trapezoid) matrix of size M x N.

    The elements of matrix R are located on and above the main diagonal of
    matrix A. The elements which are located in Tau array and below the main
    diagonal of matrix A are used to form matrix Q as follows:

    Matrix Q is represented as a product of elementary reflections

    Q = H(0)*H(2)*...*H(k-1),

    where k = min(m,n), and each H(i) is in the form

    H(i) = 1 - tau * v * (v^T)

    where tau is a scalar stored in Tau[I]; v - real vector,
    so that v(0:i-1) = 0, v(i) = 1, v(i+1:m-1) stored in A(i+1:m-1,i).

      -- ALGLIB routine --
         17.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixqr(
        ref double[,] a,
        int m,
        int n,
        out double[] tau)

### `rmatrixqrunpackq` function

    /*************************************************************************
    Partial unpacking of matrix Q from the QR decomposition of a matrix A

    Input parameters:
        A       -   matrices Q and R in compact form.
                    Output of RMatrixQR subroutine.
        M       -   number of rows in given matrix A. M>=0.
        N       -   number of columns in given matrix A. N>=0.
        Tau     -   scalar factors which are used to form Q.
                    Output of the RMatrixQR subroutine.
        QColumns -  required number of columns of matrix Q. M>=QColumns>=0.

    Output parameters:
        Q       -   first QColumns columns of matrix Q.
                    Array whose indexes range within [0..M-1, 0..QColumns-1].
                    If QColumns=0, the array remains unchanged.

      -- ALGLIB routine --
         17.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixqrunpackq(
        double[,] a,
        int m,
        int n,
        double[] tau,
        int qcolumns,
        out double[,] q)

### `rmatrixqrunpackr` function

    /*************************************************************************
    Unpacking of matrix R from the QR decomposition of a matrix A

    Input parameters:
        A       -   matrices Q and R in compact form.
                    Output of RMatrixQR subroutine.
        M       -   number of rows in given matrix A. M>=0.
        N       -   number of columns in given matrix A. N>=0.

    Output parameters:
        R       -   matrix R, array[0..M-1, 0..N-1].

      -- ALGLIB routine --
         17.02.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixqrunpackr(
        double[,] a,
        int m,
        int n,
        out double[,] r)

### `smatrixtd` function

    /*************************************************************************
    Reduction of a symmetric matrix which is given by its higher or lower
    triangular part to a tridiagonal matrix using orthogonal similarity
    transformation: Q'*A*Q=T.

    Input parameters:
        A       -   matrix to be transformed
                    array with elements [0..N-1, 0..N-1].
        N       -   size of matrix A.
        IsUpper -   storage format. If IsUpper = True, then matrix A is given
                    by its upper triangle, and the lower triangle is not used
                    and not modified by the algorithm, and vice versa
                    if IsUpper = False.

    Output parameters:
        A       -   matrices T and Q in  compact form (see lower)
        Tau     -   array of factors which are forming matrices H(i)
                    array with elements [0..N-2].
        D       -   main diagonal of symmetric matrix T.
                    array with elements [0..N-1].
        E       -   secondary diagonal of symmetric matrix T.
                    array with elements [0..N-2].

      If IsUpper=True, the matrix Q is represented as a product of elementary
      reflectors

         Q = H(n-2) . . . H(2) H(0).

      Each H(i) has the form

         H(i) = I - tau * v * v'

      where tau is a real scalar, and v is a real vector with
      v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in
      A(0:i-1,i+1), and tau in TAU(i).

      If IsUpper=False, the matrix Q is represented as a product of elementary
      reflectors

         Q = H(0) H(2) . . . H(n-2).

      Each H(i) has the form

         H(i) = I - tau * v * v'

      where tau is a real scalar, and v is a real vector with
      v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i),
      and tau in TAU(i).

      The contents of A on exit are illustrated by the following examples
      with n = 5:

      if UPLO = 'U':                       if UPLO = 'L':

        (  d   e   v1  v2  v3 )              (  d                  )
        (      d   e   v2  v3 )              (  e   d              )
        (          d   e   v3 )              (  v0  e   d          )
        (              d   e  )              (  v0  v1  e   d      )
        (                  d  )              (  v0  v1  v2  e   d  )

      where d and e denote diagonal and off-diagonal elements of T, and vi
      denotes an element of the vector defining H(i).

      -- LAPACK routine (version 3.0) --
         Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
         Courant Institute, Argonne National Lab, and Rice University
         October 31, 1992
    *************************************************************************/
    public static void smatrixtd(
        ref double[,] a,
        int n,
        bool isupper,
        out double[] tau,
        out double[] d,
        out double[] e)

### `smatrixtdunpackq` function

    /*************************************************************************
    Unpacking matrix Q which reduces symmetric matrix to a tridiagonal
    form.

    Input parameters:
        A       -   the result of a SMatrixTD subroutine
        N       -   size of matrix A.
        IsUpper -   storage format (a parameter of SMatrixTD subroutine)
        Tau     -   the result of a SMatrixTD subroutine

    Output parameters:
        Q       -   transformation matrix.
                    array with elements [0..N-1, 0..N-1].

      -- ALGLIB --
         Copyright 2005-2010 by Bochkanov Sergey
    *************************************************************************/
    public static void smatrixtdunpackq(
        double[,] a,
        int n,
        bool isupper,
        double[] tau,
        out double[,] q)

## `pca` subpackage

### Functions

pcabuildbasis

### Examples

### `pcabuildbasis` function

    /*************************************************************************
    Principal components analysis

    Subroutine  builds  orthogonal  basis  where  first  axis  corresponds  to
    direction with maximum variance, second axis maximizes variance in subspace
    orthogonal to first axis and so on.

    It should be noted that, unlike LDA, PCA does not use class labels.

    INPUT PARAMETERS:
        X           -   dataset, array[0..NPoints-1,0..NVars-1].
                        matrix contains ONLY INDEPENDENT VARIABLES.
        NPoints     -   dataset size, NPoints>=0
        NVars       -   number of independent variables, NVars>=1

    ÂÛÕÎÄÍÛÅ ÏÀÐÀÌÅÒÐÛ:
        Info        -   return code:
                        * -4, if SVD subroutine haven't converged
                        * -1, if wrong parameters has been passed (NPoints<0,
                              NVars<1)
                        *  1, if task is solved
        S2          -   array[0..NVars-1]. variance values corresponding
                        to basis vectors.
        V           -   array[0..NVars-1,0..NVars-1]
                        matrix, whose columns store basis vectors.

      -- ALGLIB --
         Copyright 25.08.2008 by Bochkanov Sergey
    *************************************************************************/
    public static void pcabuildbasis(
        double[,] x,
        int npoints,
        int nvars,
        out int info,
        out double[] s2,
        out double[,] v)

## `poissondistr` subpackage

### Functions

invpoissondistribution
poissoncdistribution
poissondistribution

### Examples

### `invpoissondistribution` function

    /*************************************************************************
    Inverse Poisson distribution

    Finds the Poisson variable x such that the integral
    from 0 to x of the Poisson density is equal to the
    given probability y.

    This is accomplished using the inverse gamma integral
    function and the relation

       m = igami( k+1, y ).

    ACCURACY:

    See inverse incomplete gamma function

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double invpoissondistribution(int k, double y)

### `poissoncdistribution` function

    /*************************************************************************
    Complemented Poisson distribution

    Returns the sum of the terms k+1 to infinity of the Poisson
    distribution:

     inf.       j
      --   -m  m
      >   e    --
      --       j!
     j=k+1

    The terms are not summed directly; instead the incomplete
    gamma integral is employed, according to the formula

    y = pdtrc( k, m ) = igam( k+1, m ).

    The arguments must both be positive.

    ACCURACY:

    See incomplete gamma function

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double poissoncdistribution(int k, double m)

### `poissondistribution` function

    /*************************************************************************
    Poisson distribution

    Returns the sum of the first k+1 terms of the Poisson
    distribution:

      k         j
      --   -m  m
      >   e    --
      --       j!
     j=0

    The terms are not summed directly; instead the incomplete
    gamma integral is employed, according to the relation

    y = pdtr( k, m ) = igamc( k+1, m ).

    The arguments must both be positive.
    ACCURACY:

    See incomplete gamma function

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double poissondistribution(int k, double m)

## `polint` subpackage

### Functions

polynomialbar2cheb
polynomialbar2pow
polynomialbuild
polynomialbuildcheb1
polynomialbuildcheb2
polynomialbuildeqdist
polynomialcalccheb1
polynomialcalccheb2
polynomialcalceqdist
polynomialcheb2bar
polynomialpow2bar

### Examples

polint_d_calcdiff |  | Interpolation and differentiation using barycentric representation
polint_d_conv |  | Conversion between power basis and barycentric representation
polint_d_spec |  | Polynomial interpolation on special grids (equidistant, Chebyshev I/II)

### `polynomialbar2cheb` function

    /*************************************************************************
    Conversion from barycentric representation to Chebyshev basis.
    This function has O(N^2) complexity.

    INPUT PARAMETERS:
        P   -   polynomial in barycentric form
        A,B -   base interval for Chebyshev polynomials (see below)
                A<>B

    OUTPUT PARAMETERS
        T   -   coefficients of Chebyshev representation;
                P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N-1 },
                where Ti - I-th Chebyshev polynomial.

    NOTES:
        barycentric interpolant passed as P may be either polynomial  obtained
        from  polynomial  interpolation/ fitting or rational function which is
        NOT polynomial. We can't distinguish between these two cases, and this
        algorithm just tries to work assuming that P IS a polynomial.  If not,
        algorithm will return results, but they won't have any meaning.

      -- ALGLIB --
         Copyright 30.09.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void polynomialbar2cheb(
        barycentricinterpolant p,
        double a,
        double b,
        out double[] t)

### `polynomialbar2pow` function

    /*************************************************************************
    Conversion from barycentric representation to power basis.
    This function has O(N^2) complexity.

    INPUT PARAMETERS:
        P   -   polynomial in barycentric form
        C   -   offset (see below); 0.0 is used as default value.
        S   -   scale (see below);  1.0 is used as default value. S<>0.

    OUTPUT PARAMETERS
        A   -   coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
        N   -   number of coefficients (polynomial degree plus 1)

    NOTES:
    1.  this function accepts offset and scale, which can be  set  to  improve
        numerical properties of polynomial. For example, if P was obtained  as
        result of interpolation on [-1,+1],  you  can  set  C=0  and  S=1  and
        represent  P  as sum of 1, x, x^2, x^3 and so on. In most cases you it
        is exactly what you need.

        However, if your interpolation model was built on [999,1001], you will
        see significant growth of numerical errors when using {1, x, x^2, x^3}
        as basis. Representing P as sum of 1, (x-1000), (x-1000)^2, (x-1000)^3
        will be better option. Such representation can be  obtained  by  using
        1000.0 as offset C and 1.0 as scale S.

    2.  power basis is ill-conditioned and tricks described above can't  solve
        this problem completely. This function  will  return  coefficients  in
        any  case,  but  for  N>8  they  will  become unreliable. However, N's
        less than 5 are pretty safe.

    3.  barycentric interpolant passed as P may be either polynomial  obtained
        from  polynomial  interpolation/ fitting or rational function which is
        NOT polynomial. We can't distinguish between these two cases, and this
        algorithm just tries to work assuming that P IS a polynomial.  If not,
        algorithm will return results, but they won't have any meaning.

      -- ALGLIB --
         Copyright 30.09.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void polynomialbar2pow(
        barycentricinterpolant p,
        out double[] a)
    public static void polynomialbar2pow(
        barycentricinterpolant p,
        double c,
        double s,
        out double[] a)

Examples: [1]

### `polynomialbuild` function

    /*************************************************************************
    Lagrange intepolant: generation of the model on the general grid.
    This function has O(N^2) complexity.

    INPUT PARAMETERS:
        X   -   abscissas, array[0..N-1]
        Y   -   function values, array[0..N-1]
        N   -   number of points, N>=1

    OUTPUT PARAMETERS
        P   -   barycentric model which represents Lagrange interpolant
                (see ratint unit info and BarycentricCalc() description for
                more information).

      -- ALGLIB --
         Copyright 02.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void polynomialbuild(
        double[] x,
        double[] y,
        out barycentricinterpolant p)
    public static void polynomialbuild(
        double[] x,
        double[] y,
        int n,
        out barycentricinterpolant p)

Examples: [1]

### `polynomialbuildcheb1` function

    /*************************************************************************
    Lagrange intepolant on Chebyshev grid (first kind).
    This function has O(N) complexity.

    INPUT PARAMETERS:
        A   -   left boundary of [A,B]
        B   -   right boundary of [A,B]
        Y   -   function values at the nodes, array[0..N-1],
                Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n)))
        N   -   number of points, N>=1
                for N=1 a constant model is constructed.

    OUTPUT PARAMETERS
        P   -   barycentric model which represents Lagrange interpolant
                (see ratint unit info and BarycentricCalc() description for
                more information).

      -- ALGLIB --
         Copyright 03.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void polynomialbuildcheb1(
        double a,
        double b,
        double[] y,
        out barycentricinterpolant p)
    public static void polynomialbuildcheb1(
        double a,
        double b,
        double[] y,
        int n,
        out barycentricinterpolant p)

Examples: [1]

### `polynomialbuildcheb2` function

    /*************************************************************************
    Lagrange intepolant on Chebyshev grid (second kind).
    This function has O(N) complexity.

    INPUT PARAMETERS:
        A   -   left boundary of [A,B]
        B   -   right boundary of [A,B]
        Y   -   function values at the nodes, array[0..N-1],
                Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1)))
        N   -   number of points, N>=1
                for N=1 a constant model is constructed.

    OUTPUT PARAMETERS
        P   -   barycentric model which represents Lagrange interpolant
                (see ratint unit info and BarycentricCalc() description for
                more information).

      -- ALGLIB --
         Copyright 03.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void polynomialbuildcheb2(
        double a,
        double b,
        double[] y,
        out barycentricinterpolant p)
    public static void polynomialbuildcheb2(
        double a,
        double b,
        double[] y,
        int n,
        out barycentricinterpolant p)

Examples: [1]

### `polynomialbuildeqdist` function

    /*************************************************************************
    Lagrange intepolant: generation of the model on equidistant grid.
    This function has O(N) complexity.

    INPUT PARAMETERS:
        A   -   left boundary of [A,B]
        B   -   right boundary of [A,B]
        Y   -   function values at the nodes, array[0..N-1]
        N   -   number of points, N>=1
                for N=1 a constant model is constructed.

    OUTPUT PARAMETERS
        P   -   barycentric model which represents Lagrange interpolant
                (see ratint unit info and BarycentricCalc() description for
                more information).

      -- ALGLIB --
         Copyright 03.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void polynomialbuildeqdist(
        double a,
        double b,
        double[] y,
        out barycentricinterpolant p)
    public static void polynomialbuildeqdist(
        double a,
        double b,
        double[] y,
        int n,
        out barycentricinterpolant p)

Examples: [1]

### `polynomialcalccheb1` function

    /*************************************************************************
    Fast polynomial interpolation function on Chebyshev points (first kind)
    with O(N) complexity.

    INPUT PARAMETERS:
        A   -   left boundary of [A,B]
        B   -   right boundary of [A,B]
        F   -   function values, array[0..N-1]
        N   -   number of points on Chebyshev grid (first kind),
                X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n))
                for N=1 a constant model is constructed.
        T   -   position where P(x) is calculated

    RESULT
        value of the Lagrange interpolant at T

    IMPORTANT
        this function provides fast interface which is not overflow-safe
        nor it is very precise.
        the best option is to use  PolIntBuildCheb1()/BarycentricCalc()
        subroutines unless you are pretty sure that your data will not result
        in overflow.

      -- ALGLIB --
         Copyright 02.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double polynomialcalccheb1(
        double a,
        double b,
        double[] f,
        double t)
    public static double polynomialcalccheb1(
        double a,
        double b,
        double[] f,
        int n,
        double t)

Examples: [1]

### `polynomialcalccheb2` function

    /*************************************************************************
    Fast polynomial interpolation function on Chebyshev points (second kind)
    with O(N) complexity.

    INPUT PARAMETERS:
        A   -   left boundary of [A,B]
        B   -   right boundary of [A,B]
        F   -   function values, array[0..N-1]
        N   -   number of points on Chebyshev grid (second kind),
                X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1))
                for N=1 a constant model is constructed.
        T   -   position where P(x) is calculated

    RESULT
        value of the Lagrange interpolant at T

    IMPORTANT
        this function provides fast interface which is not overflow-safe
        nor it is very precise.
        the best option is to use PolIntBuildCheb2()/BarycentricCalc()
        subroutines unless you are pretty sure that your data will not result
        in overflow.

      -- ALGLIB --
         Copyright 02.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double polynomialcalccheb2(
        double a,
        double b,
        double[] f,
        double t)
    public static double polynomialcalccheb2(
        double a,
        double b,
        double[] f,
        int n,
        double t)

Examples: [1]

### `polynomialcalceqdist` function

    /*************************************************************************
    Fast equidistant polynomial interpolation function with O(N) complexity

    INPUT PARAMETERS:
        A   -   left boundary of [A,B]
        B   -   right boundary of [A,B]
        F   -   function values, array[0..N-1]
        N   -   number of points on equidistant grid, N>=1
                for N=1 a constant model is constructed.
        T   -   position where P(x) is calculated

    RESULT
        value of the Lagrange interpolant at T

    IMPORTANT
        this function provides fast interface which is not overflow-safe
        nor it is very precise.
        the best option is to use  PolynomialBuildEqDist()/BarycentricCalc()
        subroutines unless you are pretty sure that your data will not result
        in overflow.

      -- ALGLIB --
         Copyright 02.12.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double polynomialcalceqdist(
        double a,
        double b,
        double[] f,
        double t)
    public static double polynomialcalceqdist(
        double a,
        double b,
        double[] f,
        int n,
        double t)

Examples: [1]

### `polynomialcheb2bar` function

    /*************************************************************************
    Conversion from Chebyshev basis to barycentric representation.
    This function has O(N^2) complexity.

    INPUT PARAMETERS:
        T   -   coefficients of Chebyshev representation;
                P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N },
                where Ti - I-th Chebyshev polynomial.
        N   -   number of coefficients:
                * if given, only leading N elements of T are used
                * if not given, automatically determined from size of T
        A,B -   base interval for Chebyshev polynomials (see above)
                A<B

    OUTPUT PARAMETERS
        P   -   polynomial in barycentric form

      -- ALGLIB --
         Copyright 30.09.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void polynomialcheb2bar(
        double[] t,
        double a,
        double b,
        out barycentricinterpolant p)
    public static void polynomialcheb2bar(
        double[] t,
        int n,
        double a,
        double b,
        out barycentricinterpolant p)

### `polynomialpow2bar` function

    /*************************************************************************
    Conversion from power basis to barycentric representation.
    This function has O(N^2) complexity.

    INPUT PARAMETERS:
        A   -   coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
        N   -   number of coefficients (polynomial degree plus 1)
                * if given, only leading N elements of A are used
                * if not given, automatically determined from size of A
        C   -   offset (see below); 0.0 is used as default value.
        S   -   scale (see below);  1.0 is used as default value. S<>0.

    OUTPUT PARAMETERS
        P   -   polynomial in barycentric form

    NOTES:
    1.  this function accepts offset and scale, which can be  set  to  improve
        numerical properties of polynomial. For example, if you interpolate on
        [-1,+1],  you  can  set C=0 and S=1 and convert from sum of 1, x, x^2,
        x^3 and so on. In most cases you it is exactly what you need.

        However, if your interpolation model was built on [999,1001], you will
        see significant growth of numerical errors when using {1, x, x^2, x^3}
        as  input  basis.  Converting  from  sum  of  1, (x-1000), (x-1000)^2,
        (x-1000)^3 will be better option (you have to specify 1000.0 as offset
        C and 1.0 as scale S).

    2.  power basis is ill-conditioned and tricks described above can't  solve
        this problem completely. This function  will  return barycentric model
        in any case, but for N>8 accuracy well degrade. However, N's less than
        5 are pretty safe.

      -- ALGLIB --
         Copyright 30.09.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void polynomialpow2bar(
        double[] a,
        out barycentricinterpolant p)
    public static void polynomialpow2bar(
        double[] a,
        int n,
        double c,
        double s,
        out barycentricinterpolant p)

Examples: [1]

### polint_d_calcdiff example

    public static int Main(string[] args)
    {
        //
        // Here we demonstrate polynomial interpolation and differentiation
        // of y=x^2-x sampled at [0,1,2]. Barycentric representation of polynomial is used.
        //
        double[] x = new double[]{0,1,2};
        double[] y = new double[]{0,0,2};
        double t = -1;
        double v;
        double dv;
        double d2v;
        alglib.barycentricinterpolant p;

        // barycentric model is created
        alglib.polynomialbuild(x, y, out p);

        // barycentric interpolation is demonstrated
        v = alglib.barycentriccalc(p, t);
        System.Console.WriteLine("{0:F4", v); // EXPECTED: 2.0

        // barycentric differentation is demonstrated
        alglib.barycentricdiff1(p, t, out v, out dv);
        System.Console.WriteLine("{0:F4", v); // EXPECTED: 2.0
        System.Console.WriteLine("{0:F4", dv); // EXPECTED: -3.0

        // second derivatives with barycentric representation
        alglib.barycentricdiff1(p, t, out v, out dv);
        System.Console.WriteLine("{0:F4", v); // EXPECTED: 2.0
        System.Console.WriteLine("{0:F4", dv); // EXPECTED: -3.0
        alglib.barycentricdiff2(p, t, out v, out dv, out d2v);
        System.Console.WriteLine("{0:F4", v); // EXPECTED: 2.0
        System.Console.WriteLine("{0:F4", dv); // EXPECTED: -3.0
        System.Console.WriteLine("{0:F4", d2v); // EXPECTED: 2.0
        System.Console.ReadLine();
        return 0;
    }

### polint_d_conv example

    public static int Main(string[] args)
    {
        //
        // Here we demonstrate conversion of y=x^2-x
        // between power basis and barycentric representation.
        //
        double[] a = new double[]{0,-1,+1};
        double t = 2;
        double[] a2;
        double v;
        alglib.barycentricinterpolant p;

        //
        // a=[0,-1,+1] is decomposition of y=x^2-x in the power basis:
        //
        //     y = 0 - 1*x + 1*x^2
        //
        // We convert it to the barycentric form.
        //
        alglib.polynomialpow2bar(a, out p);

        // now we have barycentric interpolation; we can use it for interpolation
        v = alglib.barycentriccalc(p, t);
        System.Console.WriteLine("{0:F2", v); // EXPECTED: 2.0

        // we can also convert back from barycentric representation to power basis
        alglib.polynomialbar2pow(p, out a2);
        System.Console.WriteLine("{0}", alglib.ap.format(a2,2)); // EXPECTED: [0,-1,+1]
        System.Console.ReadLine();
        return 0;
    }

### polint_d_spec example

    public static int Main(string[] args)
    {
        //
        // Temporaries:
        // * values of y=x^2-x sampled at three special grids:
        //   * equdistant grid spanning [0,2],     x[i] = 2*i/(N-1), i=0..N-1
        //   * Chebyshev-I grid spanning [-1,+1],  x[i] = 1 + Cos(PI*(2*i+1)/(2*n)), i=0..N-1
        //   * Chebyshev-II grid spanning [-1,+1], x[i] = 1 + Cos(PI*i/(n-1)), i=0..N-1
        // * barycentric interpolants for these three grids
        // * vectors to store coefficients of quadratic representation
        //
        double[] y_eqdist = new double[]{0,0,2};
        double[] y_cheb1 = new double[]{-0.116025,0.000000,1.616025};
        double[] y_cheb2 = new double[]{0,0,2};
        alglib.barycentricinterpolant p_eqdist;
        alglib.barycentricinterpolant p_cheb1;
        alglib.barycentricinterpolant p_cheb2;
        double[] a_eqdist;
        double[] a_cheb1;
        double[] a_cheb2;

        //
        // First, we demonstrate construction of barycentric interpolants on
        // special grids. We unpack power representation to ensure that
        // interpolant was built correctly.
        //
        // In all three cases we should get same quadratic function.
        //
        alglib.polynomialbuildeqdist(0.0, 2.0, y_eqdist, out p_eqdist);
        alglib.polynomialbar2pow(p_eqdist, out a_eqdist);
        System.Console.WriteLine("{0}", alglib.ap.format(a_eqdist,4)); // EXPECTED: [0,-1,+1]

        alglib.polynomialbuildcheb1(-1, +1, y_cheb1, out p_cheb1);
        alglib.polynomialbar2pow(p_cheb1, out a_cheb1);
        System.Console.WriteLine("{0}", alglib.ap.format(a_cheb1,4)); // EXPECTED: [0,-1,+1]

        alglib.polynomialbuildcheb2(-1, +1, y_cheb2, out p_cheb2);
        alglib.polynomialbar2pow(p_cheb2, out a_cheb2);
        System.Console.WriteLine("{0}", alglib.ap.format(a_cheb2,4)); // EXPECTED: [0,-1,+1]

        //
        // Now we demonstrate polynomial interpolation without construction
        // of the barycentricinterpolant structure.
        //
        // We calculate interpolant value at x=-2.
        // In all three cases we should get same f=6
        //
        double t = -2;
        double v;
        v = alglib.polynomialcalceqdist(0.0, 2.0, y_eqdist, t);
        System.Console.WriteLine("{0:F4", v); // EXPECTED: 6.0

        v = alglib.polynomialcalccheb1(-1, +1, y_cheb1, t);
        System.Console.WriteLine("{0:F4", v); // EXPECTED: 6.0

        v = alglib.polynomialcalccheb2(-1, +1, y_cheb2, t);
        System.Console.WriteLine("{0:F4", v); // EXPECTED: 6.0
        System.Console.ReadLine();
        return 0;
    }

## `psif` subpackage

### Functions

psi

### Examples

### `psi` function

    /*************************************************************************
    Psi (digamma) function

                 d      -
      psi(x)  =  -- ln | (x)
                 dx

    is the logarithmic derivative of the gamma function.
    For integer x,
                      n-1
                       -
    psi(n) = -EUL  +   >  1/k.
                       -
                      k=1

    This formula is used for 0 < n <= 10.  If x is negative, it
    is transformed to a positive argument by the reflection
    formula  psi(1-x) = psi(x) + pi cot(pi x).
    For general positive x, the argument is made greater than 10
    using the recurrence  psi(x+1) = psi(x) + 1/x.
    Then the following asymptotic expansion is applied:

                              inf.   B
                               -      2k
    psi(x) = log(x) - 1/2x -   >   -------
                               -        2k
                              k=1   2k x

    where the B2k are Bernoulli numbers.

    ACCURACY:
       Relative error (except absolute when |psi| < 1):
    arithmetic   domain     # trials      peak         rms
       IEEE      0,30        30000       1.3e-15     1.4e-16
       IEEE      -30,0       40000       1.5e-15     2.2e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double psi(double x)

## `pspline` subpackage

### Classes

pspline2interpolant
pspline3interpolant

### Functions

pspline2arclength
pspline2build
pspline2buildperiodic
pspline2calc
pspline2diff
pspline2diff2
pspline2parametervalues
pspline2tangent
pspline3arclength
pspline3build
pspline3buildperiodic
pspline3calc
pspline3diff
pspline3diff2
pspline3parametervalues
pspline3tangent

### Examples

### `pspline2interpolant` class

    /*************************************************************************
    Parametric spline inteprolant: 2-dimensional curve.

    You should not try to access its members directly - use PSpline2XXXXXXXX()
    functions instead.
    *************************************************************************/
    public class pspline2interpolant
    {
    }

### `pspline3interpolant` class

    /*************************************************************************
    Parametric spline inteprolant: 3-dimensional curve.

    You should not try to access its members directly - use PSpline3XXXXXXXX()
    functions instead.
    *************************************************************************/
    public class pspline3interpolant
    {
    }

### `pspline2arclength` function

    /*************************************************************************
    This function  calculates  arc length, i.e. length of  curve  between  t=a
    and t=b.

    INPUT PARAMETERS:
        P   -   parametric spline interpolant
        A,B -   parameter values corresponding to arc ends:
                * B>A will result in positive length returned
                * B<A will result in negative length returned

    RESULT:
        length of arc starting at T=A and ending at T=B.

      -- ALGLIB PROJECT --
         Copyright 30.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static double pspline2arclength(
        pspline2interpolant p,
        double a,
        double b)

### `pspline2build` function

    /*************************************************************************
    This function  builds  non-periodic 2-dimensional parametric spline  which
    starts at (X[0],Y[0]) and ends at (X[N-1],Y[N-1]).

    INPUT PARAMETERS:
        XY  -   points, array[0..N-1,0..1].
                XY[I,0:1] corresponds to the Ith point.
                Order of points is important!
        N   -   points count, N>=5 for Akima splines, N>=2 for other types  of
                splines.
        ST  -   spline type:
                * 0     Akima spline
                * 1     parabolically terminated Catmull-Rom spline (Tension=0)
                * 2     parabolically terminated cubic spline
        PT  -   parameterization type:
                * 0     uniform
                * 1     chord length
                * 2     centripetal

    OUTPUT PARAMETERS:
        P   -   parametric spline interpolant

    NOTES:
    * this function  assumes  that  there all consequent points  are distinct.
      I.e. (x0,y0)<>(x1,y1),  (x1,y1)<>(x2,y2),  (x2,y2)<>(x3,y3)  and  so on.
      However, non-consequent points may coincide, i.e. we can  have  (x0,y0)=
      =(x2,y2).

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline2build(
        double[,] xy,
        int n,
        int st,
        int pt,
        out pspline2interpolant p)

### `pspline2buildperiodic` function

    /*************************************************************************
    This  function  builds  periodic  2-dimensional  parametric  spline  which
    starts at (X[0],Y[0]), goes through all points to (X[N-1],Y[N-1]) and then
    back to (X[0],Y[0]).

    INPUT PARAMETERS:
        XY  -   points, array[0..N-1,0..1].
                XY[I,0:1] corresponds to the Ith point.
                XY[N-1,0:1] must be different from XY[0,0:1].
                Order of points is important!
        N   -   points count, N>=3 for other types of splines.
        ST  -   spline type:
                * 1     Catmull-Rom spline (Tension=0) with cyclic boundary conditions
                * 2     cubic spline with cyclic boundary conditions
        PT  -   parameterization type:
                * 0     uniform
                * 1     chord length
                * 2     centripetal

    OUTPUT PARAMETERS:
        P   -   parametric spline interpolant

    NOTES:
    * this function  assumes  that there all consequent points  are  distinct.
      I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2),  (x2,y2)<>(x3,y3)  and  so  on.
      However, non-consequent points may coincide, i.e. we can  have  (x0,y0)=
      =(x2,y2).
    * last point of sequence is NOT equal to the first  point.  You  shouldn't
      make curve "explicitly periodic" by making them equal.

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline2buildperiodic(
        double[,] xy,
        int n,
        int st,
        int pt,
        out pspline2interpolant p)

### `pspline2calc` function

    /*************************************************************************
    This function  calculates  the value of the parametric spline for a  given
    value of parameter T

    INPUT PARAMETERS:
        P   -   parametric spline interpolant
        T   -   point:
                * T in [0,1] corresponds to interval spanned by points
                * for non-periodic splines T<0 (or T>1) correspond to parts of
                  the curve before the first (after the last) point
                * for periodic splines T<0 (or T>1) are projected  into  [0,1]
                  by making T=T-floor(T).

    OUTPUT PARAMETERS:
        X   -   X-position
        Y   -   Y-position

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline2calc(
        pspline2interpolant p,
        double t,
        out double x,
        out double y)

### `pspline2diff` function

    /*************************************************************************
    This function calculates derivative, i.e. it returns (dX/dT,dY/dT).

    INPUT PARAMETERS:
        P   -   parametric spline interpolant
        T   -   point:
                * T in [0,1] corresponds to interval spanned by points
                * for non-periodic splines T<0 (or T>1) correspond to parts of
                  the curve before the first (after the last) point
                * for periodic splines T<0 (or T>1) are projected  into  [0,1]
                  by making T=T-floor(T).

    OUTPUT PARAMETERS:
        X   -   X-value
        DX  -   X-derivative
        Y   -   Y-value
        DY  -   Y-derivative

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline2diff(
        pspline2interpolant p,
        double t,
        out double x,
        out double dx,
        out double y,
        out double dy)

### `pspline2diff2` function

    /*************************************************************************
    This function calculates first and second derivative with respect to T.

    INPUT PARAMETERS:
        P   -   parametric spline interpolant
        T   -   point:
                * T in [0,1] corresponds to interval spanned by points
                * for non-periodic splines T<0 (or T>1) correspond to parts of
                  the curve before the first (after the last) point
                * for periodic splines T<0 (or T>1) are projected  into  [0,1]
                  by making T=T-floor(T).

    OUTPUT PARAMETERS:
        X   -   X-value
        DX  -   derivative
        D2X -   second derivative
        Y   -   Y-value
        DY  -   derivative
        D2Y -   second derivative

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline2diff2(
        pspline2interpolant p,
        double t,
        out double x,
        out double dx,
        out double d2x,
        out double y,
        out double dy,
        out double d2y)

### `pspline2parametervalues` function

    /*************************************************************************
    This function returns vector of parameter values correspoding to points.

    I.e. for P created from (X[0],Y[0])...(X[N-1],Y[N-1]) and U=TValues(P)  we
    have
        (X[0],Y[0]) = PSpline2Calc(P,U[0]),
        (X[1],Y[1]) = PSpline2Calc(P,U[1]),
        (X[2],Y[2]) = PSpline2Calc(P,U[2]),
        ...

    INPUT PARAMETERS:
        P   -   parametric spline interpolant

    OUTPUT PARAMETERS:
        N   -   array size
        T   -   array[0..N-1]

    NOTES:
    * for non-periodic splines U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]=1
    * for periodic splines     U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]<1

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline2parametervalues(
        pspline2interpolant p,
        out int n,
        out double[] t)

### `pspline2tangent` function

    /*************************************************************************
    This function  calculates  tangent vector for a given value of parameter T

    INPUT PARAMETERS:
        P   -   parametric spline interpolant
        T   -   point:
                * T in [0,1] corresponds to interval spanned by points
                * for non-periodic splines T<0 (or T>1) correspond to parts of
                  the curve before the first (after the last) point
                * for periodic splines T<0 (or T>1) are projected  into  [0,1]
                  by making T=T-floor(T).

    OUTPUT PARAMETERS:
        X    -   X-component of tangent vector (normalized)
        Y    -   Y-component of tangent vector (normalized)

    NOTE:
        X^2+Y^2 is either 1 (for non-zero tangent vector) or 0.

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline2tangent(
        pspline2interpolant p,
        double t,
        out double x,
        out double y)

### `pspline3arclength` function

    /*************************************************************************
    This function  calculates  arc length, i.e. length of  curve  between  t=a
    and t=b.

    INPUT PARAMETERS:
        P   -   parametric spline interpolant
        A,B -   parameter values corresponding to arc ends:
                * B>A will result in positive length returned
                * B<A will result in negative length returned

    RESULT:
        length of arc starting at T=A and ending at T=B.

      -- ALGLIB PROJECT --
         Copyright 30.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static double pspline3arclength(
        pspline3interpolant p,
        double a,
        double b)

### `pspline3build` function

    /*************************************************************************
    This function  builds  non-periodic 3-dimensional parametric spline  which
    starts at (X[0],Y[0],Z[0]) and ends at (X[N-1],Y[N-1],Z[N-1]).

    Same as PSpline2Build() function, but for 3D, so we  won't  duplicate  its
    description here.

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline3build(
        double[,] xy,
        int n,
        int st,
        int pt,
        out pspline3interpolant p)

### `pspline3buildperiodic` function

    /*************************************************************************
    This  function  builds  periodic  3-dimensional  parametric  spline  which
    starts at (X[0],Y[0],Z[0]), goes through all points to (X[N-1],Y[N-1],Z[N-1])
    and then back to (X[0],Y[0],Z[0]).

    Same as PSpline2Build() function, but for 3D, so we  won't  duplicate  its
    description here.

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline3buildperiodic(
        double[,] xy,
        int n,
        int st,
        int pt,
        out pspline3interpolant p)

### `pspline3calc` function

    /*************************************************************************
    This function  calculates  the value of the parametric spline for a  given
    value of parameter T.

    INPUT PARAMETERS:
        P   -   parametric spline interpolant
        T   -   point:
                * T in [0,1] corresponds to interval spanned by points
                * for non-periodic splines T<0 (or T>1) correspond to parts of
                  the curve before the first (after the last) point
                * for periodic splines T<0 (or T>1) are projected  into  [0,1]
                  by making T=T-floor(T).

    OUTPUT PARAMETERS:
        X   -   X-position
        Y   -   Y-position
        Z   -   Z-position

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline3calc(
        pspline3interpolant p,
        double t,
        out double x,
        out double y,
        out double z)

### `pspline3diff` function

    /*************************************************************************
    This function calculates derivative, i.e. it returns (dX/dT,dY/dT,dZ/dT).

    INPUT PARAMETERS:
        P   -   parametric spline interpolant
        T   -   point:
                * T in [0,1] corresponds to interval spanned by points
                * for non-periodic splines T<0 (or T>1) correspond to parts of
                  the curve before the first (after the last) point
                * for periodic splines T<0 (or T>1) are projected  into  [0,1]
                  by making T=T-floor(T).

    OUTPUT PARAMETERS:
        X   -   X-value
        DX  -   X-derivative
        Y   -   Y-value
        DY  -   Y-derivative
        Z   -   Z-value
        DZ  -   Z-derivative

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline3diff(
        pspline3interpolant p,
        double t,
        out double x,
        out double dx,
        out double y,
        out double dy,
        out double z,
        out double dz)

### `pspline3diff2` function

    /*************************************************************************
    This function calculates first and second derivative with respect to T.

    INPUT PARAMETERS:
        P   -   parametric spline interpolant
        T   -   point:
                * T in [0,1] corresponds to interval spanned by points
                * for non-periodic splines T<0 (or T>1) correspond to parts of
                  the curve before the first (after the last) point
                * for periodic splines T<0 (or T>1) are projected  into  [0,1]
                  by making T=T-floor(T).

    OUTPUT PARAMETERS:
        X   -   X-value
        DX  -   derivative
        D2X -   second derivative
        Y   -   Y-value
        DY  -   derivative
        D2Y -   second derivative
        Z   -   Z-value
        DZ  -   derivative
        D2Z -   second derivative

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline3diff2(
        pspline3interpolant p,
        double t,
        out double x,
        out double dx,
        out double d2x,
        out double y,
        out double dy,
        out double d2y,
        out double z,
        out double dz,
        out double d2z)

### `pspline3parametervalues` function

    /*************************************************************************
    This function returns vector of parameter values correspoding to points.

    Same as PSpline2ParameterValues(), but for 3D.

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline3parametervalues(
        pspline3interpolant p,
        out int n,
        out double[] t)

### `pspline3tangent` function

    /*************************************************************************
    This function  calculates  tangent vector for a given value of parameter T

    INPUT PARAMETERS:
        P   -   parametric spline interpolant
        T   -   point:
                * T in [0,1] corresponds to interval spanned by points
                * for non-periodic splines T<0 (or T>1) correspond to parts of
                  the curve before the first (after the last) point
                * for periodic splines T<0 (or T>1) are projected  into  [0,1]
                  by making T=T-floor(T).

    OUTPUT PARAMETERS:
        X    -   X-component of tangent vector (normalized)
        Y    -   Y-component of tangent vector (normalized)
        Z    -   Z-component of tangent vector (normalized)

    NOTE:
        X^2+Y^2+Z^2 is either 1 (for non-zero tangent vector) or 0.

      -- ALGLIB PROJECT --
         Copyright 28.05.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void pspline3tangent(
        pspline3interpolant p,
        double t,
        out double x,
        out double y,
        out double z)

## `ratint` subpackage

### Classes

barycentricinterpolant

### Functions

barycentricbuildfloaterhormann
barycentricbuildxyw
barycentriccalc
barycentricdiff1
barycentricdiff2
barycentriclintransx
barycentriclintransy
barycentricunpack

### Examples

### `barycentricinterpolant` class

    /*************************************************************************
    Barycentric interpolant.
    *************************************************************************/
    public class barycentricinterpolant
    {
    }

### `barycentricbuildfloaterhormann` function

    /*************************************************************************
    Rational interpolant without poles

    The subroutine constructs the rational interpolating function without real
    poles  (see  'Barycentric rational interpolation with no  poles  and  high
    rates of approximation', Michael S. Floater. and  Kai  Hormann,  for  more
    information on this subject).

    Input parameters:
        X   -   interpolation nodes, array[0..N-1].
        Y   -   function values, array[0..N-1].
        N   -   number of nodes, N>0.
        D   -   order of the interpolation scheme, 0 <= D <= N-1.
                D<0 will cause an error.
                D>=N it will be replaced with D=N-1.
                if you don't know what D to choose, use small value about 3-5.

    Output parameters:
        B   -   barycentric interpolant.

    Note:
        this algorithm always succeeds and calculates the weights  with  close
        to machine precision.

      -- ALGLIB PROJECT --
         Copyright 17.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void barycentricbuildfloaterhormann(
        double[] x,
        double[] y,
        int n,
        int d,
        out barycentricinterpolant b)

### `barycentricbuildxyw` function

    /*************************************************************************
    Rational interpolant from X/Y/W arrays

    F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i]))

    INPUT PARAMETERS:
        X   -   interpolation nodes, array[0..N-1]
        F   -   function values, array[0..N-1]
        W   -   barycentric weights, array[0..N-1]
        N   -   nodes count, N>0

    OUTPUT PARAMETERS:
        B   -   barycentric interpolant built from (X, Y, W)

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void barycentricbuildxyw(
        double[] x,
        double[] y,
        double[] w,
        int n,
        out barycentricinterpolant b)

### `barycentriccalc` function

    /*************************************************************************
    Rational interpolation using barycentric formula

    F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i]))

    Input parameters:
        B   -   barycentric interpolant built with one of model building
                subroutines.
        T   -   interpolation point

    Result:
        barycentric interpolant F(t)

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static double barycentriccalc(barycentricinterpolant b, double t)

### `barycentricdiff1` function

    /*************************************************************************
    Differentiation of barycentric interpolant: first derivative.

    Algorithm used in this subroutine is very robust and should not fail until
    provided with values too close to MaxRealNumber  (usually  MaxRealNumber/N
    or greater will overflow).

    INPUT PARAMETERS:
        B   -   barycentric interpolant built with one of model building
                subroutines.
        T   -   interpolation point

    OUTPUT PARAMETERS:
        F   -   barycentric interpolant at T
        DF  -   first derivative

    NOTE

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void barycentricdiff1(
        barycentricinterpolant b,
        double t,
        out double f,
        out double df)

### `barycentricdiff2` function

    /*************************************************************************
    Differentiation of barycentric interpolant: first/second derivatives.

    INPUT PARAMETERS:
        B   -   barycentric interpolant built with one of model building
                subroutines.
        T   -   interpolation point

    OUTPUT PARAMETERS:
        F   -   barycentric interpolant at T
        DF  -   first derivative
        D2F -   second derivative

    NOTE: this algorithm may fail due to overflow/underflor if  used  on  data
    whose values are close to MaxRealNumber or MinRealNumber.  Use more robust
    BarycentricDiff1() subroutine in such cases.

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void barycentricdiff2(
        barycentricinterpolant b,
        double t,
        out double f,
        out double df,
        out double d2f)

### `barycentriclintransx` function

    /*************************************************************************
    This subroutine performs linear transformation of the argument.

    INPUT PARAMETERS:
        B       -   rational interpolant in barycentric form
        CA, CB  -   transformation coefficients: x = CA*t + CB

    OUTPUT PARAMETERS:
        B       -   transformed interpolant with X replaced by T

      -- ALGLIB PROJECT --
         Copyright 19.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void barycentriclintransx(
        barycentricinterpolant b,
        double ca,
        double cb)

### `barycentriclintransy` function

    /*************************************************************************
    This  subroutine   performs   linear  transformation  of  the  barycentric
    interpolant.

    INPUT PARAMETERS:
        B       -   rational interpolant in barycentric form
        CA, CB  -   transformation coefficients: B2(x) = CA*B(x) + CB

    OUTPUT PARAMETERS:
        B       -   transformed interpolant

      -- ALGLIB PROJECT --
         Copyright 19.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void barycentriclintransy(
        barycentricinterpolant b,
        double ca,
        double cb)

### `barycentricunpack` function

    /*************************************************************************
    Extracts X/Y/W arrays from rational interpolant

    INPUT PARAMETERS:
        B   -   barycentric interpolant

    OUTPUT PARAMETERS:
        N   -   nodes count, N>0
        X   -   interpolation nodes, array[0..N-1]
        F   -   function values, array[0..N-1]
        W   -   barycentric weights, array[0..N-1]

      -- ALGLIB --
         Copyright 17.08.2009 by Bochkanov Sergey
    *************************************************************************/
    public static void barycentricunpack(
        barycentricinterpolant b,
        out int n,
        out double[] x,
        out double[] y,
        out double[] w)

## `rcond` subpackage

### Functions

cmatrixlurcond1
cmatrixlurcondinf
cmatrixrcond1
cmatrixrcondinf
cmatrixtrrcond1
cmatrixtrrcondinf
hpdmatrixcholeskyrcond
hpdmatrixrcond
rmatrixlurcond1
rmatrixlurcondinf
rmatrixrcond1
rmatrixrcondinf
rmatrixtrrcond1
rmatrixtrrcondinf
spdmatrixcholeskyrcond
spdmatrixrcond

### Examples

### `cmatrixlurcond1` function

    /*************************************************************************
    Estimate of the condition number of a matrix given by its LU decomposition (1-norm)

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        LUA         -   LU decomposition of a matrix in compact form. Output of
                        the CMatrixLU subroutine.
        N           -   size of matrix A.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double cmatrixlurcond1(complex[,] lua, int n)

### `cmatrixlurcondinf` function

    /*************************************************************************
    Estimate of the condition number of a matrix given by its LU decomposition
    (infinity norm).

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        LUA     -   LU decomposition of a matrix in compact form. Output of
                    the CMatrixLU subroutine.
        N       -   size of matrix A.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double cmatrixlurcondinf(complex[,] lua, int n)

### `cmatrixrcond1` function

    /*************************************************************************
    Estimate of a matrix condition number (1-norm)

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
        N   -   size of matrix A.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double cmatrixrcond1(complex[,] a, int n)

### `cmatrixrcondinf` function

    /*************************************************************************
    Estimate of a matrix condition number (infinity-norm).

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
        N   -   size of matrix A.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double cmatrixrcondinf(complex[,] a, int n)

### `cmatrixtrrcond1` function

    /*************************************************************************
    Triangular matrix: estimate of a condition number (1-norm)

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        A       -   matrix. Array[0..N-1, 0..N-1].
        N       -   size of A.
        IsUpper -   True, if the matrix is upper triangular.
        IsUnit  -   True, if the matrix has a unit diagonal.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double cmatrixtrrcond1(
        complex[,] a,
        int n,
        bool isupper,
        bool isunit)

### `cmatrixtrrcondinf` function

    /*************************************************************************
    Triangular matrix: estimate of a matrix condition number (infinity-norm).

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
        N   -   size of matrix A.
        IsUpper -   True, if the matrix is upper triangular.
        IsUnit  -   True, if the matrix has a unit diagonal.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double cmatrixtrrcondinf(
        complex[,] a,
        int n,
        bool isupper,
        bool isunit)

### `hpdmatrixcholeskyrcond` function

    /*************************************************************************
    Condition number estimate of a Hermitian positive definite matrix given by
    Cholesky decomposition.

    The algorithm calculates a lower bound of the condition number. In this
    case, the algorithm does not return a lower bound of the condition number,
    but an inverse number (to avoid an overflow in case of a singular matrix).

    It should be noted that 1-norm and inf-norm condition numbers of symmetric
    matrices are equal, so the algorithm doesn't take into account the
    differences between these types of norms.

    Input parameters:
        CD  - Cholesky decomposition of matrix A,
              output of SMatrixCholesky subroutine.
        N   - size of matrix A.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double hpdmatrixcholeskyrcond(
        complex[,] a,
        int n,
        bool isupper)

### `hpdmatrixrcond` function

    /*************************************************************************
    Condition number estimate of a Hermitian positive definite matrix.

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    It should be noted that 1-norm and inf-norm of condition numbers of symmetric
    matrices are equal, so the algorithm doesn't take into account the
    differences between these types of norms.

    Input parameters:
        A       -   Hermitian positive definite matrix which is given by its
                    upper or lower triangle depending on the value of
                    IsUpper. Array with elements [0..N-1, 0..N-1].
        N       -   size of matrix A.
        IsUpper -   storage format.

    Result:
        1/LowerBound(cond(A)), if matrix A is positive definite,
       -1, if matrix A is not positive definite, and its condition number
        could not be found by this algorithm.

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double hpdmatrixrcond(complex[,] a, int n, bool isupper)

### `rmatrixlurcond1` function

    /*************************************************************************
    Estimate of the condition number of a matrix given by its LU decomposition (1-norm)

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        LUA         -   LU decomposition of a matrix in compact form. Output of
                        the RMatrixLU subroutine.
        N           -   size of matrix A.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double rmatrixlurcond1(double[,] lua, int n)

### `rmatrixlurcondinf` function

    /*************************************************************************
    Estimate of the condition number of a matrix given by its LU decomposition
    (infinity norm).

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        LUA     -   LU decomposition of a matrix in compact form. Output of
                    the RMatrixLU subroutine.
        N       -   size of matrix A.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double rmatrixlurcondinf(double[,] lua, int n)

### `rmatrixrcond1` function

    /*************************************************************************
    Estimate of a matrix condition number (1-norm)

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
        N   -   size of matrix A.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double rmatrixrcond1(double[,] a, int n)

### `rmatrixrcondinf` function

    /*************************************************************************
    Estimate of a matrix condition number (infinity-norm).

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
        N   -   size of matrix A.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double rmatrixrcondinf(double[,] a, int n)

### `rmatrixtrrcond1` function

    /*************************************************************************
    Triangular matrix: estimate of a condition number (1-norm)

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        A       -   matrix. Array[0..N-1, 0..N-1].
        N       -   size of A.
        IsUpper -   True, if the matrix is upper triangular.
        IsUnit  -   True, if the matrix has a unit diagonal.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double rmatrixtrrcond1(
        double[,] a,
        int n,
        bool isupper,
        bool isunit)

### `rmatrixtrrcondinf` function

    /*************************************************************************
    Triangular matrix: estimate of a matrix condition number (infinity-norm).

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    Input parameters:
        A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
        N   -   size of matrix A.
        IsUpper -   True, if the matrix is upper triangular.
        IsUnit  -   True, if the matrix has a unit diagonal.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double rmatrixtrrcondinf(
        double[,] a,
        int n,
        bool isupper,
        bool isunit)

### `spdmatrixcholeskyrcond` function

    /*************************************************************************
    Condition number estimate of a symmetric positive definite matrix given by
    Cholesky decomposition.

    The algorithm calculates a lower bound of the condition number. In this
    case, the algorithm does not return a lower bound of the condition number,
    but an inverse number (to avoid an overflow in case of a singular matrix).

    It should be noted that 1-norm and inf-norm condition numbers of symmetric
    matrices are equal, so the algorithm doesn't take into account the
    differences between these types of norms.

    Input parameters:
        CD  - Cholesky decomposition of matrix A,
              output of SMatrixCholesky subroutine.
        N   - size of matrix A.

    Result: 1/LowerBound(cond(A))

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double spdmatrixcholeskyrcond(
        double[,] a,
        int n,
        bool isupper)

### `spdmatrixrcond` function

    /*************************************************************************
    Condition number estimate of a symmetric positive definite matrix.

    The algorithm calculates a lower bound of the condition number. In this case,
    the algorithm does not return a lower bound of the condition number, but an
    inverse number (to avoid an overflow in case of a singular matrix).

    It should be noted that 1-norm and inf-norm of condition numbers of symmetric
    matrices are equal, so the algorithm doesn't take into account the
    differences between these types of norms.

    Input parameters:
        A       -   symmetric positive definite matrix which is given by its
                    upper or lower triangle depending on the value of
                    IsUpper. Array with elements [0..N-1, 0..N-1].
        N       -   size of matrix A.
        IsUpper -   storage format.

    Result:
        1/LowerBound(cond(A)), if matrix A is positive definite,
       -1, if matrix A is not positive definite, and its condition number
        could not be found by this algorithm.

    NOTE:
        if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
        0.0 is returned in such cases.
    *************************************************************************/
    public static double spdmatrixrcond(double[,] a, int n, bool isupper)

## `schur` subpackage

### Functions

rmatrixschur

### Examples

### `rmatrixschur` function

    /*************************************************************************
    Subroutine performing the Schur decomposition of a general matrix by using
    the QR algorithm with multiple shifts.

    The source matrix A is represented as S'*A*S = T, where S is an orthogonal
    matrix (Schur vectors), T - upper quasi-triangular matrix (with blocks of
    sizes 1x1 and 2x2 on the main diagonal).

    Input parameters:
        A   -   matrix to be decomposed.
                Array whose indexes range within [0..N-1, 0..N-1].
        N   -   size of A, N>=0.

    Output parameters:
        A   -   contains matrix T.
                Array whose indexes range within [0..N-1, 0..N-1].
        S   -   contains Schur vectors.
                Array whose indexes range within [0..N-1, 0..N-1].

    Note 1:
        The block structure of matrix T can be easily recognized: since all
        the elements below the blocks are zeros, the elements a[i+1,i] which
        are equal to 0 show the block border.

    Note 2:
        The algorithm performance depends on the value of the internal parameter
        NS of the InternalSchurDecomposition subroutine which defines the number
        of shifts in the QR algorithm (similarly to the block width in block-matrix
        algorithms in linear algebra). If you require maximum performance on
        your machine, it is recommended to adjust this parameter manually.

    Result:
        True,
            if the algorithm has converged and parameters A and S contain the result.
        False,
            if the algorithm has not converged.

    Algorithm implemented on the basis of the DHSEQR subroutine (LAPACK 3.0 library).
    *************************************************************************/
    public static bool rmatrixschur(ref double[,] a, int n, out double[,] s)

## `spdgevd` subpackage

### Functions

smatrixgevd
smatrixgevdreduce

### Examples

### `smatrixgevd` function

    /*************************************************************************
    Algorithm for solving the following generalized symmetric positive-definite
    eigenproblem:
        A*x = lambda*B*x (1) or
        A*B*x = lambda*x (2) or
        B*A*x = lambda*x (3).
    where A is a symmetric matrix, B - symmetric positive-definite matrix.
    The problem is solved by reducing it to an ordinary  symmetric  eigenvalue
    problem.

    Input parameters:
        A           -   symmetric matrix which is given by its upper or lower
                        triangular part.
                        Array whose indexes range within [0..N-1, 0..N-1].
        N           -   size of matrices A and B.
        IsUpperA    -   storage format of matrix A.
        B           -   symmetric positive-definite matrix which is given by
                        its upper or lower triangular part.
                        Array whose indexes range within [0..N-1, 0..N-1].
        IsUpperB    -   storage format of matrix B.
        ZNeeded     -   if ZNeeded is equal to:
                         * 0, the eigenvectors are not returned;
                         * 1, the eigenvectors are returned.
        ProblemType -   if ProblemType is equal to:
                         * 1, the following problem is solved: A*x = lambda*B*x;
                         * 2, the following problem is solved: A*B*x = lambda*x;
                         * 3, the following problem is solved: B*A*x = lambda*x.

    Output parameters:
        D           -   eigenvalues in ascending order.
                        Array whose index ranges within [0..N-1].
        Z           -   if ZNeeded is equal to:
                         * 0, Z hasn’t changed;
                         * 1, Z contains eigenvectors.
                        Array whose indexes range within [0..N-1, 0..N-1].
                        The eigenvectors are stored in matrix columns. It should
                        be noted that the eigenvectors in such problems do not
                        form an orthogonal system.

    Result:
        True, if the problem was solved successfully.
        False, if the error occurred during the Cholesky decomposition of matrix
        B (the matrix isn’t positive-definite) or during the work of the iterative
        algorithm for solving the symmetric eigenproblem.

    See also the GeneralizedSymmetricDefiniteEVDReduce subroutine.

      -- ALGLIB --
         Copyright 1.28.2006 by Bochkanov Sergey
    *************************************************************************/
    public static bool smatrixgevd(
        double[,] a,
        int n,
        bool isuppera,
        double[,] b,
        bool isupperb,
        int zneeded,
        int problemtype,
        out double[] d,
        out double[,] z)

### `smatrixgevdreduce` function

    /*************************************************************************
    Algorithm for reduction of the following generalized symmetric positive-
    definite eigenvalue problem:
        A*x = lambda*B*x (1) or
        A*B*x = lambda*x (2) or
        B*A*x = lambda*x (3)
    to the symmetric eigenvalues problem C*y = lambda*y (eigenvalues of this and
    the given problems are the same, and the eigenvectors of the given problem
    could be obtained by multiplying the obtained eigenvectors by the
    transformation matrix x = R*y).

    Here A is a symmetric matrix, B - symmetric positive-definite matrix.

    Input parameters:
        A           -   symmetric matrix which is given by its upper or lower
                        triangular part.
                        Array whose indexes range within [0..N-1, 0..N-1].
        N           -   size of matrices A and B.
        IsUpperA    -   storage format of matrix A.
        B           -   symmetric positive-definite matrix which is given by
                        its upper or lower triangular part.
                        Array whose indexes range within [0..N-1, 0..N-1].
        IsUpperB    -   storage format of matrix B.
        ProblemType -   if ProblemType is equal to:
                         * 1, the following problem is solved: A*x = lambda*B*x;
                         * 2, the following problem is solved: A*B*x = lambda*x;
                         * 3, the following problem is solved: B*A*x = lambda*x.

    Output parameters:
        A           -   symmetric matrix which is given by its upper or lower
                        triangle depending on IsUpperA. Contains matrix C.
                        Array whose indexes range within [0..N-1, 0..N-1].
        R           -   upper triangular or low triangular transformation matrix
                        which is used to obtain the eigenvectors of a given problem
                        as the product of eigenvectors of C (from the right) and
                        matrix R (from the left). If the matrix is upper
                        triangular, the elements below the main diagonal
                        are equal to 0 (and vice versa). Thus, we can perform
                        the multiplication without taking into account the
                        internal structure (which is an easier though less
                        effective way).
                        Array whose indexes range within [0..N-1, 0..N-1].
        IsUpperR    -   type of matrix R (upper or lower triangular).

    Result:
        True, if the problem was reduced successfully.
        False, if the error occurred during the Cholesky decomposition of
            matrix B (the matrix is not positive-definite).

      -- ALGLIB --
         Copyright 1.28.2006 by Bochkanov Sergey
    *************************************************************************/
    public static bool smatrixgevdreduce(
        ref double[,] a,
        int n,
        bool isuppera,
        double[,] b,
        bool isupperb,
        int problemtype,
        out double[,] r,
        out bool isupperr)

## `spline1d` subpackage

### Classes

spline1dinterpolant

### Functions

spline1dbuildakima
spline1dbuildcatmullrom
spline1dbuildcubic
spline1dbuildhermite
spline1dbuildlinear
spline1dcalc
spline1dconvcubic
spline1dconvdiff2cubic
spline1dconvdiffcubic
spline1ddiff
spline1dgriddiff2cubic
spline1dgriddiffcubic
spline1dintegrate
spline1dlintransx
spline1dlintransy
spline1dunpack

### Examples

spline1d_d_convdiff |  | Resampling using cubic splines
spline1d_d_cubic |  | Cubic spline interpolation
spline1d_d_griddiff |  | Differentiation on the grid using cubic splines
spline1d_d_linear |  | Piecewise linear spline interpolation

### `spline1dinterpolant` class

    /*************************************************************************
    1-dimensional spline inteprolant
    *************************************************************************/
    public class spline1dinterpolant
    {
    }

### `spline1dbuildakima` function

    /*************************************************************************
    This subroutine builds Akima spline interpolant

    INPUT PARAMETERS:
        X           -   spline nodes, array[0..N-1]
        Y           -   function values, array[0..N-1]
        N           -   points count (optional):
                        * N>=5
                        * if given, only first N points are used to build spline
                        * if not given, automatically detected from X/Y sizes
                          (len(X) must be equal to len(Y))

    OUTPUT PARAMETERS:
        C           -   spline interpolant

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller may pass unsorted array.

      -- ALGLIB PROJECT --
         Copyright 24.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dbuildakima(
        double[] x,
        double[] y,
        out spline1dinterpolant c)
    public static void spline1dbuildakima(
        double[] x,
        double[] y,
        int n,
        out spline1dinterpolant c)

### `spline1dbuildcatmullrom` function

    /*************************************************************************
    This subroutine builds Catmull-Rom spline interpolant.

    INPUT PARAMETERS:
        X           -   spline nodes, array[0..N-1].
        Y           -   function values, array[0..N-1].

    OPTIONAL PARAMETERS:
        N           -   points count:
                        * N>=2
                        * if given, only first N points are used to build spline
                        * if not given, automatically detected from X/Y sizes
                          (len(X) must be equal to len(Y))
        BoundType   -   boundary condition type:
                        * -1 for periodic boundary condition
                        *  0 for parabolically terminated spline (default)
        Tension     -   tension parameter:
                        * tension=0   corresponds to classic Catmull-Rom spline (default)
                        * 0<tension<1 corresponds to more general form - cardinal spline

    OUTPUT PARAMETERS:
        C           -   spline interpolant

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller may pass unsorted array.

    PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

    Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
    However, this subroutine doesn't require you to specify equal  values  for
    the first and last points - it automatically forces them  to  be  equal by
    copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
    Y[last_point]. However it is recommended to pass consistent values of Y[],
    i.e. to make Y[first_point]=Y[last_point].

      -- ALGLIB PROJECT --
         Copyright 23.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dbuildcatmullrom(
        double[] x,
        double[] y,
        out spline1dinterpolant c)
    public static void spline1dbuildcatmullrom(
        double[] x,
        double[] y,
        int n,
        int boundtype,
        double tension,
        out spline1dinterpolant c)

### `spline1dbuildcubic` function

    /*************************************************************************
    This subroutine builds cubic spline interpolant.

    INPUT PARAMETERS:
        X           -   spline nodes, array[0..N-1].
        Y           -   function values, array[0..N-1].

    OPTIONAL PARAMETERS:
        N           -   points count:
                        * N>=2
                        * if given, only first N points are used to build spline
                        * if not given, automatically detected from X/Y sizes
                          (len(X) must be equal to len(Y))
        BoundLType  -   boundary condition type for the left boundary
        BoundL      -   left boundary condition (first or second derivative,
                        depending on the BoundLType)
        BoundRType  -   boundary condition type for the right boundary
        BoundR      -   right boundary condition (first or second derivative,
                        depending on the BoundRType)

    OUTPUT PARAMETERS:
        C           -   spline interpolant

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller may pass unsorted array.

    SETTING BOUNDARY VALUES:

    The BoundLType/BoundRType parameters can have the following values:
        * -1, which corresonds to the periodic (cyclic) boundary conditions.
              In this case:
              * both BoundLType and BoundRType must be equal to -1.
              * BoundL/BoundR are ignored
              * Y[last] is ignored (it is assumed to be equal to Y[first]).
        *  0, which  corresponds  to  the  parabolically   terminated  spline
              (BoundL and/or BoundR are ignored).
        *  1, which corresponds to the first derivative boundary condition
        *  2, which corresponds to the second derivative boundary condition
        *  by default, BoundType=0 is used

    PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

    Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
    However, this subroutine doesn't require you to specify equal  values  for
    the first and last points - it automatically forces them  to  be  equal by
    copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
    Y[last_point]. However it is recommended to pass consistent values of Y[],
    i.e. to make Y[first_point]=Y[last_point].

      -- ALGLIB PROJECT --
         Copyright 23.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dbuildcubic(
        double[] x,
        double[] y,
        out spline1dinterpolant c)
    public static void spline1dbuildcubic(
        double[] x,
        double[] y,
        int n,
        int boundltype,
        double boundl,
        int boundrtype,
        double boundr,
        out spline1dinterpolant c)

### `spline1dbuildhermite` function

    /*************************************************************************
    This subroutine builds Hermite spline interpolant.

    INPUT PARAMETERS:
        X           -   spline nodes, array[0..N-1]
        Y           -   function values, array[0..N-1]
        D           -   derivatives, array[0..N-1]
        N           -   points count (optional):
                        * N>=2
                        * if given, only first N points are used to build spline
                        * if not given, automatically detected from X/Y sizes
                          (len(X) must be equal to len(Y))

    OUTPUT PARAMETERS:
        C           -   spline interpolant.

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller may pass unsorted array.

      -- ALGLIB PROJECT --
         Copyright 23.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dbuildhermite(
        double[] x,
        double[] y,
        double[] d,
        out spline1dinterpolant c)
    public static void spline1dbuildhermite(
        double[] x,
        double[] y,
        double[] d,
        int n,
        out spline1dinterpolant c)

### `spline1dbuildlinear` function

    /*************************************************************************
    This subroutine builds linear spline interpolant

    INPUT PARAMETERS:
        X   -   spline nodes, array[0..N-1]
        Y   -   function values, array[0..N-1]
        N   -   points count (optional):
                * N>=2
                * if given, only first N points are used to build spline
                * if not given, automatically detected from X/Y sizes
                  (len(X) must be equal to len(Y))

    OUTPUT PARAMETERS:
        C   -   spline interpolant

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller may pass unsorted array.

      -- ALGLIB PROJECT --
         Copyright 24.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dbuildlinear(
        double[] x,
        double[] y,
        out spline1dinterpolant c)
    public static void spline1dbuildlinear(
        double[] x,
        double[] y,
        int n,
        out spline1dinterpolant c)

Examples: [1] [2]

### `spline1dcalc` function

    /*************************************************************************
    This subroutine calculates the value of the spline at the given point X.

    INPUT PARAMETERS:
        C   -   spline interpolant
        X   -   point

    Result:
        S(x)

      -- ALGLIB PROJECT --
         Copyright 23.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static double spline1dcalc(spline1dinterpolant c, double x)

Examples: [1] [2]

### `spline1dconvcubic` function

    /*************************************************************************
    This function solves following problem: given table y[] of function values
    at old nodes x[]  and new nodes  x2[],  it calculates and returns table of
    function values y2[] (calculated at x2[]).

    This function yields same result as Spline1DBuildCubic() call followed  by
    sequence of Spline1DDiff() calls, but it can be several times faster  when
    called for ordered X[] and X2[].

    INPUT PARAMETERS:
        X           -   old spline nodes
        Y           -   function values
        X2           -  new spline nodes

    OPTIONAL PARAMETERS:
        N           -   points count:
                        * N>=2
                        * if given, only first N points from X/Y are used
                        * if not given, automatically detected from X/Y sizes
                          (len(X) must be equal to len(Y))
        BoundLType  -   boundary condition type for the left boundary
        BoundL      -   left boundary condition (first or second derivative,
                        depending on the BoundLType)
        BoundRType  -   boundary condition type for the right boundary
        BoundR      -   right boundary condition (first or second derivative,
                        depending on the BoundRType)
        N2          -   new points count:
                        * N2>=2
                        * if given, only first N2 points from X2 are used
                        * if not given, automatically detected from X2 size

    OUTPUT PARAMETERS:
        F2          -   function values at X2[]

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller  may pass unsorted array.
    Function  values  are correctly reordered on  return, so F2[I]  is  always
    equal to S(X2[I]) independently of points order.

    SETTING BOUNDARY VALUES:

    The BoundLType/BoundRType parameters can have the following values:
        * -1, which corresonds to the periodic (cyclic) boundary conditions.
              In this case:
              * both BoundLType and BoundRType must be equal to -1.
              * BoundL/BoundR are ignored
              * Y[last] is ignored (it is assumed to be equal to Y[first]).
        *  0, which  corresponds  to  the  parabolically   terminated  spline
              (BoundL and/or BoundR are ignored).
        *  1, which corresponds to the first derivative boundary condition
        *  2, which corresponds to the second derivative boundary condition
        *  by default, BoundType=0 is used

    PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

    Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
    However, this subroutine doesn't require you to specify equal  values  for
    the first and last points - it automatically forces them  to  be  equal by
    copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
    Y[last_point]. However it is recommended to pass consistent values of Y[],
    i.e. to make Y[first_point]=Y[last_point].

      -- ALGLIB PROJECT --
         Copyright 03.09.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dconvcubic(
        double[] x,
        double[] y,
        double[] x2,
        out double[] y2)
    public static void spline1dconvcubic(
        double[] x,
        double[] y,
        int n,
        int boundltype,
        double boundl,
        int boundrtype,
        double boundr,
        double[] x2,
        int n2,
        out double[] y2)

Examples: [1]

### `spline1dconvdiff2cubic` function

    /*************************************************************************
    This function solves following problem: given table y[] of function values
    at old nodes x[]  and new nodes  x2[],  it calculates and returns table of
    function  values  y2[],  first  and  second  derivatives  d2[]  and  dd2[]
    (calculated at x2[]).

    This function yields same result as Spline1DBuildCubic() call followed  by
    sequence of Spline1DDiff() calls, but it can be several times faster  when
    called for ordered X[] and X2[].

    INPUT PARAMETERS:
        X           -   old spline nodes
        Y           -   function values
        X2           -  new spline nodes

    OPTIONAL PARAMETERS:
        N           -   points count:
                        * N>=2
                        * if given, only first N points from X/Y are used
                        * if not given, automatically detected from X/Y sizes
                          (len(X) must be equal to len(Y))
        BoundLType  -   boundary condition type for the left boundary
        BoundL      -   left boundary condition (first or second derivative,
                        depending on the BoundLType)
        BoundRType  -   boundary condition type for the right boundary
        BoundR      -   right boundary condition (first or second derivative,
                        depending on the BoundRType)
        N2          -   new points count:
                        * N2>=2
                        * if given, only first N2 points from X2 are used
                        * if not given, automatically detected from X2 size

    OUTPUT PARAMETERS:
        F2          -   function values at X2[]
        D2          -   first derivatives at X2[]
        DD2         -   second derivatives at X2[]

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller  may pass unsorted array.
    Function  values  are correctly reordered on  return, so F2[I]  is  always
    equal to S(X2[I]) independently of points order.

    SETTING BOUNDARY VALUES:

    The BoundLType/BoundRType parameters can have the following values:
        * -1, which corresonds to the periodic (cyclic) boundary conditions.
              In this case:
              * both BoundLType and BoundRType must be equal to -1.
              * BoundL/BoundR are ignored
              * Y[last] is ignored (it is assumed to be equal to Y[first]).
        *  0, which  corresponds  to  the  parabolically   terminated  spline
              (BoundL and/or BoundR are ignored).
        *  1, which corresponds to the first derivative boundary condition
        *  2, which corresponds to the second derivative boundary condition
        *  by default, BoundType=0 is used

    PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

    Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
    However, this subroutine doesn't require you to specify equal  values  for
    the first and last points - it automatically forces them  to  be  equal by
    copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
    Y[last_point]. However it is recommended to pass consistent values of Y[],
    i.e. to make Y[first_point]=Y[last_point].

      -- ALGLIB PROJECT --
         Copyright 03.09.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dconvdiff2cubic(
        double[] x,
        double[] y,
        double[] x2,
        out double[] y2,
        out double[] d2,
        out double[] dd2)
    public static void spline1dconvdiff2cubic(
        double[] x,
        double[] y,
        int n,
        int boundltype,
        double boundl,
        int boundrtype,
        double boundr,
        double[] x2,
        int n2,
        out double[] y2,
        out double[] d2,
        out double[] dd2)

Examples: [1]

### `spline1dconvdiffcubic` function

    /*************************************************************************
    This function solves following problem: given table y[] of function values
    at old nodes x[]  and new nodes  x2[],  it calculates and returns table of
    function values y2[] and derivatives d2[] (calculated at x2[]).

    This function yields same result as Spline1DBuildCubic() call followed  by
    sequence of Spline1DDiff() calls, but it can be several times faster  when
    called for ordered X[] and X2[].

    INPUT PARAMETERS:
        X           -   old spline nodes
        Y           -   function values
        X2           -  new spline nodes

    OPTIONAL PARAMETERS:
        N           -   points count:
                        * N>=2
                        * if given, only first N points from X/Y are used
                        * if not given, automatically detected from X/Y sizes
                          (len(X) must be equal to len(Y))
        BoundLType  -   boundary condition type for the left boundary
        BoundL      -   left boundary condition (first or second derivative,
                        depending on the BoundLType)
        BoundRType  -   boundary condition type for the right boundary
        BoundR      -   right boundary condition (first or second derivative,
                        depending on the BoundRType)
        N2          -   new points count:
                        * N2>=2
                        * if given, only first N2 points from X2 are used
                        * if not given, automatically detected from X2 size

    OUTPUT PARAMETERS:
        F2          -   function values at X2[]
        D2          -   first derivatives at X2[]

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller  may pass unsorted array.
    Function  values  are correctly reordered on  return, so F2[I]  is  always
    equal to S(X2[I]) independently of points order.

    SETTING BOUNDARY VALUES:

    The BoundLType/BoundRType parameters can have the following values:
        * -1, which corresonds to the periodic (cyclic) boundary conditions.
              In this case:
              * both BoundLType and BoundRType must be equal to -1.
              * BoundL/BoundR are ignored
              * Y[last] is ignored (it is assumed to be equal to Y[first]).
        *  0, which  corresponds  to  the  parabolically   terminated  spline
              (BoundL and/or BoundR are ignored).
        *  1, which corresponds to the first derivative boundary condition
        *  2, which corresponds to the second derivative boundary condition
        *  by default, BoundType=0 is used

    PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

    Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
    However, this subroutine doesn't require you to specify equal  values  for
    the first and last points - it automatically forces them  to  be  equal by
    copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
    Y[last_point]. However it is recommended to pass consistent values of Y[],
    i.e. to make Y[first_point]=Y[last_point].

      -- ALGLIB PROJECT --
         Copyright 03.09.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dconvdiffcubic(
        double[] x,
        double[] y,
        double[] x2,
        out double[] y2,
        out double[] d2)
    public static void spline1dconvdiffcubic(
        double[] x,
        double[] y,
        int n,
        int boundltype,
        double boundl,
        int boundrtype,
        double boundr,
        double[] x2,
        int n2,
        out double[] y2,
        out double[] d2)

Examples: [1]

### `spline1ddiff` function

    /*************************************************************************
    This subroutine differentiates the spline.

    INPUT PARAMETERS:
        C   -   spline interpolant.
        X   -   point

    Result:
        S   -   S(x)
        DS  -   S'(x)
        D2S -   S''(x)

      -- ALGLIB PROJECT --
         Copyright 24.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1ddiff(
        spline1dinterpolant c,
        double x,
        out double s,
        out double ds,
        out double d2s)

### `spline1dgriddiff2cubic` function

    /*************************************************************************
    This function solves following problem: given table y[] of function values
    at  nodes  x[],  it  calculates  and  returns  tables  of first and second
    function derivatives d1[] and d2[] (calculated at the same nodes x[]).

    This function yields same result as Spline1DBuildCubic() call followed  by
    sequence of Spline1DDiff() calls, but it can be several times faster  when
    called for ordered X[] and X2[].

    INPUT PARAMETERS:
        X           -   spline nodes
        Y           -   function values

    OPTIONAL PARAMETERS:
        N           -   points count:
                        * N>=2
                        * if given, only first N points are used
                        * if not given, automatically detected from X/Y sizes
                          (len(X) must be equal to len(Y))
        BoundLType  -   boundary condition type for the left boundary
        BoundL      -   left boundary condition (first or second derivative,
                        depending on the BoundLType)
        BoundRType  -   boundary condition type for the right boundary
        BoundR      -   right boundary condition (first or second derivative,
                        depending on the BoundRType)

    OUTPUT PARAMETERS:
        D1          -   S' values at X[]
        D2          -   S'' values at X[]

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller may pass unsorted array.
    Derivative values are correctly reordered on return, so  D[I]  is  always
    equal to S'(X[I]) independently of points order.

    SETTING BOUNDARY VALUES:

    The BoundLType/BoundRType parameters can have the following values:
        * -1, which corresonds to the periodic (cyclic) boundary conditions.
              In this case:
              * both BoundLType and BoundRType must be equal to -1.
              * BoundL/BoundR are ignored
              * Y[last] is ignored (it is assumed to be equal to Y[first]).
        *  0, which  corresponds  to  the  parabolically   terminated  spline
              (BoundL and/or BoundR are ignored).
        *  1, which corresponds to the first derivative boundary condition
        *  2, which corresponds to the second derivative boundary condition
        *  by default, BoundType=0 is used

    PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

    Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
    However, this subroutine doesn't require you to specify equal  values  for
    the first and last points - it automatically forces them  to  be  equal by
    copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
    Y[last_point]. However it is recommended to pass consistent values of Y[],
    i.e. to make Y[first_point]=Y[last_point].

      -- ALGLIB PROJECT --
         Copyright 03.09.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dgriddiff2cubic(
        double[] x,
        double[] y,
        out double[] d1,
        out double[] d2)
    public static void spline1dgriddiff2cubic(
        double[] x,
        double[] y,
        int n,
        int boundltype,
        double boundl,
        int boundrtype,
        double boundr,
        out double[] d1,
        out double[] d2)

Examples: [1]

### `spline1dgriddiffcubic` function

    /*************************************************************************
    This function solves following problem: given table y[] of function values
    at nodes x[], it calculates and returns table of function derivatives  d[]
    (calculated at the same nodes x[]).

    This function yields same result as Spline1DBuildCubic() call followed  by
    sequence of Spline1DDiff() calls, but it can be several times faster  when
    called for ordered X[] and X2[].

    INPUT PARAMETERS:
        X           -   spline nodes
        Y           -   function values

    OPTIONAL PARAMETERS:
        N           -   points count:
                        * N>=2
                        * if given, only first N points are used
                        * if not given, automatically detected from X/Y sizes
                          (len(X) must be equal to len(Y))
        BoundLType  -   boundary condition type for the left boundary
        BoundL      -   left boundary condition (first or second derivative,
                        depending on the BoundLType)
        BoundRType  -   boundary condition type for the right boundary
        BoundR      -   right boundary condition (first or second derivative,
                        depending on the BoundRType)

    OUTPUT PARAMETERS:
        D           -   derivative values at X[]

    ORDER OF POINTS

    Subroutine automatically sorts points, so caller may pass unsorted array.
    Derivative values are correctly reordered on return, so  D[I]  is  always
    equal to S'(X[I]) independently of points order.

    SETTING BOUNDARY VALUES:

    The BoundLType/BoundRType parameters can have the following values:
        * -1, which corresonds to the periodic (cyclic) boundary conditions.
              In this case:
              * both BoundLType and BoundRType must be equal to -1.
              * BoundL/BoundR are ignored
              * Y[last] is ignored (it is assumed to be equal to Y[first]).
        *  0, which  corresponds  to  the  parabolically   terminated  spline
              (BoundL and/or BoundR are ignored).
        *  1, which corresponds to the first derivative boundary condition
        *  2, which corresponds to the second derivative boundary condition
        *  by default, BoundType=0 is used

    PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

    Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
    However, this subroutine doesn't require you to specify equal  values  for
    the first and last points - it automatically forces them  to  be  equal by
    copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
    Y[last_point]. However it is recommended to pass consistent values of Y[],
    i.e. to make Y[first_point]=Y[last_point].

      -- ALGLIB PROJECT --
         Copyright 03.09.2010 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dgriddiffcubic(
        double[] x,
        double[] y,
        out double[] d)
    public static void spline1dgriddiffcubic(
        double[] x,
        double[] y,
        int n,
        int boundltype,
        double boundl,
        int boundrtype,
        double boundr,
        out double[] d)

Examples: [1]

### `spline1dintegrate` function

    /*************************************************************************
    This subroutine integrates the spline.

    INPUT PARAMETERS:
        C   -   spline interpolant.
        X   -   right bound of the integration interval [a, x],
                here 'a' denotes min(x[])
    Result:
        integral(S(t)dt,a,x)

      -- ALGLIB PROJECT --
         Copyright 23.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static double spline1dintegrate(spline1dinterpolant c, double x)

### `spline1dlintransx` function

    /*************************************************************************
    This subroutine performs linear transformation of the spline argument.

    INPUT PARAMETERS:
        C   -   spline interpolant.
        A, B-   transformation coefficients: x = A*t + B
    Result:
        C   -   transformed spline

      -- ALGLIB PROJECT --
         Copyright 30.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dlintransx(
        spline1dinterpolant c,
        double a,
        double b)

### `spline1dlintransy` function

    /*************************************************************************
    This subroutine performs linear transformation of the spline.

    INPUT PARAMETERS:
        C   -   spline interpolant.
        A, B-   transformation coefficients: S2(x) = A*S(x) + B
    Result:
        C   -   transformed spline

      -- ALGLIB PROJECT --
         Copyright 30.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dlintransy(
        spline1dinterpolant c,
        double a,
        double b)

### `spline1dunpack` function

    /*************************************************************************
    This subroutine unpacks the spline into the coefficients table.

    INPUT PARAMETERS:
        C   -   spline interpolant.
        X   -   point

    Result:
        Tbl -   coefficients table, unpacked format, array[0..N-2, 0..5].
                For I = 0...N-2:
                    Tbl[I,0] = X[i]
                    Tbl[I,1] = X[i+1]
                    Tbl[I,2] = C0
                    Tbl[I,3] = C1
                    Tbl[I,4] = C2
                    Tbl[I,5] = C3
                On [x[i], x[i+1]] spline is equals to:
                    S(x) = C0 + C1*t + C2*t^2 + C3*t^3
                    t = x-x[i]

      -- ALGLIB PROJECT --
         Copyright 29.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline1dunpack(
        spline1dinterpolant c,
        out int n,
        out double[,] tbl)

### spline1d_d_convdiff example

    public static int Main(string[] args)
    {
        //
        // We use cubic spline to do resampling, i.e. having
        // values of f(x)=x^2 sampled at 5 equidistant nodes on [-1,+1]
        // we calculate values/derivatives of cubic spline on
        // another grid (equidistant with 9 nodes on [-1,+1])
        // WITHOUT CONSTRUCTION OF SPLINE OBJECT.
        //
        // There are efficient functions spline1dconvcubic(),
        // spline1dconvdiffcubic() and spline1dconvdiff2cubic()
        // for such calculations.
        //
        // We use default boundary conditions ("parabolically terminated
        // spline") because cubic spline built with such boundary conditions
        // will exactly reproduce any quadratic f(x).
        //
        // Actually, we could use natural conditions, but we feel that
        // spline which exactly reproduces f() will show us more
        // understandable results.
        //
        double[] x_old = new double[]{-1.0,-0.5,0.0,+0.5,+1.0};
        double[] y_old = new double[]{+1.0,0.25,0.0,0.25,+1.0};
        double[] x_new = new double[]{-1.00,-0.75,-0.50,-0.25,0.00,+0.25,+0.50,+0.75,+1.00};
        double[] y_new;
        double[] d1_new;
        double[] d2_new;

        //
        // First, conversion without differentiation.
        //
        //
        alglib.spline1dconvcubic(x_old, y_old, x_new, out y_new);
        System.Console.WriteLine("{0}", alglib.ap.format(y_new,3)); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]

        //
        // Then, conversion with differentiation (first derivatives only)
        //
        //
        alglib.spline1dconvdiffcubic(x_old, y_old, x_new, out y_new, out d1_new);
        System.Console.WriteLine("{0}", alglib.ap.format(y_new,3)); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]
        System.Console.WriteLine("{0}", alglib.ap.format(d1_new,3)); // EXPECTED: [-2.0, -1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0]

        //
        // Finally, conversion with first and second derivatives
        //
        //
        alglib.spline1dconvdiff2cubic(x_old, y_old, x_new, out y_new, out d1_new, out d2_new);
        System.Console.WriteLine("{0}", alglib.ap.format(y_new,3)); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]
        System.Console.WriteLine("{0}", alglib.ap.format(d1_new,3)); // EXPECTED: [-2.0, -1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0]
        System.Console.WriteLine("{0}", alglib.ap.format(d2_new,3)); // EXPECTED: [2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0]
        System.Console.ReadLine();
        return 0;
    }

### spline1d_d_cubic example

    public static int Main(string[] args)
    {
        //
        // We use cubic spline to interpolate f(x)=x^2 sampled
        // at 5 equidistant nodes on [-1,+1].
        //
        // First, we use default boundary conditions ("parabolically terminated
        // spline") because cubic spline built with such boundary conditions
        // will exactly reproduce any quadratic f(x).
        //
        // Then we try to use natural boundary conditions
        //     d2S(-1)/dx^2 = 0.0
        //     d2S(+1)/dx^2 = 0.0
        // and see that such spline interpolated f(x) with small error.
        //
        double[] x = new double[]{-1.0,-0.5,0.0,+0.5,+1.0};
        double[] y = new double[]{+1.0,0.25,0.0,0.25,+1.0};
        double t = 0.25;
        double v;
        alglib.spline1dinterpolant s;
        int natural_bound_type = 2;
        //
        // Test exact boundary conditions: build S(x), calculare S(0.25)
        // (almost same as original function)
        //
        alglib.spline1dbuildcubic(x, y, out s);
        v = alglib.spline1dcalc(s, t);
        System.Console.WriteLine("{0:F4", v); // EXPECTED: 0.0625

        //
        // Test natural boundary conditions: build S(x), calculare S(0.25)
        // (small interpolation error)
        //
        alglib.spline1dbuildcubic(x, y, 5, natural_bound_type, 0.0, natural_bound_type, 0.0, out s);
        v = alglib.spline1dcalc(s, t);
        System.Console.WriteLine("{0:F3", v); // EXPECTED: 0.0580
        System.Console.ReadLine();
        return 0;
    }

### spline1d_d_griddiff example

    public static int Main(string[] args)
    {
        //
        // We use cubic spline to do grid differentiation, i.e. having
        // values of f(x)=x^2 sampled at 5 equidistant nodes on [-1,+1]
        // we calculate derivatives of cubic spline at nodes WITHOUT
        // CONSTRUCTION OF SPLINE OBJECT.
        //
        // There are efficient functions spline1dgriddiffcubic() and
        // spline1dgriddiff2cubic() for such calculations.
        //
        // We use default boundary conditions ("parabolically terminated
        // spline") because cubic spline built with such boundary conditions
        // will exactly reproduce any quadratic f(x).
        //
        // Actually, we could use natural conditions, but we feel that
        // spline which exactly reproduces f() will show us more
        // understandable results.
        //
        double[] x = new double[]{-1.0,-0.5,0.0,+0.5,+1.0};
        double[] y = new double[]{+1.0,0.25,0.0,0.25,+1.0};
        double[] d1;
        double[] d2;

        //
        // We calculate first derivatives: they must be equal to 2*x
        //
        alglib.spline1dgriddiffcubic(x, y, out d1);
        System.Console.WriteLine("{0}", alglib.ap.format(d1,3)); // EXPECTED: [-2.0, -1.0, 0.0, +1.0, +2.0]

        //
        // Now test griddiff2, which returns first AND second derivatives.
        // First derivative is 2*x, second is equal to 2.0
        //
        alglib.spline1dgriddiff2cubic(x, y, out d1, out d2);
        System.Console.WriteLine("{0}", alglib.ap.format(d1,3)); // EXPECTED: [-2.0, -1.0, 0.0, +1.0, +2.0]
        System.Console.WriteLine("{0}", alglib.ap.format(d2,3)); // EXPECTED: [ 2.0,  2.0, 2.0,  2.0,  2.0]
        System.Console.ReadLine();
        return 0;
    }

### spline1d_d_linear example

    public static int Main(string[] args)
    {
        //
        // We use piecewise linear spline to interpolate f(x)=x^2 sampled
        // at 5 equidistant nodes on [-1,+1].
        //
        double[] x = new double[]{-1.0,-0.5,0.0,+0.5,+1.0};
        double[] y = new double[]{+1.0,0.25,0.0,0.25,+1.0};
        double t = 0.25;
        double v;
        alglib.spline1dinterpolant s;

        // build spline
        alglib.spline1dbuildlinear(x, y, out s);

        // calculate S(0.25) - it is quite different from 0.25^2=0.0625
        v = alglib.spline1dcalc(s, t);
        System.Console.WriteLine("{0:F4", v); // EXPECTED: 0.125
        System.Console.ReadLine();
        return 0;
    }

## `spline2d` subpackage

### Classes

spline2dinterpolant

### Functions

spline2dbuildbicubic
spline2dbuildbilinear
spline2dcalc
spline2ddiff
spline2dlintransf
spline2dlintransxy
spline2dresamplebicubic
spline2dresamplebilinear
spline2dunpack

### Examples

### `spline2dinterpolant` class

    /*************************************************************************
    2-dimensional spline inteprolant
    *************************************************************************/
    public class spline2dinterpolant
    {
    }

### `spline2dbuildbicubic` function

    /*************************************************************************
    This subroutine builds bicubic spline coefficients table.

    Input parameters:
        X   -   spline abscissas, array[0..N-1]
        Y   -   spline ordinates, array[0..M-1]
        F   -   function values, array[0..M-1,0..N-1]
        M,N -   grid size, M>=2, N>=2

    Output parameters:
        C   -   spline interpolant

      -- ALGLIB PROJECT --
         Copyright 05.07.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline2dbuildbicubic(
        double[] x,
        double[] y,
        double[,] f,
        int m,
        int n,
        out spline2dinterpolant c)

### `spline2dbuildbilinear` function

    /*************************************************************************
    This subroutine builds bilinear spline coefficients table.

    Input parameters:
        X   -   spline abscissas, array[0..N-1]
        Y   -   spline ordinates, array[0..M-1]
        F   -   function values, array[0..M-1,0..N-1]
        M,N -   grid size, M>=2, N>=2

    Output parameters:
        C   -   spline interpolant

      -- ALGLIB PROJECT --
         Copyright 05.07.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline2dbuildbilinear(
        double[] x,
        double[] y,
        double[,] f,
        int m,
        int n,
        out spline2dinterpolant c)

### `spline2dcalc` function

    /*************************************************************************
    This subroutine calculates the value of the bilinear or bicubic spline  at
    the given point X.

    Input parameters:
        C   -   coefficients table.
                Built by BuildBilinearSpline or BuildBicubicSpline.
        X, Y-   point

    Result:
        S(x,y)

      -- ALGLIB PROJECT --
         Copyright 05.07.2007 by Bochkanov Sergey
    *************************************************************************/
    public static double spline2dcalc(
        spline2dinterpolant c,
        double x,
        double y)

### `spline2ddiff` function

    /*************************************************************************
    This subroutine calculates the value of the bilinear or bicubic spline  at
    the given point X and its derivatives.

    Input parameters:
        C   -   spline interpolant.
        X, Y-   point

    Output parameters:
        F   -   S(x,y)
        FX  -   dS(x,y)/dX
        FY  -   dS(x,y)/dY
        FXY -   d2S(x,y)/dXdY

      -- ALGLIB PROJECT --
         Copyright 05.07.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline2ddiff(
        spline2dinterpolant c,
        double x,
        double y,
        out double f,
        out double fx,
        out double fy,
        out double fxy)

### `spline2dlintransf` function

    /*************************************************************************
    This subroutine performs linear transformation of the spline.

    Input parameters:
        C   -   spline interpolant.
        A, B-   transformation coefficients: S2(x,y) = A*S(x,y) + B

    Output parameters:
        C   -   transformed spline

      -- ALGLIB PROJECT --
         Copyright 30.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline2dlintransf(
        spline2dinterpolant c,
        double a,
        double b)

### `spline2dlintransxy` function

    /*************************************************************************
    This subroutine performs linear transformation of the spline argument.

    Input parameters:
        C       -   spline interpolant
        AX, BX  -   transformation coefficients: x = A*t + B
        AY, BY  -   transformation coefficients: y = A*u + B
    Result:
        C   -   transformed spline

      -- ALGLIB PROJECT --
         Copyright 30.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline2dlintransxy(
        spline2dinterpolant c,
        double ax,
        double bx,
        double ay,
        double by)

### `spline2dresamplebicubic` function

    /*************************************************************************
    Bicubic spline resampling

    Input parameters:
        A           -   function values at the old grid,
                        array[0..OldHeight-1, 0..OldWidth-1]
        OldHeight   -   old grid height, OldHeight>1
        OldWidth    -   old grid width, OldWidth>1
        NewHeight   -   new grid height, NewHeight>1
        NewWidth    -   new grid width, NewWidth>1

    Output parameters:
        B           -   function values at the new grid,
                        array[0..NewHeight-1, 0..NewWidth-1]

      -- ALGLIB routine --
         15 May, 2007
         Copyright by Bochkanov Sergey
    *************************************************************************/
    public static void spline2dresamplebicubic(
        double[,] a,
        int oldheight,
        int oldwidth,
        out double[,] b,
        int newheight,
        int newwidth)

### `spline2dresamplebilinear` function

    /*************************************************************************
    Bilinear spline resampling

    Input parameters:
        A           -   function values at the old grid,
                        array[0..OldHeight-1, 0..OldWidth-1]
        OldHeight   -   old grid height, OldHeight>1
        OldWidth    -   old grid width, OldWidth>1
        NewHeight   -   new grid height, NewHeight>1
        NewWidth    -   new grid width, NewWidth>1

    Output parameters:
        B           -   function values at the new grid,
                        array[0..NewHeight-1, 0..NewWidth-1]

      -- ALGLIB routine --
         09.07.2007
         Copyright by Bochkanov Sergey
    *************************************************************************/
    public static void spline2dresamplebilinear(
        double[,] a,
        int oldheight,
        int oldwidth,
        out double[,] b,
        int newheight,
        int newwidth)

### `spline2dunpack` function

    /*************************************************************************
    This subroutine unpacks two-dimensional spline into the coefficients table

    Input parameters:
        C   -   spline interpolant.

    Result:
        M, N-   grid size (x-axis and y-axis)
        Tbl -   coefficients table, unpacked format,
                [0..(N-1)*(M-1)-1, 0..19].
                For I = 0...M-2, J=0..N-2:
                    K =  I*(N-1)+J
                    Tbl[K,0] = X[j]
                    Tbl[K,1] = X[j+1]
                    Tbl[K,2] = Y[i]
                    Tbl[K,3] = Y[i+1]
                    Tbl[K,4] = C00
                    Tbl[K,5] = C01
                    Tbl[K,6] = C02
                    Tbl[K,7] = C03
                    Tbl[K,8] = C10
                    Tbl[K,9] = C11
                    ...
                    Tbl[K,19] = C33
                On each grid square spline is equals to:
                    S(x) = SUM(c[i,j]*(x^i)*(y^j), i=0..3, j=0..3)
                    t = x-x[j]
                    u = y-y[i]

      -- ALGLIB PROJECT --
         Copyright 29.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void spline2dunpack(
        spline2dinterpolant c,
        out int m,
        out int n,
        out double[,] tbl)

## `stest` subpackage

### Functions

onesamplesigntest

### Examples

### `onesamplesigntest` function

    /*************************************************************************
    Sign test

    This test checks three hypotheses about the median of  the  given  sample.
    The following tests are performed:
        * two-tailed test (null hypothesis - the median is equal to the  given
          value)
        * left-tailed test (null hypothesis - the median is  greater  than  or
          equal to the given value)
        * right-tailed test (null hypothesis - the  median  is  less  than  or
          equal to the given value)

    Requirements:
        * the scale of measurement should be ordinal, interval or ratio  (i.e.
          the test could not be applied to nominal variables).

    The test is non-parametric and doesn't require distribution X to be normal

    Input parameters:
        X       -   sample. Array whose index goes from 0 to N-1.
        N       -   size of the sample.
        Median  -   assumed median value.

    Output parameters:
        BothTails   -   p-value for two-tailed test.
                        If BothTails is less than the given significance level
                        the null hypothesis is rejected.
        LeftTail    -   p-value for left-tailed test.
                        If LeftTail is less than the given significance level,
                        the null hypothesis is rejected.
        RightTail   -   p-value for right-tailed test.
                        If RightTail is less than the given significance level
                        the null hypothesis is rejected.

    While   calculating   p-values   high-precision   binomial    distribution
    approximation is used, so significance levels have about 15 exact digits.

      -- ALGLIB --
         Copyright 08.09.2006 by Bochkanov Sergey
    *************************************************************************/
    public static void onesamplesigntest(
        double[] x,
        int n,
        double median,
        out double bothtails,
        out double lefttail,
        out double righttail)

## `studenttdistr` subpackage

### Functions

invstudenttdistribution
studenttdistribution

### Examples

### `invstudenttdistribution` function

    /*************************************************************************
    Functional inverse of Student's t distribution

    Given probability p, finds the argument t such that stdtr(k,t)
    is equal to p.

    ACCURACY:

    Tested at random 1 <= k <= 100.  The "domain" refers to p:
                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE    .001,.999     25000       5.7e-15     8.0e-16
       IEEE    10^-6,.001    25000       2.0e-12     2.9e-14

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double invstudenttdistribution(int k, double p)

### `studenttdistribution` function

    /*************************************************************************
    Student's t distribution

    Computes the integral from minus infinity to t of the Student
    t distribution with integer k > 0 degrees of freedom:

                                         t
                                         -
                                        | |
                 -                      |         2   -(k+1)/2
                | ( (k+1)/2 )           |  (     x   )
          ----------------------        |  ( 1 + --- )        dx
                        -               |  (      k  )
          sqrt( k pi ) | ( k/2 )        |
                                      | |
                                       -
                                      -inf.

    Relation to incomplete beta integral:

           1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
    where
           z = k/(k + t**2).

    For t < -2, this is the method of computation.  For higher t,
    a direct method is derived from integration by parts.
    Since the function is symmetric about t=0, the area under the
    right tail of the density is found by calling the function
    with -t instead of t.

    ACCURACY:

    Tested at random 1 <= k <= 25.  The "domain" refers to t.
                         Relative error:
    arithmetic   domain     # trials      peak         rms
       IEEE     -100,-2      50000       5.9e-15     1.4e-15
       IEEE     -2,100      500000       2.7e-15     4.9e-17

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static double studenttdistribution(int k, double t)

## `studentttests` subpackage

### Functions

studentttest1
studentttest2
unequalvariancettest

### Examples

### `studentttest1` function

    /*************************************************************************
    One-sample t-test

    This test checks three hypotheses about the mean of the given sample.  The
    following tests are performed:
        * two-tailed test (null hypothesis - the mean is equal  to  the  given
          value)
        * left-tailed test (null hypothesis - the  mean  is  greater  than  or
          equal to the given value)
        * right-tailed test (null hypothesis - the mean is less than or  equal
          to the given value).

    The test is based on the assumption that  a  given  sample  has  a  normal
    distribution and  an  unknown  dispersion.  If  the  distribution  sharply
    differs from normal, the test will work incorrectly.

    Input parameters:
        X       -   sample. Array whose index goes from 0 to N-1.
        N       -   size of sample.
        Mean    -   assumed value of the mean.

    Output parameters:
        BothTails   -   p-value for two-tailed test.
                        If BothTails is less than the given significance level
                        the null hypothesis is rejected.
        LeftTail    -   p-value for left-tailed test.
                        If LeftTail is less than the given significance level,
                        the null hypothesis is rejected.
        RightTail   -   p-value for right-tailed test.
                        If RightTail is less than the given significance level
                        the null hypothesis is rejected.

      -- ALGLIB --
         Copyright 08.09.2006 by Bochkanov Sergey
    *************************************************************************/
    public static void studentttest1(
        double[] x,
        int n,
        double mean,
        out double bothtails,
        out double lefttail,
        out double righttail)

### `studentttest2` function

    /*************************************************************************
    Two-sample pooled test

    This test checks three hypotheses about the mean of the given samples. The
    following tests are performed:
        * two-tailed test (null hypothesis - the means are equal)
        * left-tailed test (null hypothesis - the mean of the first sample  is
          greater than or equal to the mean of the second sample)
        * right-tailed test (null hypothesis - the mean of the first sample is
          less than or equal to the mean of the second sample).

    Test is based on the following assumptions:
        * given samples have normal distributions
        * dispersions are equal
        * samples are independent.

    Input parameters:
        X       -   sample 1. Array whose index goes from 0 to N-1.
        N       -   size of sample.
        Y       -   sample 2. Array whose index goes from 0 to M-1.
        M       -   size of sample.

    Output parameters:
        BothTails   -   p-value for two-tailed test.
                        If BothTails is less than the given significance level
                        the null hypothesis is rejected.
        LeftTail    -   p-value for left-tailed test.
                        If LeftTail is less than the given significance level,
                        the null hypothesis is rejected.
        RightTail   -   p-value for right-tailed test.
                        If RightTail is less than the given significance level
                        the null hypothesis is rejected.

      -- ALGLIB --
         Copyright 18.09.2006 by Bochkanov Sergey
    *************************************************************************/
    public static void studentttest2(
        double[] x,
        int n,
        double[] y,
        int m,
        out double bothtails,
        out double lefttail,
        out double righttail)

### `unequalvariancettest` function

    /*************************************************************************
    Two-sample unpooled test

    This test checks three hypotheses about the mean of the given samples. The
    following tests are performed:
        * two-tailed test (null hypothesis - the means are equal)
        * left-tailed test (null hypothesis - the mean of the first sample  is
          greater than or equal to the mean of the second sample)
        * right-tailed test (null hypothesis - the mean of the first sample is
          less than or equal to the mean of the second sample).

    Test is based on the following assumptions:
        * given samples have normal distributions
        * samples are independent.
    Dispersion equality is not required

    Input parameters:
        X - sample 1. Array whose index goes from 0 to N-1.
        N - size of the sample.
        Y - sample 2. Array whose index goes from 0 to M-1.
        M - size of the sample.

    Output parameters:
        BothTails   -   p-value for two-tailed test.
                        If BothTails is less than the given significance level
                        the null hypothesis is rejected.
        LeftTail    -   p-value for left-tailed test.
                        If LeftTail is less than the given significance level,
                        the null hypothesis is rejected.
        RightTail   -   p-value for right-tailed test.
                        If RightTail is less than the given significance level
                        the null hypothesis is rejected.

      -- ALGLIB --
         Copyright 18.09.2006 by Bochkanov Sergey
    *************************************************************************/
    public static void unequalvariancettest(
        double[] x,
        int n,
        double[] y,
        int m,
        out double bothtails,
        out double lefttail,
        out double righttail)

## `svd` subpackage

### Functions

rmatrixsvd

### Examples

### `rmatrixsvd` function

    /*************************************************************************
    Singular value decomposition of a rectangular matrix.

    The algorithm calculates the singular value decomposition of a matrix of
    size MxN: A = U * S * V^T

    The algorithm finds the singular values and, optionally, matrices U and V^T.
    The algorithm can find both first min(M,N) columns of matrix U and rows of
    matrix V^T (singular vectors), and matrices U and V^T wholly (of sizes MxM
    and NxN respectively).

    Take into account that the subroutine does not return matrix V but V^T.

    Input parameters:
        A           -   matrix to be decomposed.
                        Array whose indexes range within [0..M-1, 0..N-1].
        M           -   number of rows in matrix A.
        N           -   number of columns in matrix A.
        UNeeded     -   0, 1 or 2. See the description of the parameter U.
        VTNeeded    -   0, 1 or 2. See the description of the parameter VT.
        AdditionalMemory -
                        If the parameter:
                         * equals 0, the algorithm doesn’t use additional
                           memory (lower requirements, lower performance).
                         * equals 1, the algorithm uses additional
                           memory of size min(M,N)*min(M,N) of real numbers.
                           It often speeds up the algorithm.
                         * equals 2, the algorithm uses additional
                           memory of size M*min(M,N) of real numbers.
                           It allows to get a maximum performance.
                        The recommended value of the parameter is 2.

    Output parameters:
        W           -   contains singular values in descending order.
        U           -   if UNeeded=0, U isn't changed, the left singular vectors
                        are not calculated.
                        if Uneeded=1, U contains left singular vectors (first
                        min(M,N) columns of matrix U). Array whose indexes range
                        within [0..M-1, 0..Min(M,N)-1].
                        if UNeeded=2, U contains matrix U wholly. Array whose
                        indexes range within [0..M-1, 0..M-1].
        VT          -   if VTNeeded=0, VT isn’t changed, the right singular vectors
                        are not calculated.
                        if VTNeeded=1, VT contains right singular vectors (first
                        min(M,N) rows of matrix V^T). Array whose indexes range
                        within [0..min(M,N)-1, 0..N-1].
                        if VTNeeded=2, VT contains matrix V^T wholly. Array whose
                        indexes range within [0..N-1, 0..N-1].

      -- ALGLIB --
         Copyright 2005 by Bochkanov Sergey
    *************************************************************************/
    public static bool rmatrixsvd(
        double[,] a,
        int m,
        int n,
        int uneeded,
        int vtneeded,
        int additionalmemory,
        out double[] w,
        out double[,] u,
        out double[,] vt)

## `trfac` subpackage

### Functions

cmatrixlu
hpdmatrixcholesky
rmatrixlu
spdmatrixcholesky

### Examples

### `cmatrixlu` function

    /*************************************************************************
    LU decomposition of a general complex matrix with row pivoting

    A is represented as A = P*L*U, where:
    * L is lower unitriangular matrix
    * U is upper triangular matrix
    * P = P0*P1*...*PK, K=min(M,N)-1,
      Pi - permutation matrix for I and Pivots[I]

    This is cache-oblivous implementation of LU decomposition. It is optimized
    for square matrices. As for rectangular matrices:
    * best case - M>>N
    * worst case - N>>M, small M, large N, matrix does not fit in CPU cache

    INPUT PARAMETERS:
        A       -   array[0..M-1, 0..N-1].
        M       -   number of rows in matrix A.
        N       -   number of columns in matrix A.

    OUTPUT PARAMETERS:
        A       -   matrices L and U in compact form:
                    * L is stored under main diagonal
                    * U is stored on and above main diagonal
        Pivots  -   permutation matrix in compact form.
                    array[0..Min(M-1,N-1)].

      -- ALGLIB routine --
         10.01.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void cmatrixlu(
        ref complex[,] a,
        int m,
        int n,
        out int[] pivots)

### `hpdmatrixcholesky` function

    /*************************************************************************
    Cache-oblivious Cholesky decomposition

    The algorithm computes Cholesky decomposition  of  a  Hermitian  positive-
    definite matrix. The result of an algorithm is a representation  of  A  as
    A=U'*U  or A=L*L' (here X' detones conj(X^T)).

    INPUT PARAMETERS:
        A       -   upper or lower triangle of a factorized matrix.
                    array with elements [0..N-1, 0..N-1].
        N       -   size of matrix A.
        IsUpper -   if IsUpper=True, then A contains an upper triangle of
                    a symmetric matrix, otherwise A contains a lower one.

    OUTPUT PARAMETERS:
        A       -   the result of factorization. If IsUpper=True, then
                    the upper triangle contains matrix U, so that A = U'*U,
                    and the elements below the main diagonal are not modified.
                    Similarly, if IsUpper = False.

    RESULT:
        If  the  matrix  is  positive-definite,  the  function  returns  True.
        Otherwise, the function returns False. Contents of A is not determined
        in such case.

      -- ALGLIB routine --
         15.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static bool hpdmatrixcholesky(
        ref complex[,] a,
        int n,
        bool isupper)

### `rmatrixlu` function

    /*************************************************************************
    LU decomposition of a general real matrix with row pivoting

    A is represented as A = P*L*U, where:
    * L is lower unitriangular matrix
    * U is upper triangular matrix
    * P = P0*P1*...*PK, K=min(M,N)-1,
      Pi - permutation matrix for I and Pivots[I]

    This is cache-oblivous implementation of LU decomposition.
    It is optimized for square matrices. As for rectangular matrices:
    * best case - M>>N
    * worst case - N>>M, small M, large N, matrix does not fit in CPU cache

    INPUT PARAMETERS:
        A       -   array[0..M-1, 0..N-1].
        M       -   number of rows in matrix A.
        N       -   number of columns in matrix A.

    OUTPUT PARAMETERS:
        A       -   matrices L and U in compact form:
                    * L is stored under main diagonal
                    * U is stored on and above main diagonal
        Pivots  -   permutation matrix in compact form.
                    array[0..Min(M-1,N-1)].

      -- ALGLIB routine --
         10.01.2010
         Bochkanov Sergey
    *************************************************************************/
    public static void rmatrixlu(
        ref double[,] a,
        int m,
        int n,
        out int[] pivots)

### `spdmatrixcholesky` function

    /*************************************************************************
    Cache-oblivious Cholesky decomposition

    The algorithm computes Cholesky decomposition  of  a  symmetric  positive-
    definite matrix. The result of an algorithm is a representation  of  A  as
    A=U^T*U  or A=L*L^T

    INPUT PARAMETERS:
        A       -   upper or lower triangle of a factorized matrix.
                    array with elements [0..N-1, 0..N-1].
        N       -   size of matrix A.
        IsUpper -   if IsUpper=True, then A contains an upper triangle of
                    a symmetric matrix, otherwise A contains a lower one.

    OUTPUT PARAMETERS:
        A       -   the result of factorization. If IsUpper=True, then
                    the upper triangle contains matrix U, so that A = U^T*U,
                    and the elements below the main diagonal are not modified.
                    Similarly, if IsUpper = False.

    RESULT:
        If  the  matrix  is  positive-definite,  the  function  returns  True.
        Otherwise, the function returns False. Contents of A is not determined
        in such case.

      -- ALGLIB routine --
         15.12.2009
         Bochkanov Sergey
    *************************************************************************/
    public static bool spdmatrixcholesky(ref double[,] a, int n, bool isupper)

## `trigintegrals` subpackage

### Functions

hyperbolicsinecosineintegrals
sinecosineintegrals

### Examples

### `hyperbolicsinecosineintegrals` function

    /*************************************************************************
    Hyperbolic sine and cosine integrals

    Approximates the integrals

                               x
                               -
                              | |   cosh t - 1
      Chi(x) = eul + ln x +   |    -----------  dt,
                            | |          t
                             -
                             0

                  x
                  -
                 | |  sinh t
      Shi(x) =   |    ------  dt
               | |       t
                -
                0

    where eul = 0.57721566490153286061 is Euler's constant.
    The integrals are evaluated by power series for x < 8
    and by Chebyshev expansions for x between 8 and 88.
    For large x, both functions approach exp(x)/2x.
    Arguments greater than 88 in magnitude return MAXNUM.

    ACCURACY:

    Test interval 0 to 88.
                         Relative error:
    arithmetic   function  # trials      peak         rms
       IEEE         Shi      30000       6.9e-16     1.6e-16
           Absolute error, except relative when |Chi| > 1:
       IEEE         Chi      30000       8.4e-16     1.4e-16

    Cephes Math Library Release 2.8:  June, 2000
    Copyright 1984, 1987, 2000 by Stephen L. Moshier
    *************************************************************************/
    public static void hyperbolicsinecosineintegrals(
        double x,
        out double shi,
        out double chi)

### `sinecosineintegrals` function

    /*************************************************************************
    Sine and cosine integrals

    Evaluates the integrals

                             x
                             -
                            |  cos t - 1
      Ci(x) = eul + ln x +  |  --------- dt,
                            |      t
                           -
                            0
                x
                -
               |  sin t
      Si(x) =  |  ----- dt
               |    t
              -
               0

    where eul = 0.57721566490153286061 is Euler's constant.
    The integrals are approximated by rational functions.
    For x > 8 auxiliary functions f(x) and g(x) are employed
    such that

    Ci(x) = f(x) sin(x) - g(x) cos(x)
    Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)

    ACCURACY:
       Test interval = [0,50].
    Absolute error, except relative when > 1:
    arithmetic   function   # trials      peak         rms
       IEEE        Si        30000       4.4e-16     7.3e-17
       IEEE        Ci        30000       6.9e-16     5.1e-17

    Cephes Math Library Release 2.1:  January, 1989
    Copyright 1984, 1987, 1989 by Stephen L. Moshier
    *************************************************************************/
    public static void sinecosineintegrals(
        double x,
        out double si,
        out double ci)

## `variancetests` subpackage

### Functions

ftest
onesamplevariancetest

### Examples

### `ftest` function

    /*************************************************************************
    Two-sample F-test

    This test checks three hypotheses about dispersions of the given  samples.
    The following tests are performed:
        * two-tailed test (null hypothesis - the dispersions are equal)
        * left-tailed test (null hypothesis  -  the  dispersion  of  the first
          sample is greater than or equal to  the  dispersion  of  the  second
          sample).
        * right-tailed test (null hypothesis - the  dispersion  of  the  first
          sample is less than or equal to the dispersion of the second sample)

    The test is based on the following assumptions:
        * the given samples have normal distributions
        * the samples are independent.

    Input parameters:
        X   -   sample 1. Array whose index goes from 0 to N-1.
        N   -   sample size.
        Y   -   sample 2. Array whose index goes from 0 to M-1.
        M   -   sample size.

    Output parameters:
        BothTails   -   p-value for two-tailed test.
                        If BothTails is less than the given significance level
                        the null hypothesis is rejected.
        LeftTail    -   p-value for left-tailed test.
                        If LeftTail is less than the given significance level,
                        the null hypothesis is rejected.
        RightTail   -   p-value for right-tailed test.
                        If RightTail is less than the given significance level
                        the null hypothesis is rejected.

      -- ALGLIB --
         Copyright 19.09.2006 by Bochkanov Sergey
    *************************************************************************/
    public static void ftest(
        double[] x,
        int n,
        double[] y,
        int m,
        out double bothtails,
        out double lefttail,
        out double righttail)

### `onesamplevariancetest` function

    /*************************************************************************
    One-sample chi-square test

    This test checks three hypotheses about the dispersion of the given sample
    The following tests are performed:
        * two-tailed test (null hypothesis - the dispersion equals  the  given
          number)
        * left-tailed test (null hypothesis - the dispersion is  greater  than
          or equal to the given number)
        * right-tailed test (null hypothesis  -  dispersion is  less  than  or
          equal to the given number).

    Test is based on the following assumptions:
        * the given sample has a normal distribution.

    Input parameters:
        X           -   sample 1. Array whose index goes from 0 to N-1.
        N           -   size of the sample.
        Variance    -   dispersion value to compare with.

    Output parameters:
        BothTails   -   p-value for two-tailed test.
                        If BothTails is less than the given significance level
                        the null hypothesis is rejected.
        LeftTail    -   p-value for left-tailed test.
                        If LeftTail is less than the given significance level,
                        the null hypothesis is rejected.
        RightTail   -   p-value for right-tailed test.
                        If RightTail is less than the given significance level
                        the null hypothesis is rejected.

      -- ALGLIB --
         Copyright 19.09.2006 by Bochkanov Sergey
    *************************************************************************/
    public static void onesamplevariancetest(
        double[] x,
        int n,
        double variance,
        out double bothtails,
        out double lefttail,
        out double righttail)

## `wsr` subpackage

### Functions

wilcoxonsignedranktest

### Examples

### `wilcoxonsignedranktest` function

    /*************************************************************************
    Wilcoxon signed-rank test

    This test checks three hypotheses about the median  of  the  given sample.
    The following tests are performed:
        * two-tailed test (null hypothesis - the median is equal to the  given
          value)
        * left-tailed test (null hypothesis - the median is  greater  than  or
          equal to the given value)
        * right-tailed test (null hypothesis  -  the  median  is  less than or
          equal to the given value)

    Requirements:
        * the scale of measurement should be ordinal, interval or  ratio (i.e.
          the test could not be applied to nominal variables).
        * the distribution should be continuous and symmetric relative to  its
          median.
        * number of distinct values in the X array should be greater than 4

    The test is non-parametric and doesn't require distribution X to be normal

    Input parameters:
        X       -   sample. Array whose index goes from 0 to N-1.
        N       -   size of the sample.
        Median  -   assumed median value.

    Output parameters:
        BothTails   -   p-value for two-tailed test.
                        If BothTails is less than the given significance level
                        the null hypothesis is rejected.
        LeftTail    -   p-value for left-tailed test.
                        If LeftTail is less than the given significance level,
                        the null hypothesis is rejected.
        RightTail   -   p-value for right-tailed test.
                        If RightTail is less than the given significance level
                        the null hypothesis is rejected.

    To calculate p-values, special approximation is used. This method lets  us
    calculate p-values with two decimal places in interval [0.0001, 1].

    "Two decimal places" does not sound very impressive, but in  practice  the
    relative error of less than 1% is enough to make a decision.

    There is no approximation outside the [0.0001, 1] interval. Therefore,  if
    the significance level outlies this interval, the test returns 0.0001.

      -- ALGLIB --
         Copyright 08.09.2006 by Bochkanov Sergey
    *************************************************************************/
    public static void wilcoxonsignedranktest(
        double[] x,
        int n,
        double e,
        out double bothtails,
        out double lefttail,
        out double righttail)