File: solvers.h

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/*************************************************************************
Copyright (c) Sergey Bochkanov (ALGLIB project).

>>> SOURCE LICENSE >>>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation (www.fsf.org); either version 2 of the
License, or (at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

A copy of the GNU General Public License is available at
http://www.fsf.org/licensing/licenses
>>> END OF LICENSE >>>
*************************************************************************/
#ifndef _solvers_pkg_h
#define _solvers_pkg_h
#include "ap.h"
#include "alglibinternal.h"
#include "linalg.h"
#include "alglibmisc.h"

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (DATATYPES)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
typedef struct
{
    double r1;
    double rinf;
} densesolverreport;
typedef struct
{
    double r2;
    ae_matrix cx;
    ae_int_t n;
    ae_int_t k;
} densesolverlsreport;
typedef struct
{
    normestimatorstate nes;
    ae_vector rx;
    ae_vector b;
    ae_int_t n;
    ae_int_t m;
    ae_int_t prectype;
    ae_vector ui;
    ae_vector uip1;
    ae_vector vi;
    ae_vector vip1;
    ae_vector omegai;
    ae_vector omegaip1;
    double alphai;
    double alphaip1;
    double betai;
    double betaip1;
    double phibari;
    double phibarip1;
    double phii;
    double rhobari;
    double rhobarip1;
    double rhoi;
    double ci;
    double si;
    double theta;
    double lambdai;
    ae_vector d;
    double anorm;
    double bnorm2;
    double dnorm;
    double r2;
    ae_vector x;
    ae_vector mv;
    ae_vector mtv;
    double epsa;
    double epsb;
    double epsc;
    ae_int_t maxits;
    ae_bool xrep;
    ae_bool xupdated;
    ae_bool needmv;
    ae_bool needmtv;
    ae_bool needmv2;
    ae_bool needvmv;
    ae_bool needprec;
    ae_int_t repiterationscount;
    ae_int_t repnmv;
    ae_int_t repterminationtype;
    ae_bool running;
    ae_vector tmpd;
    ae_vector tmpx;
    rcommstate rstate;
} linlsqrstate;
typedef struct
{
    ae_int_t iterationscount;
    ae_int_t nmv;
    ae_int_t terminationtype;
} linlsqrreport;
typedef struct
{
    ae_vector rx;
    ae_vector b;
    ae_int_t n;
    ae_int_t prectype;
    ae_vector cx;
    ae_vector cr;
    ae_vector cz;
    ae_vector p;
    ae_vector r;
    ae_vector z;
    double alpha;
    double beta;
    double r2;
    double meritfunction;
    ae_vector x;
    ae_vector mv;
    ae_vector pv;
    double vmv;
    ae_vector startx;
    double epsf;
    ae_int_t maxits;
    ae_int_t itsbeforerestart;
    ae_int_t itsbeforerupdate;
    ae_bool xrep;
    ae_bool xupdated;
    ae_bool needmv;
    ae_bool needmtv;
    ae_bool needmv2;
    ae_bool needvmv;
    ae_bool needprec;
    ae_int_t repiterationscount;
    ae_int_t repnmv;
    ae_int_t repterminationtype;
    ae_bool running;
    ae_vector tmpd;
    rcommstate rstate;
} lincgstate;
typedef struct
{
    ae_int_t iterationscount;
    ae_int_t nmv;
    ae_int_t terminationtype;
    double r2;
} lincgreport;
typedef struct
{
    ae_int_t n;
    ae_int_t m;
    double epsf;
    ae_int_t maxits;
    ae_bool xrep;
    double stpmax;
    ae_vector x;
    double f;
    ae_vector fi;
    ae_matrix j;
    ae_bool needf;
    ae_bool needfij;
    ae_bool xupdated;
    rcommstate rstate;
    ae_int_t repiterationscount;
    ae_int_t repnfunc;
    ae_int_t repnjac;
    ae_int_t repterminationtype;
    ae_vector xbase;
    double fbase;
    double fprev;
    ae_vector candstep;
    ae_vector rightpart;
    ae_vector cgbuf;
} nleqstate;
typedef struct
{
    ae_int_t iterationscount;
    ae_int_t nfunc;
    ae_int_t njac;
    ae_int_t terminationtype;
} nleqreport;

}

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS C++ INTERFACE
//
/////////////////////////////////////////////////////////////////////////
namespace alglib
{

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

*************************************************************************/
class _densesolverreport_owner
{
public:
    _densesolverreport_owner();
    _densesolverreport_owner(const _densesolverreport_owner &rhs);
    _densesolverreport_owner& operator=(const _densesolverreport_owner &rhs);
    virtual ~_densesolverreport_owner();
    alglib_impl::densesolverreport* c_ptr();
    alglib_impl::densesolverreport* c_ptr() const;
protected:
    alglib_impl::densesolverreport *p_struct;
};
class densesolverreport : public _densesolverreport_owner
{
public:
    densesolverreport();
    densesolverreport(const densesolverreport &rhs);
    densesolverreport& operator=(const densesolverreport &rhs);
    virtual ~densesolverreport();
    double &r1;
    double &rinf;

};


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

*************************************************************************/
class _densesolverlsreport_owner
{
public:
    _densesolverlsreport_owner();
    _densesolverlsreport_owner(const _densesolverlsreport_owner &rhs);
    _densesolverlsreport_owner& operator=(const _densesolverlsreport_owner &rhs);
    virtual ~_densesolverlsreport_owner();
    alglib_impl::densesolverlsreport* c_ptr();
    alglib_impl::densesolverlsreport* c_ptr() const;
protected:
    alglib_impl::densesolverlsreport *p_struct;
};
class densesolverlsreport : public _densesolverlsreport_owner
{
public:
    densesolverlsreport();
    densesolverlsreport(const densesolverlsreport &rhs);
    densesolverlsreport& operator=(const densesolverlsreport &rhs);
    virtual ~densesolverlsreport();
    double &r2;
    real_2d_array cx;
    ae_int_t &n;
    ae_int_t &k;

};

/*************************************************************************
This object stores state of the LinLSQR method.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class _linlsqrstate_owner
{
public:
    _linlsqrstate_owner();
    _linlsqrstate_owner(const _linlsqrstate_owner &rhs);
    _linlsqrstate_owner& operator=(const _linlsqrstate_owner &rhs);
    virtual ~_linlsqrstate_owner();
    alglib_impl::linlsqrstate* c_ptr();
    alglib_impl::linlsqrstate* c_ptr() const;
protected:
    alglib_impl::linlsqrstate *p_struct;
};
class linlsqrstate : public _linlsqrstate_owner
{
public:
    linlsqrstate();
    linlsqrstate(const linlsqrstate &rhs);
    linlsqrstate& operator=(const linlsqrstate &rhs);
    virtual ~linlsqrstate();

};


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

*************************************************************************/
class _linlsqrreport_owner
{
public:
    _linlsqrreport_owner();
    _linlsqrreport_owner(const _linlsqrreport_owner &rhs);
    _linlsqrreport_owner& operator=(const _linlsqrreport_owner &rhs);
    virtual ~_linlsqrreport_owner();
    alglib_impl::linlsqrreport* c_ptr();
    alglib_impl::linlsqrreport* c_ptr() const;
protected:
    alglib_impl::linlsqrreport *p_struct;
};
class linlsqrreport : public _linlsqrreport_owner
{
public:
    linlsqrreport();
    linlsqrreport(const linlsqrreport &rhs);
    linlsqrreport& operator=(const linlsqrreport &rhs);
    virtual ~linlsqrreport();
    ae_int_t &iterationscount;
    ae_int_t &nmv;
    ae_int_t &terminationtype;

};

/*************************************************************************
This object stores state of the linear CG method.

You should use ALGLIB functions to work with this object.
Never try to access its fields directly!
*************************************************************************/
class _lincgstate_owner
{
public:
    _lincgstate_owner();
    _lincgstate_owner(const _lincgstate_owner &rhs);
    _lincgstate_owner& operator=(const _lincgstate_owner &rhs);
    virtual ~_lincgstate_owner();
    alglib_impl::lincgstate* c_ptr();
    alglib_impl::lincgstate* c_ptr() const;
protected:
    alglib_impl::lincgstate *p_struct;
};
class lincgstate : public _lincgstate_owner
{
public:
    lincgstate();
    lincgstate(const lincgstate &rhs);
    lincgstate& operator=(const lincgstate &rhs);
    virtual ~lincgstate();

};


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

*************************************************************************/
class _lincgreport_owner
{
public:
    _lincgreport_owner();
    _lincgreport_owner(const _lincgreport_owner &rhs);
    _lincgreport_owner& operator=(const _lincgreport_owner &rhs);
    virtual ~_lincgreport_owner();
    alglib_impl::lincgreport* c_ptr();
    alglib_impl::lincgreport* c_ptr() const;
protected:
    alglib_impl::lincgreport *p_struct;
};
class lincgreport : public _lincgreport_owner
{
public:
    lincgreport();
    lincgreport(const lincgreport &rhs);
    lincgreport& operator=(const lincgreport &rhs);
    virtual ~lincgreport();
    ae_int_t &iterationscount;
    ae_int_t &nmv;
    ae_int_t &terminationtype;
    double &r2;

};

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

*************************************************************************/
class _nleqstate_owner
{
public:
    _nleqstate_owner();
    _nleqstate_owner(const _nleqstate_owner &rhs);
    _nleqstate_owner& operator=(const _nleqstate_owner &rhs);
    virtual ~_nleqstate_owner();
    alglib_impl::nleqstate* c_ptr();
    alglib_impl::nleqstate* c_ptr() const;
protected:
    alglib_impl::nleqstate *p_struct;
};
class nleqstate : public _nleqstate_owner
{
public:
    nleqstate();
    nleqstate(const nleqstate &rhs);
    nleqstate& operator=(const nleqstate &rhs);
    virtual ~nleqstate();
    ae_bool &needf;
    ae_bool &needfij;
    ae_bool &xupdated;
    double &f;
    real_1d_array fi;
    real_2d_array j;
    real_1d_array x;

};


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

*************************************************************************/
class _nleqreport_owner
{
public:
    _nleqreport_owner();
    _nleqreport_owner(const _nleqreport_owner &rhs);
    _nleqreport_owner& operator=(const _nleqreport_owner &rhs);
    virtual ~_nleqreport_owner();
    alglib_impl::nleqreport* c_ptr();
    alglib_impl::nleqreport* c_ptr() const;
protected:
    alglib_impl::nleqreport *p_struct;
};
class nleqreport : public _nleqreport_owner
{
public:
    nleqreport();
    nleqreport(const nleqreport &rhs);
    nleqreport& operator=(const nleqreport &rhs);
    virtual ~nleqreport();
    ae_int_t &iterationscount;
    ae_int_t &nfunc;
    ae_int_t &njac;
    ae_int_t &terminationtype;

};

/*************************************************************************
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
*************************************************************************/
void rmatrixsolve(const real_2d_array &a, const ae_int_t n, const real_1d_array &b, ae_int_t &info, densesolverreport &rep, real_1d_array &x);


/*************************************************************************
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
*************************************************************************/
void rmatrixsolvem(const real_2d_array &a, const ae_int_t n, const real_2d_array &b, const ae_int_t m, const bool rfs, ae_int_t &info, densesolverreport &rep, real_2d_array &x);


/*************************************************************************
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
*************************************************************************/
void rmatrixlusolve(const real_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const real_1d_array &b, ae_int_t &info, densesolverreport &rep, real_1d_array &x);


/*************************************************************************
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
*************************************************************************/
void rmatrixlusolvem(const real_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const real_2d_array &b, const ae_int_t m, ae_int_t &info, densesolverreport &rep, real_2d_array &x);


/*************************************************************************
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
*************************************************************************/
void rmatrixmixedsolve(const real_2d_array &a, const real_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const real_1d_array &b, ae_int_t &info, densesolverreport &rep, real_1d_array &x);


/*************************************************************************
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
*************************************************************************/
void rmatrixmixedsolvem(const real_2d_array &a, const real_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const real_2d_array &b, const ae_int_t m, ae_int_t &info, densesolverreport &rep, real_2d_array &x);


/*************************************************************************
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
*************************************************************************/
void cmatrixsolvem(const complex_2d_array &a, const ae_int_t n, const complex_2d_array &b, const ae_int_t m, const bool rfs, ae_int_t &info, densesolverreport &rep, complex_2d_array &x);


/*************************************************************************
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
*************************************************************************/
void cmatrixsolve(const complex_2d_array &a, const ae_int_t n, const complex_1d_array &b, ae_int_t &info, densesolverreport &rep, complex_1d_array &x);


/*************************************************************************
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
*************************************************************************/
void cmatrixlusolvem(const complex_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const complex_2d_array &b, const ae_int_t m, ae_int_t &info, densesolverreport &rep, complex_2d_array &x);


/*************************************************************************
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
*************************************************************************/
void cmatrixlusolve(const complex_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const complex_1d_array &b, ae_int_t &info, densesolverreport &rep, complex_1d_array &x);


/*************************************************************************
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
*************************************************************************/
void cmatrixmixedsolvem(const complex_2d_array &a, const complex_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const complex_2d_array &b, const ae_int_t m, ae_int_t &info, densesolverreport &rep, complex_2d_array &x);


/*************************************************************************
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
*************************************************************************/
void cmatrixmixedsolve(const complex_2d_array &a, const complex_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const complex_1d_array &b, ae_int_t &info, densesolverreport &rep, complex_1d_array &x);


/*************************************************************************
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
*************************************************************************/
void spdmatrixsolvem(const real_2d_array &a, const ae_int_t n, const bool isupper, const real_2d_array &b, const ae_int_t m, ae_int_t &info, densesolverreport &rep, real_2d_array &x);


/*************************************************************************
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
*************************************************************************/
void spdmatrixsolve(const real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &b, ae_int_t &info, densesolverreport &rep, real_1d_array &x);


/*************************************************************************
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
*************************************************************************/
void spdmatrixcholeskysolvem(const real_2d_array &cha, const ae_int_t n, const bool isupper, const real_2d_array &b, const ae_int_t m, ae_int_t &info, densesolverreport &rep, real_2d_array &x);


/*************************************************************************
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
*************************************************************************/
void spdmatrixcholeskysolve(const real_2d_array &cha, const ae_int_t n, const bool isupper, const real_1d_array &b, ae_int_t &info, densesolverreport &rep, real_1d_array &x);


/*************************************************************************
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
*************************************************************************/
void hpdmatrixsolvem(const complex_2d_array &a, const ae_int_t n, const bool isupper, const complex_2d_array &b, const ae_int_t m, ae_int_t &info, densesolverreport &rep, complex_2d_array &x);


/*************************************************************************
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
*************************************************************************/
void hpdmatrixsolve(const complex_2d_array &a, const ae_int_t n, const bool isupper, const complex_1d_array &b, ae_int_t &info, densesolverreport &rep, complex_1d_array &x);


/*************************************************************************
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
*************************************************************************/
void hpdmatrixcholeskysolvem(const complex_2d_array &cha, const ae_int_t n, const bool isupper, const complex_2d_array &b, const ae_int_t m, ae_int_t &info, densesolverreport &rep, complex_2d_array &x);


/*************************************************************************
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
*************************************************************************/
void hpdmatrixcholeskysolve(const complex_2d_array &cha, const ae_int_t n, const bool isupper, const complex_1d_array &b, ae_int_t &info, densesolverreport &rep, complex_1d_array &x);


/*************************************************************************
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 (and correct handling!) 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 (even for singular A)
                * zeros, if SVD subroutine failed

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
*************************************************************************/
void rmatrixsolvels(const real_2d_array &a, const ae_int_t nrows, const ae_int_t ncols, const real_1d_array &b, const double threshold, ae_int_t &info, densesolverlsreport &rep, real_1d_array &x);

/*************************************************************************
This function initializes linear LSQR Solver. This solver is used to solve
non-symmetric (and, possibly, non-square) problems. Least squares solution
is returned for non-compatible systems.

USAGE:
1. User initializes algorithm state with LinLSQRCreate() call
2. User tunes solver parameters with  LinLSQRSetCond() and other functions
3. User  calls  LinLSQRSolveSparse()  function which takes algorithm state
   and SparseMatrix object.
4. User calls LinLSQRResults() to get solution
5. Optionally, user may call LinLSQRSolveSparse() again to  solve  another
   problem  with different matrix and/or right part without reinitializing
   LinLSQRState structure.

INPUT PARAMETERS:
    M       -   number of rows in A
    N       -   number of variables, N>0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrcreate(const ae_int_t m, const ae_int_t n, linlsqrstate &state);


/*************************************************************************
This  function  changes  preconditioning  settings of LinLSQQSolveSparse()
function. By default, SolveSparse() uses diagonal preconditioner,  but  if
you want to use solver without preconditioning, you can call this function
which forces solver to use unit matrix for preconditioning.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void linlsqrsetprecunit(const linlsqrstate &state);


/*************************************************************************
This  function  changes  preconditioning  settings  of  LinCGSolveSparse()
function.  LinCGSolveSparse() will use diagonal of the  system  matrix  as
preconditioner. This preconditioning mode is active by default.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void linlsqrsetprecdiag(const linlsqrstate &state);


/*************************************************************************
This function sets optional Tikhonov regularization coefficient.
It is zero by default.

INPUT PARAMETERS:
    LambdaI -   regularization factor, LambdaI>=0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrsetlambdai(const linlsqrstate &state, const double lambdai);


/*************************************************************************
Procedure for solution of A*x=b with sparse A.

INPUT PARAMETERS:
    State   -   algorithm state
    A       -   sparse M*N matrix in the CRS format (you MUST contvert  it
                to CRS format  by  calling  SparseConvertToCRS()  function
                BEFORE you pass it to this function).
    B       -   right part, array[M]

RESULT:
    This function returns no result.
    You can get solution by calling LinCGResults()

NOTE: this function uses lightweight preconditioning -  multiplication  by
      inverse of diag(A). If you want, you can turn preconditioning off by
      calling LinLSQRSetPrecUnit(). However, preconditioning cost is   low
      and preconditioner is very important for solution  of  badly  scaled
      problems.

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrsolvesparse(const linlsqrstate &state, const sparsematrix &a, const real_1d_array &b);


/*************************************************************************
This function sets stopping criteria.

INPUT PARAMETERS:
    EpsA    -   algorithm will be stopped if ||A^T*Rk||/(||A||*||Rk||)<=EpsA.
    EpsB    -   algorithm will be stopped if ||Rk||<=EpsB*||B||
    MaxIts  -   algorithm will be stopped if number of iterations
                more than MaxIts.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE: if EpsA,EpsB,EpsC and MaxIts are zero then these variables will
be setted as default values.

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrsetcond(const linlsqrstate &state, const double epsa, const double epsb, const ae_int_t maxits);


/*************************************************************************
LSQR solver: results.

This function must be called after LinLSQRSolve

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[N], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    *  1    ||Rk||<=EpsB*||B||
                    *  4    ||A^T*Rk||/(||A||*||Rk||)<=EpsA
                    *  5    MaxIts steps was taken
                    *  7    rounding errors prevent further progress,
                            X contains best point found so far.
                            (sometimes returned on singular systems)
                * Rep.IterationsCount contains iterations count
                * NMV countains number of matrix-vector calculations

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrresults(const linlsqrstate &state, real_1d_array &x, linlsqrreport &rep);


/*************************************************************************
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 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrsetxrep(const linlsqrstate &state, const bool needxrep);

/*************************************************************************
This function initializes linear CG Solver. This solver is used  to  solve
symmetric positive definite problems. If you want  to  solve  nonsymmetric
(or non-positive definite) problem you may use LinLSQR solver provided  by
ALGLIB.

USAGE:
1. User initializes algorithm state with LinCGCreate() call
2. User tunes solver parameters with  LinCGSetCond() and other functions
3. Optionally, user sets starting point with LinCGSetStartingPoint()
4. User  calls LinCGSolveSparse() function which takes algorithm state and
   SparseMatrix object.
5. User calls LinCGResults() to get solution
6. Optionally, user may call LinCGSolveSparse()  again  to  solve  another
   problem  with different matrix and/or right part without reinitializing
   LinCGState structure.

INPUT PARAMETERS:
    N       -   problem dimension, N>0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgcreate(const ae_int_t n, lincgstate &state);


/*************************************************************************
This function sets starting point.
By default, zero starting point is used.

INPUT PARAMETERS:
    X       -   starting point, array[N]

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsetstartingpoint(const lincgstate &state, const real_1d_array &x);


/*************************************************************************
This  function  changes  preconditioning  settings  of  LinCGSolveSparse()
function. By default, SolveSparse() uses diagonal preconditioner,  but  if
you want to use solver without preconditioning, you can call this function
which forces solver to use unit matrix for preconditioning.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void lincgsetprecunit(const lincgstate &state);


/*************************************************************************
This  function  changes  preconditioning  settings  of  LinCGSolveSparse()
function.  LinCGSolveSparse() will use diagonal of the  system  matrix  as
preconditioner. This preconditioning mode is active by default.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void lincgsetprecdiag(const lincgstate &state);


/*************************************************************************
This function sets stopping criteria.

INPUT PARAMETERS:
    EpsF    -   algorithm will be stopped if norm of residual is less than
                EpsF*||b||.
    MaxIts  -   algorithm will be stopped if number of iterations is  more
                than MaxIts.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
If  both  EpsF  and  MaxIts  are  zero then small EpsF will be set to small
value.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsetcond(const lincgstate &state, const double epsf, const ae_int_t maxits);


/*************************************************************************
Procedure for solution of A*x=b with sparse A.

INPUT PARAMETERS:
    State   -   algorithm state
    A       -   sparse matrix in the CRS format (you MUST contvert  it  to
                CRS format by calling SparseConvertToCRS() function).
    IsUpper -   whether upper or lower triangle of A is used:
                * IsUpper=True  => only upper triangle is used and lower
                                   triangle is not referenced at all
                * IsUpper=False => only lower triangle is used and upper
                                   triangle is not referenced at all
    B       -   right part, array[N]

RESULT:
    This function returns no result.
    You can get solution by calling LinCGResults()

NOTE: this function uses lightweight preconditioning -  multiplication  by
      inverse of diag(A). If you want, you can turn preconditioning off by
      calling LinCGSetPrecUnit(). However, preconditioning cost is low and
      preconditioner  is  very  important  for  solution  of  badly scaled
      problems.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsolvesparse(const lincgstate &state, const sparsematrix &a, const bool isupper, const real_1d_array &b);


/*************************************************************************
CG-solver: results.

This function must be called after LinCGSolve

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[N], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    * -5    input matrix is either not positive definite,
                            too large or too small
                    * -4    overflow/underflow during solution
                            (ill conditioned problem)
                    *  1    ||residual||<=EpsF*||b||
                    *  5    MaxIts steps was taken
                    *  7    rounding errors prevent further progress,
                            best point found is returned
                * Rep.IterationsCount contains iterations count
                * NMV countains number of matrix-vector calculations

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgresults(const lincgstate &state, real_1d_array &x, lincgreport &rep);


/*************************************************************************
This function sets restart frequency. By default, algorithm  is  restarted
after N subsequent iterations.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsetrestartfreq(const lincgstate &state, const ae_int_t srf);


/*************************************************************************
This function sets frequency of residual recalculations.

Algorithm updates residual r_k using iterative formula,  but  recalculates
it from scratch after each 10 iterations. It is done to avoid accumulation
of numerical errors and to stop algorithm when r_k starts to grow.

Such low update frequence (1/10) gives very  little  overhead,  but  makes
algorithm a bit more robust against numerical errors. However, you may
change it

INPUT PARAMETERS:
    Freq    -   desired update frequency, Freq>=0.
                Zero value means that no updates will be done.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsetrupdatefreq(const lincgstate &state, const ae_int_t freq);


/*************************************************************************
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 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsetxrep(const lincgstate &state, const bool needxrep);

/*************************************************************************
                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
*************************************************************************/
void nleqcreatelm(const ae_int_t n, const ae_int_t m, const real_1d_array &x, nleqstate &state);
void nleqcreatelm(const ae_int_t m, const real_1d_array &x, nleqstate &state);


/*************************************************************************
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
*************************************************************************/
void nleqsetcond(const nleqstate &state, const double epsf, const ae_int_t maxits);


/*************************************************************************
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
*************************************************************************/
void nleqsetxrep(const nleqstate &state, const bool needxrep);


/*************************************************************************
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
*************************************************************************/
void nleqsetstpmax(const nleqstate &state, const double stpmax);


/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool nleqiteration(const nleqstate &state);


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

*************************************************************************/
void nleqsolve(nleqstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL);


/*************************************************************************
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
*************************************************************************/
void nleqresults(const nleqstate &state, real_1d_array &x, nleqreport &rep);


/*************************************************************************
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
*************************************************************************/
void nleqresultsbuf(const nleqstate &state, real_1d_array &x, nleqreport &rep);


/*************************************************************************
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
*************************************************************************/
void nleqrestartfrom(const nleqstate &state, const real_1d_array &x);
}

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (FUNCTIONS)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
void rmatrixsolve(/* Real    */ ae_matrix* a,
     ae_int_t n,
     /* Real    */ ae_vector* b,
     ae_int_t* info,
     densesolverreport* rep,
     /* Real    */ ae_vector* x,
     ae_state *_state);
void rmatrixsolvem(/* Real    */ ae_matrix* a,
     ae_int_t n,
     /* Real    */ ae_matrix* b,
     ae_int_t m,
     ae_bool rfs,
     ae_int_t* info,
     densesolverreport* rep,
     /* Real    */ ae_matrix* x,
     ae_state *_state);
void rmatrixlusolve(/* Real    */ ae_matrix* lua,
     /* Integer */ ae_vector* p,
     ae_int_t n,
     /* Real    */ ae_vector* b,
     ae_int_t* info,
     densesolverreport* rep,
     /* Real    */ ae_vector* x,
     ae_state *_state);
void rmatrixlusolvem(/* Real    */ ae_matrix* lua,
     /* Integer */ ae_vector* p,
     ae_int_t n,
     /* Real    */ ae_matrix* b,
     ae_int_t m,
     ae_int_t* info,
     densesolverreport* rep,
     /* Real    */ ae_matrix* x,
     ae_state *_state);
void rmatrixmixedsolve(/* Real    */ ae_matrix* a,
     /* Real    */ ae_matrix* lua,
     /* Integer */ ae_vector* p,
     ae_int_t n,
     /* Real    */ ae_vector* b,
     ae_int_t* info,
     densesolverreport* rep,
     /* Real    */ ae_vector* x,
     ae_state *_state);
void rmatrixmixedsolvem(/* Real    */ ae_matrix* a,
     /* Real    */ ae_matrix* lua,
     /* Integer */ ae_vector* p,
     ae_int_t n,
     /* Real    */ ae_matrix* b,
     ae_int_t m,
     ae_int_t* info,
     densesolverreport* rep,
     /* Real    */ ae_matrix* x,
     ae_state *_state);
void cmatrixsolvem(/* Complex */ ae_matrix* a,
     ae_int_t n,
     /* Complex */ ae_matrix* b,
     ae_int_t m,
     ae_bool rfs,
     ae_int_t* info,
     densesolverreport* rep,
     /* Complex */ ae_matrix* x,
     ae_state *_state);
void cmatrixsolve(/* Complex */ ae_matrix* a,
     ae_int_t n,
     /* Complex */ ae_vector* b,
     ae_int_t* info,
     densesolverreport* rep,
     /* Complex */ ae_vector* x,
     ae_state *_state);
void cmatrixlusolvem(/* Complex */ ae_matrix* lua,
     /* Integer */ ae_vector* p,
     ae_int_t n,
     /* Complex */ ae_matrix* b,
     ae_int_t m,
     ae_int_t* info,
     densesolverreport* rep,
     /* Complex */ ae_matrix* x,
     ae_state *_state);
void cmatrixlusolve(/* Complex */ ae_matrix* lua,
     /* Integer */ ae_vector* p,
     ae_int_t n,
     /* Complex */ ae_vector* b,
     ae_int_t* info,
     densesolverreport* rep,
     /* Complex */ ae_vector* x,
     ae_state *_state);
void cmatrixmixedsolvem(/* Complex */ ae_matrix* a,
     /* Complex */ ae_matrix* lua,
     /* Integer */ ae_vector* p,
     ae_int_t n,
     /* Complex */ ae_matrix* b,
     ae_int_t m,
     ae_int_t* info,
     densesolverreport* rep,
     /* Complex */ ae_matrix* x,
     ae_state *_state);
void cmatrixmixedsolve(/* Complex */ ae_matrix* a,
     /* Complex */ ae_matrix* lua,
     /* Integer */ ae_vector* p,
     ae_int_t n,
     /* Complex */ ae_vector* b,
     ae_int_t* info,
     densesolverreport* rep,
     /* Complex */ ae_vector* x,
     ae_state *_state);
void spdmatrixsolvem(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ ae_matrix* b,
     ae_int_t m,
     ae_int_t* info,
     densesolverreport* rep,
     /* Real    */ ae_matrix* x,
     ae_state *_state);
void spdmatrixsolve(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ ae_vector* b,
     ae_int_t* info,
     densesolverreport* rep,
     /* Real    */ ae_vector* x,
     ae_state *_state);
void spdmatrixcholeskysolvem(/* Real    */ ae_matrix* cha,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ ae_matrix* b,
     ae_int_t m,
     ae_int_t* info,
     densesolverreport* rep,
     /* Real    */ ae_matrix* x,
     ae_state *_state);
void spdmatrixcholeskysolve(/* Real    */ ae_matrix* cha,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ ae_vector* b,
     ae_int_t* info,
     densesolverreport* rep,
     /* Real    */ ae_vector* x,
     ae_state *_state);
void hpdmatrixsolvem(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Complex */ ae_matrix* b,
     ae_int_t m,
     ae_int_t* info,
     densesolverreport* rep,
     /* Complex */ ae_matrix* x,
     ae_state *_state);
void hpdmatrixsolve(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Complex */ ae_vector* b,
     ae_int_t* info,
     densesolverreport* rep,
     /* Complex */ ae_vector* x,
     ae_state *_state);
void hpdmatrixcholeskysolvem(/* Complex */ ae_matrix* cha,
     ae_int_t n,
     ae_bool isupper,
     /* Complex */ ae_matrix* b,
     ae_int_t m,
     ae_int_t* info,
     densesolverreport* rep,
     /* Complex */ ae_matrix* x,
     ae_state *_state);
void hpdmatrixcholeskysolve(/* Complex */ ae_matrix* cha,
     ae_int_t n,
     ae_bool isupper,
     /* Complex */ ae_vector* b,
     ae_int_t* info,
     densesolverreport* rep,
     /* Complex */ ae_vector* x,
     ae_state *_state);
void rmatrixsolvels(/* Real    */ ae_matrix* a,
     ae_int_t nrows,
     ae_int_t ncols,
     /* Real    */ ae_vector* b,
     double threshold,
     ae_int_t* info,
     densesolverlsreport* rep,
     /* Real    */ ae_vector* x,
     ae_state *_state);
ae_bool _densesolverreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _densesolverreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _densesolverreport_clear(void* _p);
void _densesolverreport_destroy(void* _p);
ae_bool _densesolverlsreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _densesolverlsreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _densesolverlsreport_clear(void* _p);
void _densesolverlsreport_destroy(void* _p);
void linlsqrcreate(ae_int_t m,
     ae_int_t n,
     linlsqrstate* state,
     ae_state *_state);
void linlsqrsetb(linlsqrstate* state,
     /* Real    */ ae_vector* b,
     ae_state *_state);
void linlsqrsetprecunit(linlsqrstate* state, ae_state *_state);
void linlsqrsetprecdiag(linlsqrstate* state, ae_state *_state);
void linlsqrsetlambdai(linlsqrstate* state,
     double lambdai,
     ae_state *_state);
ae_bool linlsqriteration(linlsqrstate* state, ae_state *_state);
void linlsqrsolvesparse(linlsqrstate* state,
     sparsematrix* a,
     /* Real    */ ae_vector* b,
     ae_state *_state);
void linlsqrsetcond(linlsqrstate* state,
     double epsa,
     double epsb,
     ae_int_t maxits,
     ae_state *_state);
void linlsqrresults(linlsqrstate* state,
     /* Real    */ ae_vector* x,
     linlsqrreport* rep,
     ae_state *_state);
void linlsqrsetxrep(linlsqrstate* state,
     ae_bool needxrep,
     ae_state *_state);
void linlsqrrestart(linlsqrstate* state, ae_state *_state);
ae_bool _linlsqrstate_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _linlsqrstate_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _linlsqrstate_clear(void* _p);
void _linlsqrstate_destroy(void* _p);
ae_bool _linlsqrreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _linlsqrreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _linlsqrreport_clear(void* _p);
void _linlsqrreport_destroy(void* _p);
void lincgcreate(ae_int_t n, lincgstate* state, ae_state *_state);
void lincgsetstartingpoint(lincgstate* state,
     /* Real    */ ae_vector* x,
     ae_state *_state);
void lincgsetb(lincgstate* state,
     /* Real    */ ae_vector* b,
     ae_state *_state);
void lincgsetprecunit(lincgstate* state, ae_state *_state);
void lincgsetprecdiag(lincgstate* state, ae_state *_state);
void lincgsetcond(lincgstate* state,
     double epsf,
     ae_int_t maxits,
     ae_state *_state);
ae_bool lincgiteration(lincgstate* state, ae_state *_state);
void lincgsolvesparse(lincgstate* state,
     sparsematrix* a,
     ae_bool isupper,
     /* Real    */ ae_vector* b,
     ae_state *_state);
void lincgresults(lincgstate* state,
     /* Real    */ ae_vector* x,
     lincgreport* rep,
     ae_state *_state);
void lincgsetrestartfreq(lincgstate* state,
     ae_int_t srf,
     ae_state *_state);
void lincgsetrupdatefreq(lincgstate* state,
     ae_int_t freq,
     ae_state *_state);
void lincgsetxrep(lincgstate* state, ae_bool needxrep, ae_state *_state);
void lincgrestart(lincgstate* state, ae_state *_state);
ae_bool _lincgstate_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _lincgstate_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _lincgstate_clear(void* _p);
void _lincgstate_destroy(void* _p);
ae_bool _lincgreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _lincgreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _lincgreport_clear(void* _p);
void _lincgreport_destroy(void* _p);
void nleqcreatelm(ae_int_t n,
     ae_int_t m,
     /* Real    */ ae_vector* x,
     nleqstate* state,
     ae_state *_state);
void nleqsetcond(nleqstate* state,
     double epsf,
     ae_int_t maxits,
     ae_state *_state);
void nleqsetxrep(nleqstate* state, ae_bool needxrep, ae_state *_state);
void nleqsetstpmax(nleqstate* state, double stpmax, ae_state *_state);
ae_bool nleqiteration(nleqstate* state, ae_state *_state);
void nleqresults(nleqstate* state,
     /* Real    */ ae_vector* x,
     nleqreport* rep,
     ae_state *_state);
void nleqresultsbuf(nleqstate* state,
     /* Real    */ ae_vector* x,
     nleqreport* rep,
     ae_state *_state);
void nleqrestartfrom(nleqstate* state,
     /* Real    */ ae_vector* x,
     ae_state *_state);
ae_bool _nleqstate_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _nleqstate_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _nleqstate_clear(void* _p);
void _nleqstate_destroy(void* _p);
ae_bool _nleqreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _nleqreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _nleqreport_clear(void* _p);
void _nleqreport_destroy(void* _p);

}
#endif