/*************************************************************************
ALGLIB 4.07.0 (source code generated 2025-12-29)
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 _linalg_pkg_h
#define _linalg_pkg_h
#include "ap.h"
#include "alglibinternal.h"
#include "alglibmisc.h"

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (DATATYPES)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
#if defined(AE_COMPILE_ABLAS) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_ORTFAC) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_MATGEN) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SPARSE) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
    ae_vector d;
    ae_vector u;
    sparsematrix s;
} sparsebuffers;
#endif
#if defined(AE_COMPILE_HSSCHUR) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_EVD) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
    ae_int_t n;
    ae_int_t k;
    ae_int_t nwork;
    ae_int_t maxits;
    double eps;
    ae_int_t eigenvectorsneeded;
    ae_int_t solvermode;
    ae_bool usewarmstart;
    ae_bool firstcall;
    hqrndstate rs;
    ae_bool running;
    ae_vector tau;
    ae_matrix q0;
    ae_matrix qcur;
    ae_matrix qnew;
    ae_matrix znew;
    ae_matrix r;
    ae_matrix rz;
    ae_matrix tz;
    ae_matrix rq;
    ae_matrix dummy;
    ae_vector rw;
    ae_vector tw;
    ae_vector tmprow;
    ae_vector wcur;
    ae_vector wprev;
    ae_vector wrank;
    apbuffers buf;
    ae_matrix x;
    ae_matrix ax;
    ae_int_t requesttype;
    ae_int_t requestsize;
    ae_int_t repiterationscount;
    rcommstate rstate;
} eigsubspacestate;
typedef struct
{
    ae_int_t iterationscount;
} eigsubspacereport;
#endif
#if defined(AE_COMPILE_DLU) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SPTRF) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
    ae_int_t nfixed;
    ae_int_t ndynamic;
    ae_vector idxfirst;
    ae_vector strgidx;
    ae_vector strgval;
    ae_int_t nallocated;
    ae_int_t nused;
} sluv2list1matrix;
typedef struct
{
    ae_int_t n;
    ae_int_t k;
    ae_vector nzc;
    ae_int_t maxwrkcnt;
    ae_int_t maxwrknz;
    ae_int_t wrkcnt;
    ae_vector wrkset;
    ae_vector colid;
    ae_vector isdensified;
    ae_vector slscolptr;
    ae_vector slsrowptr;
    ae_vector slsidx;
    ae_vector slsval;
    ae_int_t slsused;
    ae_vector tmp0;
} sluv2sparsetrail;
typedef struct
{
    ae_int_t n;
    ae_int_t ndense;
    ae_matrix d;
    ae_vector did;
} sluv2densetrail;
typedef struct
{
    ae_int_t n;
    sparsematrix sparsel;
    sparsematrix sparseut;
    sluv2list1matrix bleft;
    sluv2list1matrix bupper;
    sluv2sparsetrail strail;
    sluv2densetrail dtrail;
    ae_vector rowpermrawidx;
    ae_matrix dbuf;
    ae_vector v0i;
    ae_vector v1i;
    ae_vector v0r;
    ae_vector v1r;
    ae_vector tmp0;
    ae_vector tmpi;
    ae_vector tmpp;
} sluv2buffer;
#endif
#if defined(AE_COMPILE_AMDORDERING) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
    ae_int_t k;
    ae_int_t n;
    ae_vector flagarray;
    ae_vector vbegin;
    ae_vector vallocated;
    ae_vector vcnt;
    ae_vector data;
    ae_int_t dataused;
    ae_int_t iterrow;
    ae_int_t iteridx;
} amdknset;
typedef struct
{
    ae_int_t n;
    ae_bool checkexactdegrees;
    ae_int_t smallestdegree;
    ae_vector approxd;
    ae_vector optionalexactd;
    ae_vector isvertex;
    ae_vector eligible;
    ae_vector vbegin;
    ae_vector vprev;
    ae_vector vnext;
} amdvertexset;
typedef struct
{
    ae_int_t n;
    ae_vector vbegin;
    ae_vector vcolcnt;
    ae_vector entries;
    ae_int_t entriesinitialized;
} amdllmatrix;
typedef struct
{
    ae_int_t n;
    ae_bool extendeddebug;
    ae_bool checkexactdegrees;
    ae_vector iseliminated;
    ae_vector issupernode;
    ae_vector iseligible;
    amdknset setsuper;
    amdknset seta;
    amdknset sete;
    amdllmatrix mtxl;
    amdvertexset vertexdegrees;
    niset setq;
    ae_vector perm;
    ae_vector invperm;
    ae_vector columnswaps;
    niset setp;
    niset lp;
    niset setrp;
    niset ep;
    niset adji;
    niset adjj;
    ae_vector ls;
    ae_int_t lscnt;
    niset setqsupercand;
    niset exactdegreetmp0;
    amdknset hashbuckets;
    niset nonemptybuckets;
    ae_vector sncandidates;
    ae_vector tmp0;
    ae_vector arrwe;
    ae_matrix dbga;
} amdbuffer;
#endif
#if defined(AE_COMPILE_SPCHOL) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
    ae_vector tmpperm;
    ae_vector invtmpperm;
    amdbuffer amdtmp;
    sparsematrix tmpa2;
    sparsematrix tmpbottomt;
    sparsematrix tmpupdate;
    sparsematrix tmpupdatet;
    sparsematrix tmpnewtailt;
} priorityamdbuffers;
typedef struct
{
    ae_vector rowbegin;
    ae_vector rowend;
    ae_vector idx;
    ae_vector urow0;
    ae_vector uwidth;
    ae_vector uflop;
    ae_vector nflop;
    ae_vector sflop;
} spcholadj;
typedef struct
{
    ae_int_t tasktype;
    ae_int_t n;
    ae_bool unitd;
    ae_bool dotrace;
    ae_bool pblrequested;
    ae_bool pblused;
    ae_opaque_object pbl;
    ae_bool pblneedsl;
    sparsematrix pbla;
    ae_bool pblachanged;
    ae_vector fillinperm;
    ae_vector invfillinperm;
    ae_vector effectiveperm;
    ae_vector inveffectiveperm;
    ae_int_t permtype;
    ae_int_t modtype;
    double modparam0;
    double modparam1;
    double modparam2;
    double modparam3;
    ae_vector bsigns;
    ae_bool debugblocksupernodal;
    ae_bool extendeddebug;
    ae_bool dotracescheduler;
    ae_bool dotracesupernodalstructure;
    ae_vector rpivotsigns;
    ae_vector referenceridx;
    ae_int_t nsuper;
    ae_vector parentsupernode;
    ae_vector childsupernodesridx;
    ae_vector childsupernodesidx;
    ae_vector supercolrange;
    ae_vector superrowridx;
    ae_vector superrowidx;
    ae_bool useparallelism;
    ae_vector superperm;
    ae_vector invsuperperm;
    ae_bool istopologicalordering;
    ae_bool applypermutationtooutput;
    spcholadj ladj;
    ae_vector outrowcounts;
    ae_vector inputstorage;
    ae_vector outputstorage;
    ae_vector rowstrides;
    ae_vector rowoffsets;
    ae_vector diagd;
    ae_nxpool n1booleanpool;
    ae_nxpool ns1booleanpool;
    ae_nxpool n1integerpool;
    ae_nxpool ns1integerpool;
    ae_nxpool nrealpool;
    ae_nxpool ns1realpool;
    ae_vector curladjrowbegin;
    ae_vector flagarray;
    ae_vector curpriorities;
    ae_vector tmpparent;
    ae_vector node2supernode;
    ae_smart_ptr _ptramdtmp;
    amdbuffer *ptramdtmp;
    ae_smart_ptr _ptrpamdtmp;
    priorityamdbuffers *ptrpamdtmp;
    ae_vector tmp0;
    ae_vector tmp1;
    ae_vector tmp2;
    ae_vector tmp3;
    ae_vector tmp4;
    ae_vector raw2smap;
    sparsematrix tmpa;
    sparsematrix tmpat;
    ae_vector tmpx;
    ae_vector simdbuf;
} spcholanalysis;
#endif
#if defined(AE_COMPILE_TRFAC) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
    ae_int_t n;
    ae_int_t facttype;
    ae_int_t permtype;
    spcholanalysis analysis;
    sparsematrix wrka;
    sparsematrix wrkat;
    sparsematrix crsa;
    sparsematrix crsat;
} sparsedecompositionanalysis;
#endif
#if defined(AE_COMPILE_BDSVD) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SVD) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_RCOND) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_FBLS) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
    double e1;
    double e2;
    ae_vector x;
    ae_vector ax;
    double xax;
    ae_int_t n;
    ae_vector rk;
    ae_vector rk1;
    ae_vector xk;
    ae_vector xk1;
    ae_vector pk;
    ae_vector pk1;
    ae_vector b;
    rcommstate rstate;
    ae_vector tmp2;
} fblslincgstate;
typedef struct
{
    ae_vector b;
    ae_vector x;
    ae_vector ax;
    ae_vector xs;
    ae_matrix qi;
    ae_matrix aqi;
    ae_matrix h;
    ae_matrix hq;
    ae_matrix hr;
    ae_vector hqb;
    ae_vector ys;
    ae_vector tmp0;
    ae_vector tmp1;
    ae_int_t n;
    ae_int_t itscnt;
    double epsort;
    double epsres;
    double epsred;
    double epsdiag;
    ae_int_t itsperformed;
    ae_int_t retcode;
    double reprelres;
    rcommstate rstate;
} fblsgmresstate;
#endif
#if defined(AE_COMPILE_NORMESTIMATOR) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
    ae_int_t n;
    ae_int_t m;
    ae_int_t nstart;
    ae_int_t nits;
    ae_int_t seedval;
    ae_vector x0;
    ae_vector x1;
    ae_vector t;
    ae_vector xbest;
    hqrndstate r;
    ae_vector x;
    ae_vector mv;
    ae_vector mtv;
    ae_bool needmv;
    ae_bool needmtv;
    double repnorm;
    rcommstate rstate;
} normestimatorstate;
#endif
#if defined(AE_COMPILE_MATINV) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
    ae_int_t terminationtype;
    double r1;
    double rinf;
} matinvreport;
#endif
#if defined(AE_COMPILE_INVERSEUPDATE) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SCHUR) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SPDGEVD) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_MATDET) || !defined(AE_PARTIAL_BUILD)
#endif

}

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

#if defined(AE_COMPILE_ABLAS) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_ORTFAC) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_MATGEN) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_SPARSE) || !defined(AE_PARTIAL_BUILD)
class _sparsebuffers_owner;
class sparsebuffers;


/*************************************************************************
Temporary buffers for sparse matrix operations.

You should pass an instance of this structure to factorization  functions.
It allows to reuse memory during repeated sparse  factorizations.  You  do
not have to call some initialization function - simply passing an instance
to factorization function is enough.
*************************************************************************/
class _sparsebuffers_owner
{
public:
    _sparsebuffers_owner();
    _sparsebuffers_owner(alglib_impl::sparsebuffers *attach_to);
    _sparsebuffers_owner(const _sparsebuffers_owner &rhs);
    _sparsebuffers_owner& operator=(const _sparsebuffers_owner &rhs);
    virtual ~_sparsebuffers_owner();
    alglib_impl::sparsebuffers* c_ptr();
    const alglib_impl::sparsebuffers* c_ptr() const;
protected:
    alglib_impl::sparsebuffers *p_struct;
    bool is_attached;
};
class sparsebuffers : public _sparsebuffers_owner
{
public:
    sparsebuffers();
    sparsebuffers(alglib_impl::sparsebuffers *attach_to);
    sparsebuffers(const sparsebuffers &rhs);
    sparsebuffers& operator=(const sparsebuffers &rhs);
    virtual ~sparsebuffers();


};
#endif

#if defined(AE_COMPILE_HSSCHUR) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_EVD) || !defined(AE_PARTIAL_BUILD)
class _eigsubspacestate_owner;
class eigsubspacestate;
class _eigsubspacereport_owner;
class eigsubspacereport;


/*************************************************************************
This object stores state of the subspace iteration algorithm.

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


};


/*************************************************************************
This object stores state of the subspace iteration algorithm.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class _eigsubspacereport_owner
{
public:
    _eigsubspacereport_owner();
    _eigsubspacereport_owner(alglib_impl::eigsubspacereport *attach_to);
    _eigsubspacereport_owner(const _eigsubspacereport_owner &rhs);
    _eigsubspacereport_owner& operator=(const _eigsubspacereport_owner &rhs);
    virtual ~_eigsubspacereport_owner();
    alglib_impl::eigsubspacereport* c_ptr();
    const alglib_impl::eigsubspacereport* c_ptr() const;
protected:
    alglib_impl::eigsubspacereport *p_struct;
    bool is_attached;
};
class eigsubspacereport : public _eigsubspacereport_owner
{
public:
    eigsubspacereport();
    eigsubspacereport(alglib_impl::eigsubspacereport *attach_to);
    eigsubspacereport(const eigsubspacereport &rhs);
    eigsubspacereport& operator=(const eigsubspacereport &rhs);
    virtual ~eigsubspacereport();
    ae_int_t &iterationscount;


};
#endif

#if defined(AE_COMPILE_DLU) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_SPTRF) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_AMDORDERING) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_SPCHOL) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_TRFAC) || !defined(AE_PARTIAL_BUILD)
class _sparsedecompositionanalysis_owner;
class sparsedecompositionanalysis;


/*************************************************************************
An analysis of the sparse matrix decomposition, performed prior to  actual
numerical factorization. You should not directly  access  fields  of  this
object - use appropriate ALGLIB functions to work with this object.
*************************************************************************/
class _sparsedecompositionanalysis_owner
{
public:
    _sparsedecompositionanalysis_owner();
    _sparsedecompositionanalysis_owner(alglib_impl::sparsedecompositionanalysis *attach_to);
    _sparsedecompositionanalysis_owner(const _sparsedecompositionanalysis_owner &rhs);
    _sparsedecompositionanalysis_owner& operator=(const _sparsedecompositionanalysis_owner &rhs);
    virtual ~_sparsedecompositionanalysis_owner();
    alglib_impl::sparsedecompositionanalysis* c_ptr();
    const alglib_impl::sparsedecompositionanalysis* c_ptr() const;
protected:
    alglib_impl::sparsedecompositionanalysis *p_struct;
    bool is_attached;
};
class sparsedecompositionanalysis : public _sparsedecompositionanalysis_owner
{
public:
    sparsedecompositionanalysis();
    sparsedecompositionanalysis(alglib_impl::sparsedecompositionanalysis *attach_to);
    sparsedecompositionanalysis(const sparsedecompositionanalysis &rhs);
    sparsedecompositionanalysis& operator=(const sparsedecompositionanalysis &rhs);
    virtual ~sparsedecompositionanalysis();


};
#endif

#if defined(AE_COMPILE_BDSVD) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_SVD) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_RCOND) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_FBLS) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_NORMESTIMATOR) || !defined(AE_PARTIAL_BUILD)
class _normestimatorstate_owner;
class normestimatorstate;


/*************************************************************************
This object stores state of the iterative norm estimation algorithm.

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


};
#endif

#if defined(AE_COMPILE_MATINV) || !defined(AE_PARTIAL_BUILD)
class _matinvreport_owner;
class matinvreport;


/*************************************************************************
Matrix inverse report:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for failure due to the matrix being singular or
                         nearly-singular
* r1                reciprocal of condition number in 1-norm
* rinf              reciprocal of condition number in inf-norm
*************************************************************************/
class _matinvreport_owner
{
public:
    _matinvreport_owner();
    _matinvreport_owner(alglib_impl::matinvreport *attach_to);
    _matinvreport_owner(const _matinvreport_owner &rhs);
    _matinvreport_owner& operator=(const _matinvreport_owner &rhs);
    virtual ~_matinvreport_owner();
    alglib_impl::matinvreport* c_ptr();
    const alglib_impl::matinvreport* c_ptr() const;
protected:
    alglib_impl::matinvreport *p_struct;
    bool is_attached;
};
class matinvreport : public _matinvreport_owner
{
public:
    matinvreport();
    matinvreport(alglib_impl::matinvreport *attach_to);
    matinvreport(const matinvreport &rhs);
    matinvreport& operator=(const matinvreport &rhs);
    virtual ~matinvreport();
    ae_int_t &terminationtype;
    double &r1;
    double &rinf;


};
#endif

#if defined(AE_COMPILE_INVERSEUPDATE) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_SCHUR) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_SPDGEVD) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_MATDET) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_ABLAS) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void cmatrixtranspose(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, complex_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void rmatrixtranspose(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This code enforces symmetricy of the matrix by copying Upper part to lower
one (or vice versa).

INPUT PARAMETERS:
    A   -   matrix
    N   -   number of rows/columns
    IsUpper - whether we want to copy upper triangle to lower one (True)
            or vice versa (False).
*************************************************************************/
void rmatrixenforcesymmetricity(real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);


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

Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void cmatrixcopy(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, complex_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);


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

Input parameters:
    N   -   subvector size
    A   -   source vector, N elements are copied
    IA  -   source offset (first element index)
    B   -   destination vector, must be large enough to store result
    IB  -   destination offset (first element index)
*************************************************************************/
void rvectorcopy(const ae_int_t n, const real_1d_array &a, const ae_int_t ia, real_1d_array &b, const ae_int_t ib, const xparams _xparams = alglib::xdefault);


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

Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void rmatrixcopy(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Performs generalized copy: B := Beta*B + Alpha*A.

If Beta=0, then previous contents of B is simply ignored. If Alpha=0, then
A is ignored and not referenced. If both Alpha and Beta  are  zero,  B  is
filled by zeros.

Input parameters:
    M   -   number of rows
    N   -   number of columns
    Alpha-  coefficient
    A   -   source matrix, MxN submatrix is copied
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    Beta-   coefficient
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void rmatrixgencopy(const ae_int_t m, const ae_int_t n, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const double beta, real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);


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

NOTE: this  function  expects  A  to  be  large enough to store result. No
      automatic preallocation happens for  smaller  arrays.  No  integrity
      checks is performed for sizes of 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)
    Alpha-  coefficient
    U   -   vector #1
    IU  -   subvector offset
    V   -   vector #2
    IV  -   subvector offset


  -- ALGLIB routine --

     16.10.2017
     Bochkanov Sergey
*************************************************************************/
void rmatrixger(const ae_int_t m, const ae_int_t n, real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const double alpha, const real_1d_array &u, const ae_int_t iu, const real_1d_array &v, const ae_int_t iv, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void cmatrixrank1(const ae_int_t m, const ae_int_t n, complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const complex_1d_array &u, const ae_int_t iu, const complex_1d_array &v, const ae_int_t iv, const xparams _xparams = alglib::xdefault);


/*************************************************************************
IMPORTANT: this function is deprecated since ALGLIB 3.13. Use RMatrixGER()
           which is more generic version of this 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
*************************************************************************/
void rmatrixrank1(const ae_int_t m, const ae_int_t n, real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const real_1d_array &u, const ae_int_t iu, const real_1d_array &v, const ae_int_t iv, const xparams _xparams = alglib::xdefault);


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

*************************************************************************/
void rmatrixgemv(const ae_int_t m, const ae_int_t n, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t opa, const real_1d_array &x, const ae_int_t ix, const double beta, real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
    Y   -   preallocated matrix, must be large enough to store result

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
*************************************************************************/
void cmatrixmv(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t opa, const complex_1d_array &x, const ae_int_t ix, complex_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);


/*************************************************************************
IMPORTANT: this function is deprecated since ALGLIB 3.13. Use RMatrixGEMV()
           which is more generic version of this 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
    Y   -   preallocated matrix, must be large enough to store result

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
*************************************************************************/
void rmatrixmv(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t opa, const real_1d_array &x, const ae_int_t ix, real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);


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

*************************************************************************/
void rmatrixsymv(const ae_int_t n, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const bool isupper, const real_1d_array &x, const ae_int_t ix, const double beta, real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);


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

*************************************************************************/
double rmatrixsyvmv(const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const bool isupper, const real_1d_array &x, const ae_int_t ix, real_1d_array &tmp, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This subroutine solves linear system op(A)*x=b where:
* A is NxN upper/lower triangular/unitriangular matrix
* X and B are Nx1 vectors
* "op" may be identity transformation or transposition

Solution replaces X.

IMPORTANT: * no overflow/underflow/denegeracy tests is performed.
           * no integrity checks for operand sizes, out-of-bounds accesses
             and so on is performed

INPUT PARAMETERS
    N   -   matrix size, N>=0
    A       -   matrix, actial matrix is stored in A[IA:IA+N-1,JA:JA+N-1]
    IA      -   submatrix offset
    JA      -   submatrix offset
    IsUpper -   whether matrix is upper triangular
    IsUnit  -   whether matrix is unitriangular
    OpType  -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    X       -   right part, actual vector is stored in X[IX:IX+N-1]
    IX      -   offset

OUTPUT PARAMETERS
    X       -   solution replaces elements X[IX:IX+N-1]

  -- ALGLIB routine / remastering of LAPACK's DTRSV --
     (c) 2017 Bochkanov Sergey - converted to ALGLIB
     (c) 2016 Reference BLAS level1 routine (LAPACK version 3.7.0)
     Reference BLAS is a software package provided by Univ. of Tennessee,
     Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.
*************************************************************************/
void rmatrixtrsv(const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const bool isupper, const bool isunit, const ae_int_t optype, real_1d_array &x, const ae_int_t ix, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

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
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     20.01.2018
     Bochkanov Sergey
*************************************************************************/
void cmatrixrighttrsm(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, complex_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

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
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void cmatrixlefttrsm(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, complex_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
Multiplication result replaces X.

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
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void rmatrixrighttrsm(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, real_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
Multiplication result replaces X.

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
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void rmatrixlefttrsm(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, real_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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:
* 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 (row index)
    JA      -   submatrix offset (column index)
    OpTypeA -   multiplication type:
                * 0 - A*A^H is calculated
                * 2 - A^H*A is calculated
    Beta    -   coefficient
    C       -   preallocated input/output matrix
    IC      -   submatrix offset (row index)
    JC      -   submatrix offset (column index)
    IsUpper -   whether upper or lower triangle of C is updated;
                this function updates only one half of C, leaving
                other half unchanged (not referenced at all).

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     16.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void cmatrixherk(const ae_int_t n, const ae_int_t k, const double alpha, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const double beta, complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper, const xparams _xparams = alglib::xdefault);


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

Additional info:
* 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 (row index)
    JA      -   submatrix offset (column index)
    OpTypeA -   multiplication type:
                * 0 - A*A^T is calculated
                * 2 - A^T*A is calculated
    Beta    -   coefficient
    C       -   preallocated input/output matrix
    IC      -   submatrix offset (row index)
    JC      -   submatrix offset (column index)
    IsUpper -   whether C is upper triangular or lower triangular

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     16.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void rmatrixsyrk(const ae_int_t n, const ae_int_t k, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const double beta, real_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

IMPORTANT:

This function does NOT preallocate output matrix C, it MUST be preallocated
by caller prior to calling this function. In case C does not have  enough
space to store result, exception will be generated.

INPUT PARAMETERS
    M       -   matrix size, M>0
    N       -   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 (PREALLOCATED, large enough to store result)
    IC      -   submatrix offset
    JC      -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     2009-2019
     Bochkanov Sergey
*************************************************************************/
void cmatrixgemm(const ae_int_t m, const ae_int_t n, const ae_int_t k, const alglib::complex alpha, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const complex_2d_array &b, const ae_int_t ib, const ae_int_t jb, const ae_int_t optypeb, const alglib::complex beta, complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const xparams _xparams = alglib::xdefault);


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

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.

IMPORTANT:

This function does NOT preallocate output matrix C, it MUST be preallocated
by caller prior to calling this function. In case C does not have  enough
space to store result, exception will be generated.

INPUT PARAMETERS
    M       -   matrix size, M>0
    N       -   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
    B       -   matrix
    IB      -   submatrix offset
    JB      -   submatrix offset
    OpTypeB -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    Beta    -   coefficient
    C       -   PREALLOCATED output matrix, large enough to store result
    IC      -   submatrix offset
    JC      -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     2009-2019
     Bochkanov Sergey
*************************************************************************/
void rmatrixgemm(const ae_int_t m, const ae_int_t n, const ae_int_t k, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const ae_int_t optypeb, const double beta, real_2d_array &c, const ae_int_t ic, const ae_int_t jc, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This subroutine is an older version of CMatrixHERK(), one with wrong  name
(it is HErmitian update, not SYmmetric). It  is  left  here  for  backward
compatibility.

  -- ALGLIB routine --
     16.12.2009
     Bochkanov Sergey
*************************************************************************/
void cmatrixsyrk(const ae_int_t n, const ae_int_t k, const double alpha, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const double beta, complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_ORTFAC) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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).

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixqr(real_2d_array &a, const ae_int_t m, const ae_int_t n, real_1d_array &tau, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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).

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixlq(real_2d_array &a, const ae_int_t m, const ae_int_t n, real_1d_array &tau, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- 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
*************************************************************************/
void cmatrixqr(complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_1d_array &tau, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- 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
*************************************************************************/
void cmatrixlq(complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_1d_array &tau, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixqrunpackq(const real_2d_array &a, const ae_int_t m, const ae_int_t n, const real_1d_array &tau, const ae_int_t qcolumns, real_2d_array &q, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixqrunpackr(const real_2d_array &a, const ae_int_t m, const ae_int_t n, real_2d_array &r, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixlqunpackq(const real_2d_array &a, const ae_int_t m, const ae_int_t n, const real_1d_array &tau, const ae_int_t qrows, real_2d_array &q, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixlqunpackl(const real_2d_array &a, const ae_int_t m, const ae_int_t n, real_2d_array &l, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixqrunpackq(const complex_2d_array &a, const ae_int_t m, const ae_int_t n, const complex_1d_array &tau, const ae_int_t qcolumns, complex_2d_array &q, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void cmatrixqrunpackr(const complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_2d_array &r, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixlqunpackq(const complex_2d_array &a, const ae_int_t m, const ae_int_t n, const complex_1d_array &tau, const ae_int_t qrows, complex_2d_array &q, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void cmatrixlqunpackl(const complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_2d_array &l, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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^T).

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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.
*************************************************************************/
void rmatrixbd(real_2d_array &a, const ae_int_t m, const ae_int_t n, real_1d_array &tauq, real_1d_array &taup, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Unpacking matrix Q which reduces a matrix to bidiagonal form.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
void rmatrixbdunpackq(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &tauq, const ae_int_t qcolumns, real_2d_array &q, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Multiplication by matrix Q which reduces matrix A to  bidiagonal form.

The algorithm allows pre- or post-multiply by Q or Q'.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
void rmatrixbdmultiplybyq(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &tauq, real_2d_array &z, const ae_int_t zrows, const ae_int_t zcolumns, const bool fromtheright, const bool dotranspose, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixbdunpackpt(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &taup, const ae_int_t ptrows, real_2d_array &pt, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixbdmultiplybyp(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &taup, real_2d_array &z, const ae_int_t zrows, const ae_int_t zcolumns, const bool fromtheright, const bool dotranspose, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixbdunpackdiagonals(const real_2d_array &b, const ae_int_t m, const ae_int_t n, bool &isupper, real_1d_array &d, real_1d_array &e, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Reduction of a square matrix to  upper Hessenberg form: Q'*A*Q = H,
where Q is an orthogonal matrix, H - Hessenberg matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
void rmatrixhessenberg(real_2d_array &a, const ae_int_t n, real_1d_array &tau, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Unpacking matrix Q which reduces matrix A to upper Hessenberg form

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
void rmatrixhessenbergunpackq(const real_2d_array &a, const ae_int_t n, const real_1d_array &tau, real_2d_array &q, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixhessenbergunpackh(const real_2d_array &a, const ae_int_t n, real_2d_array &h, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
void smatrixtd(real_2d_array &a, const ae_int_t n, const bool isupper, real_1d_array &tau, real_1d_array &d, real_1d_array &e, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Unpacking matrix Q which reduces symmetric matrix to a tridiagonal
form.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
void smatrixtdunpackq(const real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &tau, real_2d_array &q, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
void hmatrixtd(complex_2d_array &a, const ae_int_t n, const bool isupper, complex_1d_array &tau, real_1d_array &d, real_1d_array &e, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Unpacking matrix Q which reduces a Hermitian matrix to a real  tridiagonal
form.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
void hmatrixtdunpackq(const complex_2d_array &a, const ae_int_t n, const bool isupper, const complex_1d_array &tau, complex_2d_array &q, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_MATGEN) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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]

NOTE: this function uses algorithm  described  in  Stewart, G. W.  (1980),
      "The Efficient Generation of  Random  Orthogonal  Matrices  with  an
      Application to Condition Estimators".

      Speaking short, to generate an (N+1)x(N+1) orthogonal matrix, it:
      * takes an NxN one
      * takes uniformly distributed unit vector of dimension N+1.
      * constructs a Householder reflection from the vector, then applies
        it to the smaller matrix (embedded in the larger size with a 1 at
        the bottom right corner).

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void rmatrixrndorthogonal(const ae_int_t n, real_2d_array &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixrndcond(const ae_int_t n, const double c, real_2d_array &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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]

NOTE: this function uses algorithm  described  in  Stewart, G. W.  (1980),
      "The Efficient Generation of  Random  Orthogonal  Matrices  with  an
      Application to Condition Estimators".

      Speaking short, to generate an (N+1)x(N+1) orthogonal matrix, it:
      * takes an NxN one
      * takes uniformly distributed unit vector of dimension N+1.
      * constructs a Householder reflection from the vector, then applies
        it to the smaller matrix (embedded in the larger size with a 1 at
        the bottom right corner).

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void cmatrixrndorthogonal(const ae_int_t n, complex_2d_array &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void cmatrixrndcond(const ae_int_t n, const double c, complex_2d_array &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void smatrixrndcond(const ae_int_t n, const double c, real_2d_array &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void spdmatrixrndcond(const ae_int_t n, const double c, real_2d_array &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void hmatrixrndcond(const ae_int_t n, const double c, complex_2d_array &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void hpdmatrixrndcond(const ae_int_t n, const double c, complex_2d_array &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixrndorthogonalfromtheright(real_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixrndorthogonalfromtheleft(real_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void cmatrixrndorthogonalfromtheright(complex_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void cmatrixrndorthogonalfromtheleft(complex_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void smatrixrndmultiply(real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void hmatrixrndmultiply(complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_SPARSE) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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 a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the 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 it in C# one, 
  and vice versa.
*************************************************************************/
void sparseserialize(const sparsematrix &obj, std::string &s_out);


/*************************************************************************
This function serializes data structure to C++ stream.

Data stream generated by this function is same as  string  representation
generated  by  string  version  of  serializer - alphanumeric characters,
dots, underscores, minus signs, which are grouped into words separated by
spaces and CR+LF.

We recommend you to read comments on string version of serializer to find
out more about serialization of AlGLIB objects.
*************************************************************************/
void sparseserialize(const sparsematrix &obj, std::ostream &s_out);


/*************************************************************************
This function unserializes data structure from string.
*************************************************************************/
void sparseunserialize(const std::string &s_in, sparsematrix &obj);


/*************************************************************************
This function unserializes data structure from stream.
*************************************************************************/
void sparseunserialize(const std::istream &s_in, sparsematrix &obj);


/*************************************************************************
This function creates sparse matrix in a Hash-Table format.

This function creates Hast-Table matrix, which can be  converted  to  CRS
format after its initialization is over. Typical  usage  scenario  for  a
sparse matrix is:
1. creation in a Hash-Table format
2. insertion of the matrix elements
3. conversion to the CRS representation
4. matrix is passed to some linear algebra algorithm

Some  information  about  different matrix formats can be found below, in
the "NOTES" section.

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    K           -   K>=0, expected number of non-zero elements in a matrix.
                    K can be inexact approximation, can be less than actual
                    number  of  elements  (table will grow when needed) or
                    even zero).
                    It is important to understand that although hash-table
                    may grow automatically, it is better to  provide  good
                    estimate of data size.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    All elements of the matrix are zero.

NOTE 1

Hash-tables use memory inefficiently, and they have to keep  some  amount
of the "spare memory" in order to have good performance. Hash  table  for
matrix with K non-zero elements will  need  C*K*(8+2*sizeof(int))  bytes,
where C is a small constant, about 1.5-2 in magnitude.

CRS storage, from the other side, is  more  memory-efficient,  and  needs
just K*(8+sizeof(int))+M*sizeof(int) bytes, where M is a number  of  rows
in a matrix.

When you convert from the Hash-Table to CRS  representation, all unneeded
memory will be freed.

NOTE 2

Comments of SparseMatrix structure outline  information  about  different
sparse storage formats. We recommend you to read them before starting  to
use ALGLIB sparse matrices.

NOTE 3

This function completely  overwrites S with new sparse matrix. Previously
allocated storage is NOT reused. If you  want  to reuse already allocated
memory, call SparseCreateBuf function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecreate(const ae_int_t m, const ae_int_t n, const ae_int_t k, sparsematrix &s, const xparams _xparams = alglib::xdefault);
void sparsecreate(const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This version of SparseCreate function creates sparse matrix in Hash-Table
format, reusing previously allocated storage as much  as  possible.  Read
comments for SparseCreate() for more information.

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    K           -   K>=0, expected number of non-zero elements in a matrix.
                    K can be inexact approximation, can be less than actual
                    number  of  elements  (table will grow when needed) or
                    even zero).
                    It is important to understand that although hash-table
                    may grow automatically, it is better to  provide  good
                    estimate of data size.
    S           -   SparseMatrix structure which MAY contain some  already
                    allocated storage.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    All elements of the matrix are zero.
                    Previously allocated storage is reused, if  its  size
                    is compatible with expected number of non-zeros K.

  -- ALGLIB PROJECT --
     Copyright 14.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatebuf(const ae_int_t m, const ae_int_t n, const ae_int_t k, sparsematrix &s, const xparams _xparams = alglib::xdefault);
void sparsecreatebuf(const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function creates sparse matrix in a CRS format - the least  flexible
but the most efficient format implemented in ALGLIB.

This function creates CRS matrix. Typical usage scenario for a CRS matrix
is:
1. creation (you have to tell the number of non-zero elements at each row
   at this moment)
2. initialization of the matrix elements (row by row, from left to right)
3. the matrix is passed to some linear algebra algorithm

This function is a memory-efficient alternative to SparseCreate(), but it
is more complex because it requires you to know in advance how large your
matrix is. Some  information about  different matrix formats can be found
in comments on SparseMatrix structure.  We recommend  you  to  read  them
before starting to use ALGLIB sparse matrices.

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    NER         -   number of elements at each row, array[M], NER[I]>=0

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in CRS representation.
                    You have to fill ALL non-zero elements by calling
                    SparseSet() BEFORE you try to use this matrix.

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call SparseCreateCRSBuf function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrs(const ae_int_t m, const ae_int_t n, const integer_1d_array &ner, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function creates sparse matrix in a CRS format (expert function  for
situations when you are running out  of  memory).  This  version  of  CRS
matrix creation function may reuse memory already allocated in S.

This function creates CRS matrix. Typical usage scenario for a CRS matrix
is:
1. creation (you have to tell number of non-zero elements at each row  at
   this moment)
2. insertion of the matrix elements (row by row, from left to right)
3. matrix is passed to some linear algebra algorithm

This function is a memory-efficient alternative to SparseCreate(), but it
is more complex because it requires you to know in advance how large your
matrix is. Some  information about  different matrix formats can be found
in comments on SparseMatrix structure.  We recommend  you  to  read  them
before starting to use ALGLIB sparse matrices..

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    NER         -   number of elements at each row, array[M], NER[I]>=0
    S           -   sparse matrix structure with possibly preallocated
                    memory.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in CRS representation.
                    You have to fill ALL non-zero elements by calling
                    SparseSet() BEFORE you try to use this matrix.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsbuf(const ae_int_t m, const ae_int_t n, const integer_1d_array &ner, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function creates a CRS-based sparse matrix from  the  dense  matrix.

This function is intended for situations when you already  have  a  dense
matrix and need a convenient way of converting it to the CRS format.

INPUT PARAMETERS
    A           -   array[M,N]. If larger, only leading MxN submatrix
                    will be used.
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1

OUTPUT PARAMETERS
    S           -   sparse M*N matrix A in the CRS format

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call SparseCreateCRSFromDenseBuf function.

  -- ALGLIB PROJECT --
     Copyright 16.06.2023 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsfromdense(const real_2d_array &a, const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);
void sparsecreatecrsfromdense(const real_2d_array &a, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function creates a CRS-based sparse matrix from  the  dense  matrix.
A buffered version which reused memory already allocated in S as much  as
possible.

This function is intended for situations when you already  have  a  dense
matrix and need a convenient way of converting it to the CRS format.

INPUT PARAMETERS
    A           -   array[M,N]. If larger, only leading MxN submatrix
                    will be used.
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    S           -   an already allocated structure; if it already has
                    enough memory to store the matrix, no new memory
                    will be allocated.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix A in the CRS format.

  -- ALGLIB PROJECT --
     Copyright 16.06.2023 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsfromdensebuf(const real_2d_array &a, const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);
void sparsecreatecrsfromdensebuf(const real_2d_array &a, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function creates a  CRS-based  sparse  matrix  from  a  dense  vector
which stores a dense 1-dimensional representation of a dense M*N matrix.

This function is intended for situations when you already  have  a  dense
vector and need a convenient way of converting it to the CRS format.

INPUT PARAMETERS
    A           -   array[M*N]. If larger, only leading M*N elements
                    will be used.
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1

OUTPUT PARAMETERS
    S           -   sparse M*N matrix A in the CRS format

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call SparseCreateCRSFromDenseBuf function.

  -- ALGLIB PROJECT --
     Copyright 17.02.2024 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsfromdensev(const real_1d_array &a, const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function creates a  CRS-based  sparse  matrix  from  a  dense vector
which stores a dense 1-dimensional representation of a dense M*N  matrix.
A buffered version which reused memory already allocated in S as much  as
possible.

This function is intended for situations when you already  have  a  dense
vector and need a convenient way of converting it to the CRS format.

INPUT PARAMETERS
    A           -   array[M*N]. If larger, only leading M*N elements
                    will be used.
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    S           -   an already allocated structure; if it already has
                    enough memory to store the matrix, no new memory
                    will be allocated.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix A in the CRS format.

  -- ALGLIB PROJECT --
     Copyright 16.06.2023 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsfromdensevbuf(const real_1d_array &a, const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function creates sparse matrix in  a  SKS  format  (skyline  storage
format). In most cases you do not need this function - CRS format  better
suits most use cases.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    D           -   "bottom" bandwidths, array[M], D[I]>=0.
                    I-th element stores number of non-zeros at I-th  row,
                    below the diagonal (diagonal itself is not  included)
    U           -   "top" bandwidths, array[N], U[I]>=0.
                    I-th element stores number of non-zeros  at I-th row,
                    above the diagonal (diagonal itself  is not included)

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to change their values.

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call SparseCreateSKSBuf function.

  -- ALGLIB PROJECT --
     Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesks(const ae_int_t m, const ae_int_t n, const integer_1d_array &d, const integer_1d_array &u, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This is "buffered"  version  of  SparseCreateSKS()  which  reuses  memory
previously allocated in S (of course, memory is reallocated if needed).

This function creates sparse matrix in  a  SKS  format  (skyline  storage
format). In most cases you do not need this function - CRS format  better
suits most use cases.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    D           -   "bottom" bandwidths, array[M], 0<=D[I]<=I.
                    I-th element stores number of non-zeros at I-th row,
                    below the diagonal (diagonal itself is not included)
    U           -   "top" bandwidths, array[N], 0<=U[I]<=I.
                    I-th element stores number of non-zeros at I-th row,
                    above the diagonal (diagonal itself is not included)

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to change their values.

  -- ALGLIB PROJECT --
     Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesksbuf(const ae_int_t m, const ae_int_t n, const integer_1d_array &d, const integer_1d_array &u, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function creates sparse matrix in  a  SKS  format  (skyline  storage
format). Unlike more general  sparsecreatesks(),  this  function  creates
sparse matrix with constant bandwidth.

You may want to use this function instead of sparsecreatesks() when  your
matrix has  constant  or  nearly-constant  bandwidth,  and  you  want  to
simplify source code.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    BW          -   matrix bandwidth, BW>=0

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to  change  their values.

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call sparsecreatesksbandbuf function.

  -- ALGLIB PROJECT --
     Copyright 25.12.2017 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesksband(const ae_int_t m, const ae_int_t n, const ae_int_t bw, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This is "buffered" version  of  sparsecreatesksband() which reuses memory
previously allocated in S (of course, memory is reallocated if needed).

You may want to use this function instead  of  sparsecreatesksbuf()  when
your matrix has  constant or nearly-constant  bandwidth,  and you want to
simplify source code.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    BW          -   bandwidth, BW>=0

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to change their values.

  -- ALGLIB PROJECT --
     Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesksbandbuf(const ae_int_t m, const ae_int_t n, const ae_int_t bw, sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function copies S0 to S1.
This function completely deallocates memory owned by S1 before creating a
copy of S0. If you want to reuse memory, use SparseCopyBuf.

NOTE:  this  function  does  not verify its arguments, it just copies all
fields of the structure.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecopy(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function efficiently swaps contents of S0 and S1.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparseswap(sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function adds value to S[i,j] - element of the sparse matrix. Matrix
must be in a Hash-Table mode.

In case S[i,j] already exists in the table, V i added to  its  value.  In
case  S[i,j]  is  non-existent,  it  is  inserted  in  the  table.  Table
automatically grows when necessary.

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    Exception will be thrown for CRS matrix.
    I           -   row index of the element to modify, 0<=I<M
    J           -   column index of the element to modify, 0<=J<N
    V           -   value to add, must be finite number

OUTPUT PARAMETERS
    S           -   modified matrix

NOTE 1:  when  S[i,j]  is exactly zero after modification, it is  deleted
from the table.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparseadd(sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function modifies S[i,j] - element of the sparse matrix.

For Hash-based storage format:
* this function can be called at any moment - during matrix initialization
  or later
* new value can be zero or non-zero.  In case new value of S[i,j] is zero,
  this element is deleted from the table.
* this  function  has  no  effect when called with zero V for non-existent
  element.

For CRS-bases storage format:
* this function can be called ONLY DURING MATRIX INITIALIZATION
* zero values are stored in the matrix similarly to non-zero ones
* elements must be initialized in correct order -  from top row to bottom,
  within row - from left to right.

For SKS storage:
* this function can be called at any moment - during matrix initialization
  or later
* zero values are stored in the matrix similarly to non-zero ones
* this function CAN NOT be called for non-existent (outside  of  the  band
  specified during SKS matrix creation) elements. Say, if you created  SKS
  matrix  with  bandwidth=2  and  tried to call sparseset(s,0,10,VAL),  an
  exception will be generated.

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table, SKS or CRS format.
    I           -   row index of the element to modify, 0<=I<M
    J           -   column index of the element to modify, 0<=J<N
    V           -   value to set, must be finite number, can be zero

OUTPUT PARAMETERS
    S           -   modified matrix

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparseset(sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function returns S[i,j] - element of the sparse matrix.  Matrix  can
be in any mode (Hash-Table, CRS, SKS), but this function is less efficient
for CRS matrices. Hash-Table and SKS matrices can find  element  in  O(1)
time, while  CRS  matrices need O(log(RS)) time, where RS is an number of
non-zero elements in a row.

INPUT PARAMETERS
    S           -   sparse M*N matrix
    I           -   row index of the element to modify, 0<=I<M
    J           -   column index of the element to modify, 0<=J<N

RESULT
    value of S[I,J] or zero (in case no element with such index is found)

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
double sparseget(const sparsematrix &s, const ae_int_t i, const ae_int_t j, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function checks whether S[i,j] is present in the sparse  matrix.  It
returns True even for elements  that  are  numerically  zero  (but  still
have place allocated for them).

The matrix  can be in any mode (Hash-Table, CRS, SKS), but this  function
is less efficient for CRS matrices. Hash-Table and SKS matrices can  find
element in O(1) time, while  CRS  matrices need O(log(RS)) time, where RS
is an number of non-zero elements in a row.

INPUT PARAMETERS
    S           -   sparse M*N matrix
    I           -   row index of the element to modify, 0<=I<M
    J           -   column index of the element to modify, 0<=J<N

RESULT
    whether S[I,J] is present in the data structure or not

  -- ALGLIB PROJECT --
     Copyright 14.10.2020 by Bochkanov Sergey
*************************************************************************/
bool sparseexists(const sparsematrix &s, const ae_int_t i, const ae_int_t j, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function returns I-th diagonal element of the sparse matrix.

Matrix can be in any mode (Hash-Table or CRS storage), but this  function
is most efficient for CRS matrices - it requires less than 50 CPU  cycles
to extract diagonal element. For Hash-Table matrices we still  have  O(1)
query time, but function is many times slower.

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    Exception will be thrown for CRS matrix.
    I           -   index of the element to read, 0<=I<min(M,N)

RESULT
    value of S[I,I] or zero (in case no element with such index is found)

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
double sparsegetdiagonal(const sparsematrix &s, const ae_int_t i, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function calculates matrix-vector product  S*x.  Matrix  S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    Y           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y           -   array[M], S*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemv(const sparsematrix &s, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function calculates matrix-vector product  S^T*x. Matrix S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    X           -   array[M], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least M, but we do not check for NAN's or INF's.
    Y           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y           -   array[N], S^T*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemtv(const sparsematrix &s, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function calculates generalized sparse matrix-vector product

    y := alpha*op(S)*x + beta*y

Matrix S must be stored in CRS or SKS format (exception  will  be  thrown
otherwise). op(S) can be either S or S^T.

NOTE: this  function  expects  Y  to  be  large enough to store result. No
      automatic preallocation happens for smaller arrays.

INPUT PARAMETERS
    S           -   sparse matrix in CRS or SKS format.
    Alpha       -   source coefficient
    OpS         -   operation type:
                    * OpS=0     =>  op(S) = S
                    * OpS=1     =>  op(S) = S^T
    X           -   input vector, must have at least Cols(op(S))+IX elements
    IX          -   subvector offset
    Beta        -   destination coefficient
    Y           -   preallocated output array, must have at least Rows(op(S))+IY elements
    IY          -   subvector offset

OUTPUT PARAMETERS
    Y           -   elements [IY...IY+Rows(op(S))-1] are replaced by result,
                    other elements are not modified

HANDLING OF SPECIAL CASES:
* below M=Rows(op(S)) and N=Cols(op(S)). Although current  ALGLIB  version
  does not allow you to  create  zero-sized  sparse  matrices,  internally
  ALGLIB  can  deal  with  such matrices. So, comments for M or N equal to
  zero are for internal use only.
* if M=0, then subroutine does nothing. It does not even touch arrays.
* if N=0 or Alpha=0.0, then:
  * if Beta=0, then Y is filled by zeros. S and X are  not  referenced  at
    all. Initial values of Y are ignored (we do not  multiply  Y by  zero,
    we just rewrite it by zeros)
  * if Beta<>0, then Y is replaced by Beta*Y
* if M>0, N>0, Alpha<>0, but  Beta=0, then  Y is replaced by alpha*op(S)*x
  initial state of Y  is ignored (rewritten without initial multiplication
  by zeros).

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 10.12.2019 by Bochkanov Sergey
*************************************************************************/
void sparsegemv(const sparsematrix &s, const double alpha, const ae_int_t ops, const real_1d_array &x, const ae_int_t ix, const double beta, real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function simultaneously calculates two matrix-vector products:
    S*x and S^T*x.
S must be square (non-rectangular) matrix stored in  CRS  or  SKS  format
(exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse N*N matrix in CRS or SKS format.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    Y0          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.
    Y1          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y0          -   array[N], S*x
    Y1          -   array[N], S^T*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemv2(const sparsematrix &s, const real_1d_array &x, real_1d_array &y0, real_1d_array &y1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function calculates matrix-vector product  S*x, when S is  symmetric
matrix. Matrix S  must be stored in CRS or SKS format  (exception will be
thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*M matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is given:
                    * if upper triangle is given,  only   S[i,j] for j>=i
                      are used, and lower triangle is ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for j<=i
                      are used, and upper triangle is ignored.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    Y           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y           -   array[M], S*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsesmv(const sparsematrix &s, const bool isupper, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function perform in-place multiplication of the matrix columns by  a
user-supplied vector X. The matrix S must be stored in CRS format.

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS format.
    X           -   array[N], coefficients vector.

OUTPUT PARAMETERS
    S           -   in-place multiplied by diag(X) from the right

NOTE: this function throws exception when called for  a  non-CRS  matrix.
You must convert your matrix with SparseConvertToCRS() before using  this
function.

  -- ALGLIB PROJECT --
     Copyright 17.02.2024 by Bochkanov Sergey
*************************************************************************/
void sparsemultiplycolsby(sparsematrix &s, const real_1d_array &x, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function perform in-place multiplication of the matrix rows by  a
user-supplied vector X. The matrix S must be stored in CRS format.

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS format.
    X           -   array[M], coefficients vector.

OUTPUT PARAMETERS
    S           -   in-place multiplied by diag(X) from the left

NOTE: this function throws exception when called for  a  non-CRS  matrix.
You must convert your matrix with SparseConvertToCRS() before using  this
function.

  -- ALGLIB PROJECT --
     Copyright 17.02.2024 by Bochkanov Sergey
*************************************************************************/
void sparsemultiplyrowsby(sparsematrix &s, const real_1d_array &x, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function perform in-place multiplication of the matrix rows and cols
by user-supplied vectors X and Y. The matrix S must be stored in CRS format.

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS format.
    X           -   array[M], row multipliers
    Y           -   array[N], column multipliers

OUTPUT PARAMETERS
    S           -   in-place multiplied by diag(X) from the left, and by
                    diag(Y) from the right

NOTE: this function throws exception when called for  a  non-CRS  matrix.
You must convert your matrix with SparseConvertToCRS() before using  this
function.

  -- ALGLIB PROJECT --
     Copyright 17.02.2024 by Bochkanov Sergey
*************************************************************************/
void sparsemultiplyrowscolsby(sparsematrix &s, const real_1d_array &x, const real_1d_array &y, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function calculates vector-matrix-vector product x'*S*x, where  S is
symmetric matrix. Matrix S must be stored in CRS or SKS format (exception
will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*M matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is given:
                    * if upper triangle is given,  only   S[i,j] for j>=i
                      are used, and lower triangle is ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for j<=i
                      are used, and upper triangle is ignored.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.

RESULT
    x'*S*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 27.01.2014 by Bochkanov Sergey
*************************************************************************/
double sparsevsmv(const sparsematrix &s, const bool isupper, const real_1d_array &x, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function calculates matrix-matrix product  S*A.  Matrix  S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    A           -   array[N][K], input dense matrix. For  performance reasons
                    we make only quick checks - we check that array size
                    is at least N, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B           -   array[M][K], S*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemm(const sparsematrix &s, const real_2d_array &a, const ae_int_t k, real_2d_array &b, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function calculates matrix-matrix product  S^T*A. Matrix S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    A           -   array[M][K], input dense matrix. For performance reasons
                    we make only quick checks - we check that array size  is
                    at least M, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B           -   array[N][K], S^T*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemtm(const sparsematrix &s, const real_2d_array &a, const ae_int_t k, real_2d_array &b, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function simultaneously calculates two matrix-matrix products:
    S*A and S^T*A.
S  must  be  square (non-rectangular) matrix stored in CRS or  SKS  format
(exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse N*N matrix in CRS or SKS format.
    A           -   array[N][K], input dense matrix. For performance reasons
                    we make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B0          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.
    B1          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B0          -   array[N][K], S*A
    B1          -   array[N][K], S^T*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemm2(const sparsematrix &s, const real_2d_array &a, const ae_int_t k, real_2d_array &b0, real_2d_array &b1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function calculates matrix-matrix product  S*A, when S  is  symmetric
matrix. Matrix S must be stored in CRS or SKS format  (exception  will  be
thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*M matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is given:
                    * if upper triangle is given,  only   S[i,j] for j>=i
                      are used, and lower triangle is ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for j<=i
                      are used, and upper triangle is ignored.
    A           -   array[N][K], input dense matrix. For performance reasons
                    we make only quick checks - we check that array size is
                    at least N, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B           -   array[M][K], S*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsesmm(const sparsematrix &s, const bool isupper, const real_2d_array &a, const ae_int_t k, real_2d_array &b, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function calculates matrix-vector product op(S)*x, when x is  vector,
S is symmetric triangular matrix, op(S) is transposition or no  operation.
Matrix S must be stored in CRS or SKS format  (exception  will  be  thrown
otherwise).

INPUT PARAMETERS
    S           -   sparse square matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is used:
                    * if upper triangle is given,  only   S[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    IsUnit      -   unit or non-unit diagonal:
                    * if True, diagonal elements of triangular matrix  are
                      considered equal to 1.0. Actual elements  stored  in
                      S are not referenced at all.
                    * if False, diagonal stored in S is used
    OpType      -   operation type:
                    * if 0, S*x is calculated
                    * if 1, (S^T)*x is calculated (transposition)
    X           -   array[N] which stores input  vector.  For  performance
                    reasons we make only quick  checks  -  we  check  that
                    array  size  is  at  least  N, but we do not check for
                    NAN's or INF's.
    Y           -   possibly  preallocated  input   buffer.  Automatically
                    resized if its size is too small.

OUTPUT PARAMETERS
    Y           -   array[N], op(S)*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 20.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsetrmv(const sparsematrix &s, const bool isupper, const bool isunit, const ae_int_t optype, real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function solves linear system op(S)*y=x  where  x  is  vector,  S  is
symmetric  triangular  matrix,  op(S)  is  transposition  or no operation.
Matrix S must be stored in CRS or SKS format  (exception  will  be  thrown
otherwise).

INPUT PARAMETERS
    S           -   sparse square matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is used:
                    * if upper triangle is given,  only   S[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    IsUnit      -   unit or non-unit diagonal:
                    * if True, diagonal elements of triangular matrix  are
                      considered equal to 1.0. Actual elements  stored  in
                      S are not referenced at all.
                    * if False, diagonal stored in S is used. It  is  your
                      responsibility  to  make  sure  that   diagonal   is
                      non-zero.
    OpType      -   operation type:
                    * if 0, S*x is calculated
                    * if 1, (S^T)*x is calculated (transposition)
    X           -   array[N] which stores input  vector.  For  performance
                    reasons we make only quick  checks  -  we  check  that
                    array  size  is  at  least  N, but we do not check for
                    NAN's or INF's.

OUTPUT PARAMETERS
    X           -   array[N], inv(op(S))*x

NOTE: this function throws exception when called for  non-CRS/SKS  matrix.
      You must convert your matrix  with  SparseConvertToCRS/SKS()  before
      using this function.

NOTE: no assertion or tests are done during algorithm  operation.   It  is
      your responsibility to provide invertible matrix to algorithm.

  -- ALGLIB PROJECT --
     Copyright 20.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsetrsv(const sparsematrix &s, const bool isupper, const bool isunit, const ae_int_t optype, real_1d_array &x, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function applies permutation given by permutation table P (as opposed
to product form of permutation) to sparse symmetric  matrix  A,  given  by
either upper or lower triangle: B := P*A*P'.

This function allocates completely new instance of B. Use buffered version
SparseSymmPermTblBuf() if you want to reuse already allocated structure.

INPUT PARAMETERS
    A           -   sparse square matrix in CRS format.
    IsUpper     -   whether upper or lower triangle of A is used:
                    * if upper triangle is given,  only   A[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   A[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    P           -   array[N] which stores permutation table;  P[I]=J means
                    that I-th row/column of matrix  A  is  moved  to  J-th
                    position. For performance reasons we do NOT check that
                    P[] is  a   correct   permutation  (that there  is  no
                    repetitions, just that all its elements  are  in [0,N)
                    range.

OUTPUT PARAMETERS
    B           -   permuted matrix.  Permutation  is  applied  to A  from
                    the both sides, only upper or lower triangle (depending
                    on IsUpper) is stored.

NOTE: this function throws exception when called for non-CRS  matrix.  You
      must convert your matrix with SparseConvertToCRS() before using this
      function.

  -- ALGLIB PROJECT --
     Copyright 05.10.2020 by Bochkanov Sergey.
*************************************************************************/
void sparsesymmpermtbl(const sparsematrix &a, const bool isupper, const integer_1d_array &p, sparsematrix &b, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function applies permutation given by permutation table P (as opposed
to product form of permutation) to sparse symmetric  matrix  A,  given  by
either upper or lower triangle: B := P*A*P'.

It outputs TRANSPOSED matrix, i.e. if A is given  by  the  lower  triangle
then B is given by the upper one, and vice versa.

This function allocates completely new instance of B. Use buffered version
SparseSymmPermTblTransposeBuf() if you want to reuse an already  allocated
structure.

INPUT PARAMETERS
    A           -   sparse square matrix in CRS format.
    IsUpper     -   whether upper or lower triangle of A is used:
                    * if upper triangle is given,  only   A[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   A[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    P           -   array[N] which stores permutation table;  P[I]=J means
                    that I-th row/column of matrix  A  is  moved  to  J-th
                    position. For performance reasons we do NOT check that
                    P[] is  a   correct   permutation  (that there  is  no
                    repetitions, just that all its elements  are  in [0,N)
                    range.

OUTPUT PARAMETERS
    B           -   permuted matrix.  Permutation  is  applied  to A  from
                    the both sides, only triangle OPPOSITE to that of A is
                    returned: a lower one if IsUpper=True,  and  an  upper
                    one otherwise.

NOTE: this function throws exception when called for non-CRS  matrix.  You
      must convert your matrix with SparseConvertToCRS() before using this
      function.

  -- ALGLIB PROJECT --
     Copyright 24.080.2024 by Bochkanov Sergey.
*************************************************************************/
void sparsesymmpermtbltranspose(const sparsematrix &a, const bool isupper, const integer_1d_array &p, sparsematrix &b, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function is a buffered version  of  SparseSymmPermTbl()  that  reuses
previously allocated storage in B as much as possible.

This function applies permutation given by permutation table P (as opposed
to product form of permutation) to sparse symmetric  matrix  A,  given  by
either upper or lower triangle: B := P*A*P'.

INPUT PARAMETERS
    A           -   sparse square matrix in CRS format.
    IsUpper     -   whether upper or lower triangle of A is used:
                    * if upper triangle is given,  only   A[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   A[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    P           -   array[N] which stores permutation table;  P[I]=J means
                    that I-th row/column of matrix  A  is  moved  to  J-th
                    position. For performance reasons we do NOT check that
                    P[] is  a   correct   permutation  (that there  is  no
                    repetitions, just that all its elements  are  in [0,N)
                    range.
    B           -   sparse matrix object that will hold the result.
                    Previously allocated memory will be reused as much  as
                    possible.

OUTPUT PARAMETERS
    B           -   permuted matrix.  Permutation  is  applied  to A  from
                    the both sides, only upper or lower triangle (depending
                    on IsUpper) is stored.

NOTE: this function throws exception when called for non-CRS  matrix.  You
      must convert your matrix with SparseConvertToCRS() before using this
      function.

  -- ALGLIB PROJECT --
     Copyright 05.10.2020 by Bochkanov Sergey.
*************************************************************************/
void sparsesymmpermtblbuf(const sparsematrix &a, const bool isupper, const integer_1d_array &p, sparsematrix &b, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function applies permutation given by permutation table P (as opposed
to product form of permutation) to sparse symmetric  matrix  A,  given  by
either upper or lower triangle: B := P*A*P'.

It outputs TRANSPOSED matrix, i.e. if A is given  by  the  lower  triangle
then B is given by the upper one, and vice versa.

This function reuses memory already allocated in B as much as possible.

INPUT PARAMETERS
    A           -   sparse square matrix in CRS format.
    IsUpper     -   whether upper or lower triangle of A is used:
                    * if upper triangle is given,  only   A[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   A[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    P           -   array[N] which stores permutation table;  P[I]=J means
                    that I-th row/column of matrix  A  is  moved  to  J-th
                    position. For performance reasons we do NOT check that
                    P[] is  a   correct   permutation  (that there  is  no
                    repetitions, just that all its elements  are  in [0,N)
                    range.
    B           -   sparse matrix object that will hold the result.
                    Previously allocated memory will be reused as much  as
                    possible.

OUTPUT PARAMETERS
    B           -   permuted matrix.  Permutation  is  applied  to A  from
                    the both sides, only triangle OPPOSITE to that of A is
                    returned: a lower one if IsUpper=True,  and  an  upper
                    one otherwise.

NOTE: this function throws exception when called for non-CRS  matrix.  You
      must convert your matrix with SparseConvertToCRS() before using this
      function.

  -- ALGLIB PROJECT --
     Copyright 24.080.2024 by Bochkanov Sergey.
*************************************************************************/
void sparsesymmpermtbltransposebuf(const sparsematrix &a, const bool isupper, const integer_1d_array &p, sparsematrix &b, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This procedure resizes Hash-Table matrix. It can be called when you  have
deleted too many elements from the matrix, and you want to  free unneeded
memory.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparseresizematrix(sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function  is  used  to enumerate all elements of the sparse matrix.
Before  first  call  user  initializes  T0 and T1 counters by zero. These
counters are used to remember current position in a  matrix;  after  each
call they are updated by the function.

Subsequent calls to this function return non-zero elements of the  sparse
matrix, one by one. If you enumerate CRS matrix, matrix is traversed from
left to right, from top to bottom. In case you enumerate matrix stored as
Hash table, elements are returned in random order.

EXAMPLE
    > T0=0
    > T1=0
    > while SparseEnumerate(S,T0,T1,I,J,V) do
    >     ....do something with I,J,V

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table or CRS representation.
    T0          -   internal counter
    T1          -   internal counter

OUTPUT PARAMETERS
    T0          -   new value of the internal counter
    T1          -   new value of the internal counter
    I           -   row index of non-zero element, 0<=I<M.
    J           -   column index of non-zero element, 0<=J<N
    V           -   value of the T-th element

RESULT
    True in case of success (next non-zero element was retrieved)
    False in case all non-zero elements were enumerated

NOTE: you may call SparseRewriteExisting() during enumeration, but it  is
      THE  ONLY  matrix  modification  function  you  can  call!!!  Other
      matrix modification functions should not be called during enumeration!

  -- ALGLIB PROJECT --
     Copyright 14.03.2012 by Bochkanov Sergey
*************************************************************************/
bool sparseenumerate(const sparsematrix &s, ae_int_t &t0, ae_int_t &t1, ae_int_t &i, ae_int_t &j, double &v, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function rewrites existing (non-zero) element. It  returns  True   if
element  exists  or  False,  when  it  is  called for non-existing  (zero)
element.

This function works with any kind of the matrix.

The purpose of this function is to provide convenient thread-safe  way  to
modify  sparse  matrix.  Such  modification  (already  existing element is
rewritten) is guaranteed to be thread-safe without any synchronization, as
long as different threads modify different elements.

INPUT PARAMETERS
    S           -   sparse M*N matrix in any kind of representation
                    (Hash, SKS, CRS).
    I           -   row index of non-zero element to modify, 0<=I<M
    J           -   column index of non-zero element to modify, 0<=J<N
    V           -   value to rewrite, must be finite number

OUTPUT PARAMETERS
    S           -   modified matrix
RESULT
    True in case when element exists
    False in case when element doesn't exist or it is zero

  -- ALGLIB PROJECT --
     Copyright 14.03.2012 by Bochkanov Sergey
*************************************************************************/
bool sparserewriteexisting(sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function returns I-th row of the sparse matrix. Matrix must be stored
in CRS or SKS format.

INPUT PARAMETERS:
    S           -   sparse M*N matrix in CRS format
    I           -   row index, 0<=I<M
    IRow        -   output buffer, can be  preallocated.  In  case  buffer
                    size  is  too  small  to  store  I-th   row,   it   is
                    automatically reallocated.

OUTPUT PARAMETERS:
    IRow        -   array[M], I-th row.

NOTE: this function has O(N) running time, where N is a  column  count. It
      allocates and fills N-element  array,  even  although  most  of  its
      elemets are zero.

NOTE: If you have O(non-zeros-per-row) time and memory  requirements,  use
      SparseGetCompressedRow() function. It  returns  data  in  compressed
      format.

NOTE: when  incorrect  I  (outside  of  [0,M-1]) or  matrix (non  CRS/SKS)
      is passed, this function throws exception.

  -- ALGLIB PROJECT --
     Copyright 10.12.2014 by Bochkanov Sergey
*************************************************************************/
void sparsegetrow(const sparsematrix &s, const ae_int_t i, real_1d_array &irow, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function returns I-th row of the sparse matrix IN COMPRESSED FORMAT -
only non-zero elements are returned (with their indexes). Matrix  must  be
stored in CRS or SKS format.

INPUT PARAMETERS:
    S           -   sparse M*N matrix in CRS format
    I           -   row index, 0<=I<M
    ColIdx      -   output buffer for column indexes, can be preallocated.
                    In case buffer size is too small to store I-th row, it
                    is automatically reallocated.
    Vals        -   output buffer for values, can be preallocated. In case
                    buffer size is too small to  store  I-th  row,  it  is
                    automatically reallocated.

OUTPUT PARAMETERS:
    ColIdx      -   column   indexes   of  non-zero  elements,  sorted  by
                    ascending. Symbolically non-zero elements are  counted
                    (i.e. if you allocated place for element, but  it  has
                    zero numerical value - it is counted).
    Vals        -   values. Vals[K] stores value of  matrix  element  with
                    indexes (I,ColIdx[K]). Symbolically non-zero  elements
                    are counted (i.e. if you allocated place for  element,
                    but it has zero numerical value - it is counted).
    NZCnt       -   number of symbolically non-zero elements per row.

NOTE: when  incorrect  I  (outside  of  [0,M-1]) or  matrix (non  CRS/SKS)
      is passed, this function throws exception.

NOTE: this function may allocate additional, unnecessary place for  ColIdx
      and Vals arrays. It is dictated by  performance  reasons  -  on  SKS
      matrices it is faster  to  allocate  space  at  the  beginning  with
      some "extra"-space, than performing two passes over matrix  -  first
      time to calculate exact space required for data, second  time  -  to
      store data itself.

  -- ALGLIB PROJECT --
     Copyright 10.12.2014 by Bochkanov Sergey
*************************************************************************/
void sparsegetcompressedrow(const sparsematrix &s, const ae_int_t i, integer_1d_array &colidx, real_1d_array &vals, ae_int_t &nzcnt, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function appends a compressed sparse row to a CRS matrix,  increasing
its row count by 1.

INPUT PARAMETERS:
    S           -   sparse M*N matrix in CRS format, including one created
                    with sparsecreatecrsempty().
    ColIdx      -   array[NZ], column indexes, values  in  [0,N-1]  range.
                    ColIdx[] can store non-distinct  values;  elements  of
                    Vals[] corresponding to duplicate column indexes  will
                    be summed up.
    Vals        -   array[NZ], element values.
    NZ          -   nonzeros count, NZ>=0. Both ColIdx[]  and  Vals[]  can
                    be longer than NZ, in   which  case  only  leading  NZ
                    elements are used.

OUTPUT PARAMETERS:
    S           -   (M+1)*N matrix in the CRS format.

NOTE: this function has amortized O(NZ*logNZ) cost.

  -- ALGLIB PROJECT --
     Copyright 2024.02.19 by Bochkanov Sergey
*************************************************************************/
void sparseappendcompressedrow(sparsematrix &s, const integer_1d_array &colidx, const real_1d_array &vals, const ae_int_t nz, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function appends an empty row to a CRS matrix,  increasing  its  rows
count by 1. The newly added row can be modified with sparseappendelement().
The matrix is a valid CRS matrix at any moment of the process.

INPUT PARAMETERS:
    S           -   sparse M*N matrix in CRS format, including one created
                    with sparsecreatecrsempty().

OUTPUT PARAMETERS:
    S           -   (M+1)*N matrix in the CRS format.

  -- ALGLIB PROJECT --
     Copyright 2024.02.19 by Bochkanov Sergey
*************************************************************************/
void sparseappendemptyrow(sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function appends an element to the last row  of  a  CRS  matrix.  New
elements can be added ONLY from left to right (column indexes are strictly
increasing).

INPUT PARAMETERS:
    S           -   a fully initialized sparse M*N matrix in CRS format, M>0
    K           -   column index, 0<=K<N, must be  strictly  greater  than
                    the last element in the last row.
    V           -   element value. Can be zero (zero values will also be added
                    to the matrix).

OUTPUT PARAMETERS:
    S           -   M*N matrix in the CRS format.

  -- ALGLIB PROJECT --
     Copyright 2024.02.19 by Bochkanov Sergey
*************************************************************************/
void sparseappendelement(sparsematrix &s, const ae_int_t k, const double v, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function performs efficient in-place  transpose  of  SKS  matrix.  No
additional memory is allocated during transposition.

This function supports only skyline storage format (SKS).

INPUT PARAMETERS
    S       -   sparse matrix in SKS format.

OUTPUT PARAMETERS
    S           -   sparse matrix, transposed.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsetransposesks(sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function performs transpose of CRS matrix.

INPUT PARAMETERS
    S       -   sparse matrix in CRS format.

OUTPUT PARAMETERS
    S           -   sparse matrix, transposed.

NOTE: internal  temporary  copy  is  allocated   for   the   purposes   of
      transposition. It is deallocated after transposition.

  -- ALGLIB PROJECT --
     Copyright 30.01.2018 by Bochkanov Sergey
*************************************************************************/
void sparsetransposecrs(sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function performs copying with transposition of CRS matrix.

INPUT PARAMETERS
    S0      -   sparse matrix in CRS format.

OUTPUT PARAMETERS
    S1      -   sparse matrix, transposed

  -- ALGLIB PROJECT --
     Copyright 23.07.2018 by Bochkanov Sergey
*************************************************************************/
void sparsecopytransposecrs(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function performs copying with transposition of CRS matrix  (buffered
version which reuses memory already allocated by  the  target as  much  as
possible).

INPUT PARAMETERS
    S0      -   sparse matrix in CRS format.

OUTPUT PARAMETERS
    S1      -   sparse matrix, transposed; previously allocated memory  is
                reused if possible.

  -- ALGLIB PROJECT --
     Copyright 23.07.2018 by Bochkanov Sergey
*************************************************************************/
void sparsecopytransposecrsbuf(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function performs copying with transposition of CRS matrix  (buffered
version which reuses memory already allocated by  the  target as  much  as
possible).

Unlike SparseCopyTransposeCRSBuf() it can copy/transpose only lower or
upper triangle or both, depending on Triangle parameter

INPUT PARAMETERS
    S0      -   sparse matrix in CRS format.
    Triangle-   triangle to copy:
                * -1    the lower triangle and diagonal are transposed,
                        S1 is upper triangular
                *  0    both triangles are copied
                * +1    the upper triangle and diagonal are transposed,
                        S1 is lower triangular

OUTPUT PARAMETERS
    S1      -   sparse matrix, transposed; previously allocated memory  is
                reused if possible.

  -- ALGLIB PROJECT --
     Copyright 23.07.2018 by Bochkanov Sergey
*************************************************************************/
void sparsecopytransposecrsxbuf(const sparsematrix &s0, const ae_int_t triangle, sparsematrix &s1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function  performs  in-place  conversion  to  desired sparse storage
format.

INPUT PARAMETERS
    S0      -   sparse matrix in any format.
    Fmt     -   desired storage format  of  the  output,  as  returned  by
                SparseGetMatrixType() function:
                * 0 for hash-based storage
                * 1 for CRS
                * 2 for SKS

OUTPUT PARAMETERS
    S0          -   sparse matrix in requested format.

NOTE: in-place conversion wastes a lot of memory which is  used  to  store
      temporaries.  If  you  perform  a  lot  of  repeated conversions, we
      recommend to use out-of-place buffered  conversion  functions,  like
      SparseCopyToBuf(), which can reuse already allocated memory.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparseconvertto(sparsematrix &s0, const ae_int_t fmt, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function  performs out-of-place conversion to desired sparse storage
format. S0 is copied to S1 and converted on-the-fly. Memory  allocated  in
S1 is reused to maximum extent possible.

INPUT PARAMETERS
    S0      -   sparse matrix in any format.
    Fmt     -   desired storage format  of  the  output,  as  returned  by
                SparseGetMatrixType() function:
                * 0 for hash-based storage
                * 1 for CRS
                * 2 for SKS

OUTPUT PARAMETERS
    S1          -   sparse matrix in requested format.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecopytobuf(const sparsematrix &s0, const ae_int_t fmt, sparsematrix &s1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function performs in-place conversion to Hash table storage.

INPUT PARAMETERS
    S           -   sparse matrix in CRS format.

OUTPUT PARAMETERS
    S           -   sparse matrix in Hash table format.

NOTE: this  function  has   no  effect  when  called with matrix which  is
      already in Hash table mode.

NOTE: in-place conversion involves allocation of temporary arrays. If  you
      perform a lot of repeated in- place  conversions,  it  may  lead  to
      memory fragmentation. Consider using out-of-place SparseCopyToHashBuf()
      function in this case.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparseconverttohash(sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function  performs  out-of-place  conversion  to  Hash table storage
format. S0 is copied to S1 and converted on-the-fly.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in Hash table format.

NOTE: if S0 is stored as Hash-table, it is just copied without conversion.

NOTE: this function de-allocates memory  occupied  by  S1 before  starting
      conversion. If you perform a  lot  of  repeated  conversions, it may
      lead to memory fragmentation. In this case we recommend you  to  use
      SparseCopyToHashBuf() function which re-uses memory in S1 as much as
      possible.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytohash(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function  performs  out-of-place  conversion  to  Hash table storage
format. S0 is copied to S1 and converted on-the-fly. Memory  allocated  in
S1 is reused to maximum extent possible.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in Hash table format.

NOTE: if S0 is stored as Hash-table, it is just copied without conversion.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytohashbuf(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function  performs  out-of-place  conversion  to  CRS format.  S0 is
copied to S1 and converted on-the-fly.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in CRS format.

NOTE: if S0 is stored as CRS, it is just copied without conversion.

NOTE: this function de-allocates memory occupied by S1 before starting CRS
      conversion. If you perform a lot of repeated CRS conversions, it may
      lead to memory fragmentation. In this case we recommend you  to  use
      SparseCopyToCRSBuf() function which re-uses memory in S1 as much  as
      possible.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytocrs(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function performs in-place conversion to SKS format.

INPUT PARAMETERS
    S           -   sparse matrix in any format.

OUTPUT PARAMETERS
    S           -   sparse matrix in SKS format.

NOTE: this  function  has   no  effect  when  called with matrix which  is
      already in SKS mode.

NOTE: in-place conversion involves allocation of temporary arrays. If  you
      perform a lot of repeated in- place  conversions,  it  may  lead  to
      memory fragmentation. Consider using out-of-place SparseCopyToSKSBuf()
      function in this case.

  -- ALGLIB PROJECT --
     Copyright 15.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparseconverttosks(sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function  performs  out-of-place  conversion  to SKS storage format.
S0 is copied to S1 and converted on-the-fly.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in SKS format.

NOTE: if S0 is stored as SKS, it is just copied without conversion.

NOTE: this function de-allocates memory  occupied  by  S1 before  starting
      conversion. If you perform a  lot  of  repeated  conversions, it may
      lead to memory fragmentation. In this case we recommend you  to  use
      SparseCopyToSKSBuf() function which re-uses memory in S1 as much  as
      possible.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytosks(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function  performs  out-of-place  conversion  to SKS format.  S0  is
copied to S1 and converted on-the-fly. Memory  allocated  in S1 is  reused
to maximum extent possible.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in SKS format.

NOTE: if S0 is stored as SKS, it is just copied without conversion.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytosksbuf(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function returns type of the matrix storage format.

INPUT PARAMETERS:
    S           -   sparse matrix.

RESULT:
    sparse storage format used by matrix:
        0   -   Hash-table
        1   -   CRS (compressed row storage)
        2   -   SKS (skyline)

NOTE: future  versions  of  ALGLIB  may  include additional sparse storage
      formats.


  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetmatrixtype(const sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using Hash table representation.

INPUT PARAMETERS:
    S   -   sparse matrix.

RESULT:
    True if matrix type is Hash table
    False if matrix type is not Hash table

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
bool sparseishash(const sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using SKS representation.

INPUT PARAMETERS:
    S   -   sparse matrix.

RESULT:
    True if matrix type is SKS
    False if matrix type is not SKS

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
bool sparseissks(const sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
The function frees all memory occupied by  sparse  matrix.  Sparse  matrix
structure becomes unusable after this call.

OUTPUT PARAMETERS
    S   -   sparse matrix to delete

  -- ALGLIB PROJECT --
     Copyright 24.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsefree(sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
The function returns number of strictly upper triangular non-zero elements
in  the  matrix.  It  counts  SYMBOLICALLY non-zero elements, i.e. entries
in the sparse matrix data structure. If some element  has  zero  numerical
value, it is still counted.

This function has different cost for different types of matrices:
* for hash-based matrices it involves complete pass over entire hash-table
  with O(NNZ) cost, where NNZ is number of non-zero elements
* for CRS and SKS matrix types cost of counting is O(N) (N - matrix size).

RESULT: number of non-zero elements strictly above main diagonal

  -- ALGLIB PROJECT --
     Copyright 12.02.2014 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetuppercount(const sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
The function returns number of strictly lower triangular non-zero elements
in  the  matrix.  It  counts  SYMBOLICALLY non-zero elements, i.e. entries
in the sparse matrix data structure. If some element  has  zero  numerical
value, it is still counted.

This function has different cost for different types of matrices:
* for hash-based matrices it involves complete pass over entire hash-table
  with O(NNZ) cost, where NNZ is number of non-zero elements
* for CRS and SKS matrix types cost of counting is O(N) (N - matrix size).

RESULT: number of non-zero elements strictly below main diagonal

  -- ALGLIB PROJECT --
     Copyright 12.02.2014 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetlowercount(const sparsematrix &s, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function performs an in-place matrix conditioning scaling  such  that

    A = R*Z*C

where A is an original matrix, R and C are diagonal scaling  matrices, and
Z is a scaled matrix. Z replaces A, R and C are returned as 1D arrays.

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS format.
    SclType     -   scaling type:
                    * 0     for automatically chosen scaling
                    * 1     for equilibration scaling
    ScaleRows   -   if False, rows are not scaled (R=identity)
    ScaleCols   -   if False, cols are not scaled (C=identity)
    ColsFirst   -   scale columns first. If False, rows are  scaled  prior
                    to scaling columns. Ignored for ScaleCols=False.

OUTPUT PARAMETERS
    R           -   array[M], row scales, R[i]>0
    C           -   array[N], col scales, C[i]>0

NOTE: this function throws exception when called  for  a  non-CRS  matrix.
      You must convert your matrix with SparseConvertToCRS()  before using
      this function.

NOTE: this  function  works  with  general  (nonsymmetric)  matrices.  See
      sparsesymmscale() for a symmetric version. See sparsescalebuf()  for
      a version which reuses space already present in output arrays R/C.

NOTE: if both ScaleRows=False and ScaleCols=False, this  function  returns
      an identity scaling.

NOTE: R[] and C[] are guaranteed to be strictly positive. When the  matrix
      has zero rows/cols, corresponding elements of R/C are set to 1.

  -- ALGLIB PROJECT --
     Copyright 12.11.2023 by Bochkanov Sergey
*************************************************************************/
void sparsescale(sparsematrix &s, const ae_int_t scltype, const bool scalerows, const bool scalecols, const bool colsfirst, real_1d_array &r, real_1d_array &c, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function tries to gather values from Src to Dst, zeroing out elements
of Dst not present in Src.

Both matrices must be in the CRS format and must  have  exactly  the  same
size. The sparsity pattern of Src must be a subset of that of Dst.

If Src contains non-zero elements not present in Dst, gather operation  is
stopped in the middle (leaving Dst values partially changed but  otherwise
fully functional) and False is returned.

INPUT PARAMETERS
    Dst         -   sparse M*N destination matrix in CRS format.
    Src         -   sparse M*N source matrix in CRS format.

OUTPUT PARAMETERS
    Dst         -   if  True  is  returned,  contains  elements of Src and
                    zeros in other positions. If False  is  returned,  its
                    values are only partially initialized.

RESULT:
    True if successful. False if the sparsity pattern  of  Src  is  not  a
    subset of that of Dst.

NOTE: this function throws exception when called  for  a  non-CRS  matrix.
      You must convert your matrix with SparseConvertToCRS()  before using
      this function. It also throws an exception if matrices have different
      sizes.


  -- ALGLIB PROJECT --
     Copyright 12.04.2025 by Bochkanov Sergey
*************************************************************************/
bool sparsetrygatherclear(sparsematrix &dst, const sparsematrix &src, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function merges sparsity patterns of S1 and S2 and stores result into
Dst, reusing previously allocated memory as much as possible.

Both matrices must be in the CRS format and must  have  exactly  the  same
size.

INPUT PARAMETERS
    S1, S2      -   sparse M*N source matrices in CRS format.
    Dst         -   previously allocated sparse matrix structure in any
                    storage format and of any size.

OUTPUT PARAMETERS
    Dst         -   sparse M*N matrix in CRS format, zero-initialized, has
                    sparsity pattern equal to union of S1 and S2

NOTE: this function throws exception when called  for  a  non-CRS  matrix.
      You must convert your matrix with SparseConvertToCRS()  before using
      this function. It also throws an exception if matrices have different
      sizes.


  -- ALGLIB PROJECT --
     Copyright 12.04.2025 by Bochkanov Sergey
*************************************************************************/
void sparsemergepatterns(const sparsematrix &s1, const sparsematrix &s2, sparsematrix &dst, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_HSSCHUR) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_EVD) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
This function initializes subspace iteration solver. This solver  is  used
to solve symmetric real eigenproblems where just a few (top K) eigenvalues
and corresponding eigenvectors is required.

This solver can be significantly faster than  complete  EVD  decomposition
in the following case:
* when only just a small fraction  of  top  eigenpairs  of dense matrix is
  required. When K approaches N, this solver is slower than complete dense
  EVD
* when problem matrix is sparse (and/or is not known explicitly, i.e. only
  matrix-matrix product can be performed)

USAGE (explicit dense/sparse matrix):
1. User initializes algorithm state with eigsubspacecreate() call
2. [optional] User tunes solver parameters by calling eigsubspacesetcond()
   or other functions
3. User  calls  eigsubspacesolvedense() or eigsubspacesolvesparse() methods,
   which take algorithm state and 2D array or alglib.sparsematrix object.

USAGE (out-of-core mode):
1. User initializes algorithm state with eigsubspacecreate() call
2. [optional] User tunes solver parameters by calling eigsubspacesetcond()
   or other functions
3. User activates out-of-core mode of  the  solver  and  repeatedly  calls
   communication functions in a loop like below:
   > alglib.eigsubspaceoocstart(state)
   > while alglib.eigsubspaceooccontinue(state) do
   >     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
   >     alglib.eigsubspaceoocgetrequestdata(state, out X)
   >     [calculate  Y=A*X, with X=R^NxM]
   >     alglib.eigsubspaceoocsendresult(state, in Y)
   > alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    N       -   problem dimensionality, N>0
    K       -   number of top eigenvector to calculate, 0<K<=N.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE: if you solve many similar EVD problems you may  find  it  useful  to
      reuse previous subspace as warm-start point for new EVD problem.  It
      can be done with eigsubspacesetwarmstart() function.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacecreate(const ae_int_t n, const ae_int_t k, eigsubspacestate &state, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Buffered version of constructor which aims to reuse  previously  allocated
memory as much as possible.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacecreatebuf(const ae_int_t n, const ae_int_t k, eigsubspacestate &state, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function sets stopping critera for the solver:
* error in eigenvector/value allowed by solver
* maximum number of iterations to perform

INPUT PARAMETERS:
    State       -   solver structure
    Eps         -   eps>=0,  with non-zero value used to tell solver  that
                    it can  stop  after  all  eigenvalues  converged  with
                    error  roughly  proportional  to  eps*MAX(LAMBDA_MAX),
                    where LAMBDA_MAX is a maximum eigenvalue.
                    Zero  value  means  that  no  check  for  precision is
                    performed.
    MaxIts      -   maxits>=0,  with non-zero value used  to  tell  solver
                    that it can stop after maxits  steps  (no  matter  how
                    precise current estimate is)

NOTE: passing  eps=0  and  maxits=0  results  in  automatic  selection  of
      moderate eps as stopping criteria (1.0E-6 in current implementation,
      but it may change without notice).

NOTE: very small values of eps are possible (say, 1.0E-12),  although  the
      larger problem you solve (N and/or K), the  harder  it  is  to  find
      precise eigenvectors because rounding errors tend to accumulate.

NOTE: passing non-zero eps results in  some performance  penalty,  roughly
      equal to 2N*(2K)^2 FLOPs per iteration. These additional computations
      are required in order to estimate current error in  eigenvalues  via
      Rayleigh-Ritz process.
      Most of this additional time is  spent  in  construction  of  ~2Kx2K
      symmetric  subproblem  whose  eigenvalues  are  checked  with  exact
      eigensolver.
      This additional time is negligible if you search for eigenvalues  of
      the large dense matrix, but may become noticeable on  highly  sparse
      EVD problems, where cost of matrix-matrix product is low.
      If you set eps to exactly zero,  Rayleigh-Ritz  phase  is completely
      turned off.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesetcond(eigsubspacestate &state, const double eps, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function sets warm-start mode of the solver: next call to the  solver
will reuse previous subspace as warm-start  point.  It  can  significantly
speed-up convergence when you solve many similar eigenproblems.

INPUT PARAMETERS:
    State       -   solver structure
    UseWarmStart-   either True or False

  -- ALGLIB --
     Copyright 12.11.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesetwarmstart(eigsubspacestate &state, const bool usewarmstart, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function  initiates  out-of-core  mode  of  subspace eigensolver. It
should be used in conjunction with other out-of-core-related functions  of
this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State       -   solver object
    MType       -   matrix type and solver mode:

                    * 0 =   real symmetric matrix A, products  of the form
                            A*X are computed. At every step  the  basis of
                            the  invariant  subspace  is  reorthogonalized
                            with LQ decomposition  which  makes  the  algo
                            more robust.

                            The first mode introduced in ALGLIB, the  most
                            precise and robust. However, it is  suboptimal
                            for easy problems which can be solved  in  3-5
                            iterations without LQ step.

                    * 1 =   real symmetric matrix A, products  of the form
                            A*X are computed. The  invariant  subspace  is
                            NOT reorthogonalized,  no  error  checks.  The
                            solver  stops  after   specified   number   of
                            iterations which should be small, 5 at most.

                            This mode is intended for easy  problems  with
                            extremely fast convergence.

                    Future versions of ALGLIB may  introduce  support  for
                    other  matrix   types;   for   now,   only   symmetric
                    eigenproblems are supported.


  -- ALGLIB --
     Copyright 07.06.2023 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocstart(eigsubspacestate &state, const ae_int_t mtype, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function performs subspace iteration  in  the  out-of-core  mode.  It
should be used in conjunction with other out-of-core-related functions  of
this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
bool eigsubspaceooccontinue(eigsubspacestate &state, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function is used to retrieve information  about  out-of-core  request
sent by solver to user code: request type (current version  of  the solver
sends only requests for matrix-matrix products) and request size (size  of
the matrices being multiplied).

This function returns just request metrics; in order  to  get contents  of
the matrices being multiplied, use eigsubspaceoocgetrequestdata().

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode

OUTPUT PARAMETERS:
    RequestType     -   type of the request to process:
                        * 0 - for matrix-matrix product A*X, with A  being
                          NxN matrix whose eigenvalues/vectors are needed,
                          and X being NxREQUESTSIZE one which is  returned
                          by the eigsubspaceoocgetrequestdata().
    RequestSize     -   size of the X matrix (number of columns),  usually
                        it is several times larger than number of  vectors
                        K requested by user.


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocgetrequestinfo(eigsubspacestate &state, ae_int_t &requesttype, ae_int_t &requestsize, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function is used to retrieve information  about  out-of-core  request
sent by solver to user code: matrix X (array[N,RequestSize) which have  to
be multiplied by out-of-core matrix A in a product A*X.

This function returns just request data; in order to get size of  the data
prior to processing requestm, use eigsubspaceoocgetrequestinfo().

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode
    X               -   possibly  preallocated   storage;  reallocated  if
                        needed, left unchanged, if large enough  to  store
                        request data.

OUTPUT PARAMETERS:
    X               -   array[N,RequestSize] or larger, leading  rectangle
                        is filled with dense matrix X.


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocgetrequestdata(eigsubspacestate &state, real_2d_array &x, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function is used to send user reply to out-of-core  request  sent  by
solver. Usually it is product A*X for returned by solver matrix X.

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode
    AX              -   array[N,RequestSize] or larger, leading  rectangle
                        is filled with product A*X.


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocsendresult(eigsubspacestate &state, const real_2d_array &ax, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function  finalizes out-of-core  mode  of  subspace eigensolver.  It
should be used in conjunction with other out-of-core-related functions  of
this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State       -   solver state

OUTPUT PARAMETERS:
    W           -   array[K], depending on solver settings:
                    * top  K  eigenvalues ordered  by  descending   -   if
                      eigenvectors are returned in Z
                    * zeros - if invariant subspace is returned in Z
    Z           -   array[N,K], depending on solver settings either:
                    * matrix of eigenvectors found
                    * orthogonal basis of K-dimensional invariant subspace
    Rep         -   report with additional parameters

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocstop(eigsubspacestate &state, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function runs subspace eigensolver for dense NxN symmetric matrix A,
given by its upper or lower triangle.

This function can not process nonsymmetric matrices.

INPUT PARAMETERS:
    State       -   solver state
    A           -   array[N,N], symmetric NxN matrix given by one  of  its
                    triangles
    IsUpper     -   whether upper or lower triangle of  A  is  given  (the
                    other one is not referenced at all).

OUTPUT PARAMETERS:
    W           -   array[K], top  K  eigenvalues ordered  by   descending
                    of their absolute values
    Z           -   array[N,K], matrix of eigenvectors found
    Rep         -   report with additional parameters

NOTE: internally this function allocates a copy of NxN dense A. You should
      take it into account when working with very large matrices occupying
      almost all RAM.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesolvedenses(eigsubspacestate &state, const real_2d_array &a, const bool isupper, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This  function runs eigensolver for dense NxN symmetric matrix A, given by
upper or lower triangle.

This function can not process nonsymmetric matrices.

INPUT PARAMETERS:
    State       -   solver state
    A           -   NxN symmetric matrix given by one of its triangles
    IsUpper     -   whether upper or lower triangle of  A  is  given  (the
                    other one is not referenced at all).

OUTPUT PARAMETERS:
    W           -   array[K], top  K  eigenvalues ordered  by   descending
                    of their absolute values
    Z           -   array[N,K], matrix of eigenvectors found
    Rep         -   report with additional parameters

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesolvesparses(eigsubspacestate &state, const sparsematrix &a, const bool isupper, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
bool smatrixevd(const real_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, real_1d_array &d, real_2d_array &z, const xparams _xparams = alglib::xdefault);


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

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
bool smatrixevdr(const real_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const double b1, const double b2, ae_int_t &m, real_1d_array &w, real_2d_array &z, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
bool smatrixevdi(const real_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const ae_int_t i1, const ae_int_t i2, real_1d_array &w, real_2d_array &z, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
bool hmatrixevd(const complex_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, real_1d_array &d, complex_2d_array &z, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
bool hmatrixevdr(const complex_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const double b1, const double b2, ae_int_t &m, real_1d_array &w, complex_2d_array &z, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
bool hmatrixevdi(const complex_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const ae_int_t i1, const ae_int_t i2, real_1d_array &w, complex_2d_array &z, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
bool smatrixtdevd(real_1d_array &d, const real_1d_array &e, const ae_int_t n, const ae_int_t zneeded, real_2d_array &z, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
bool smatrixtdevdr(real_1d_array &d, const real_1d_array &e, const ae_int_t n, const ae_int_t zneeded, const double a, const double b, ae_int_t &m, real_2d_array &z, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
bool smatrixtdevdi(real_1d_array &d, const real_1d_array &e, const ae_int_t n, const ae_int_t zneeded, const ae_int_t i1, const ae_int_t i2, real_2d_array &z, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Finding eigenvalues and eigenvectors of a general (unsymmetric) matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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.
*************************************************************************/
bool rmatrixevd(const real_2d_array &a, const ae_int_t n, const ae_int_t vneeded, real_1d_array &wr, real_1d_array &wi, real_2d_array &vl, real_2d_array &vr, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_DLU) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_SPTRF) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_AMDORDERING) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_SPCHOL) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_TRFAC) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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]

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)].

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.01.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixlu(real_2d_array &a, const ae_int_t m, const ae_int_t n, integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);
void rmatrixlu(real_2d_array &a, integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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]

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)].

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.01.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixlu(complex_2d_array &a, const ae_int_t m, const ae_int_t n, integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);
void cmatrixlu(complex_2d_array &a, integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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' denotes 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.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
bool hpdmatrixcholesky(complex_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
bool hpdmatrixcholesky(complex_2d_array &a, const bool isupper, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     15.12.2009
     Bochkanov Sergey
*************************************************************************/
bool spdmatrixcholesky(real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
bool spdmatrixcholesky(real_2d_array &a, const bool isupper, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Update of Cholesky decomposition: rank-1 update to original A.  "Buffered"
version which uses preallocated buffer which is saved  between  subsequent
function calls.

This function uses internally allocated buffer which is not saved  between
subsequent  calls.  So,  if  you  perform  a lot  of  subsequent  updates,
we  recommend   you   to   use   "buffered"   version   of  this function:
SPDMatrixCholeskyUpdateAdd1Buf().

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    U       -   array[N], rank-1 update to A: A_mod = A + u*u'
                Exception is thrown if array size is too small.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

NOTE: this function always succeeds, so it does not return completion code

NOTE: this function checks sizes of input arrays, but it does  NOT  checks
      for presence of infinities or NAN's.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdateadd1(real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &u, const xparams _xparams = alglib::xdefault);
void spdmatrixcholeskyupdateadd1(real_2d_array &a, const bool isupper, const real_1d_array &u, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Update of Cholesky decomposition: "fixing" some variables.

This function uses internally allocated buffer which is not saved  between
subsequent  calls.  So,  if  you  perform  a lot  of  subsequent  updates,
we  recommend   you   to   use   "buffered"   version   of  this function:
SPDMatrixCholeskyUpdateFixBuf().

"FIXING" EXPLAINED:

    Suppose we have N*N positive definite matrix A. "Fixing" some variable
    means filling corresponding row/column of  A  by  zeros,  and  setting
    diagonal element to 1.

    For example, if we fix 2nd variable in 4*4 matrix A, it becomes Af:

        ( A00  A01  A02  A03 )      ( Af00  0   Af02 Af03 )
        ( A10  A11  A12  A13 )      (  0    1    0    0   )
        ( A20  A21  A22  A23 )  =>  ( Af20  0   Af22 Af23 )
        ( A30  A31  A32  A33 )      ( Af30  0   Af32 Af33 )

    If we have Cholesky decomposition of A, it must be recalculated  after
    variables were  fixed.  However,  it  is  possible  to  use  efficient
    algorithm, which needs O(K*N^2)  time  to  "fix"  K  variables,  given
    Cholesky decomposition of original, "unfixed" A.

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    Fix     -   array[N], I-th element is True if I-th  variable  must  be
                fixed. Exception is thrown if array size is too small.
    BufR    -   possibly preallocated  buffer;  automatically  resized  if
                needed. It is recommended to  reuse  this  buffer  if  you
                perform a lot of subsequent decompositions.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

NOTE: this function always succeeds, so it does not return completion code

NOTE: this function checks sizes of input arrays, but it does  NOT  checks
      for presence of infinities or NAN's.

NOTE: this  function  is  efficient  only  for  moderate amount of updated
      variables - say, 0.1*N or 0.3*N. For larger amount of  variables  it
      will  still  work,  but  you  may  get   better   performance   with
      straightforward Cholesky.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdatefix(real_2d_array &a, const ae_int_t n, const bool isupper, const boolean_1d_array &fix, const xparams _xparams = alglib::xdefault);
void spdmatrixcholeskyupdatefix(real_2d_array &a, const bool isupper, const boolean_1d_array &fix, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Update of Cholesky decomposition: rank-1 update to original A.  "Buffered"
version which uses preallocated buffer which is saved  between  subsequent
function calls.

See comments for SPDMatrixCholeskyUpdateAdd1() for more information.

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    U       -   array[N], rank-1 update to A: A_mod = A + u*u'
                Exception is thrown if array size is too small.
    BufR    -   possibly preallocated  buffer;  automatically  resized  if
                needed. It is recommended to  reuse  this  buffer  if  you
                perform a lot of subsequent decompositions.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdateadd1buf(real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &u, real_1d_array &bufr, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Update of Cholesky  decomposition:  "fixing"  some  variables.  "Buffered"
version which uses preallocated buffer which is saved  between  subsequent
function calls.

See comments for SPDMatrixCholeskyUpdateFix() for more information.

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    Fix     -   array[N], I-th element is True if I-th  variable  must  be
                fixed. Exception is thrown if array size is too small.
    BufR    -   possibly preallocated  buffer;  automatically  resized  if
                needed. It is recommended to  reuse  this  buffer  if  you
                perform a lot of subsequent decompositions.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdatefixbuf(real_2d_array &a, const ae_int_t n, const bool isupper, const boolean_1d_array &fix, real_1d_array &bufr, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Sparse LU decomposition with column pivoting for sparsity and row pivoting
for stability. Input must be square sparse matrix stored in CRS format.

The algorithm  computes  LU  decomposition  of  a  general  square  matrix
(rectangular ones are not supported). The result  of  an  algorithm  is  a
representation of A as A = P*L*U*Q, where:
* L is lower unitriangular matrix
* U is upper triangular matrix
* P = P0*P1*...*PK, K=N-1, Pi - permutation matrix for I and P[I]
* Q = QK*...*Q1*Q0, K=N-1, Qi - permutation matrix for I and Q[I]

This function pivots columns for higher sparsity, and then pivots rows for
stability (larger element at the diagonal).

INPUT PARAMETERS:
    A       -   sparse NxN matrix in CRS format. An exception is generated
                if matrix is non-CRS or non-square.
    PivotType-  pivoting strategy:
                * 0 for best pivoting available (2 in current version)
                * 1 for row-only pivoting (NOT RECOMMENDED)
                * 2 for complete pivoting which produces most sparse outputs

OUTPUT PARAMETERS:
    A       -   the result of factorization, matrices L and U stored in
                compact form using CRS sparse storage format:
                * lower unitriangular L is stored strictly under main diagonal
                * upper triangilar U is stored ON and ABOVE main diagonal
    P       -   row permutation matrix in compact form, array[N]
    Q       -   col permutation matrix in compact form, array[N]

This function always succeeds, i.e. it ALWAYS returns valid factorization,
but for your convenience it also returns  boolean  value  which  helps  to
detect symbolically degenerate matrices:
* function returns TRUE, if the matrix was factorized AND symbolically
  non-degenerate
* function returns FALSE, if the matrix was factorized but U has strictly
  zero elements at the diagonal (the factorization is returned anyway).


  -- ALGLIB routine --
     03.09.2018
     Bochkanov Sergey
*************************************************************************/
bool sparselu(sparsematrix &a, const ae_int_t pivottype, integer_1d_array &p, integer_1d_array &q, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Sparse Cholesky decomposition for skyline matrixm using in-place algorithm
without allocating additional storage.

The algorithm computes Cholesky decomposition  of  a  symmetric  positive-
definite sparse matrix. The result of an algorithm is a representation  of
A as A=U^T*U or A=L*L^T

This function allows to perform very efficient decomposition of low-profile
matrices (average bandwidth is ~5-10 elements). For larger matrices it  is
recommended to use supernodal Cholesky decomposition: SparseCholeskyP() or
SparseCholeskyAnalyze()/SparseCholeskyFactorize().

INPUT PARAMETERS:
    A       -   sparse matrix in skyline storage (SKS) format.
    N       -   size of matrix A (can be smaller than actual size of A)
    IsUpper -   if IsUpper=True, then factorization is performed on  upper
                triangle. Another triangle is ignored (it may contant some
                data, but it is not changed).


OUTPUT PARAMETERS:
    A       -   the result of factorization, stored in SKS. If IsUpper=True,
                then the upper  triangle  contains  matrix  U,  such  that
                A = U^T*U. Lower triangle is not changed.
                Similarly, if IsUpper = False. In this case L is returned,
                and we have A = L*(L^T).
                Note that THIS function does not  perform  permutation  of
                rows to reduce bandwidth.

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.

NOTE: for  performance  reasons  this  function  does NOT check that input
      matrix  includes  only  finite  values. It is your responsibility to
      make sure that there are no infinite or NAN values in the matrix.

  -- ALGLIB routine --
     16.01.2014
     Bochkanov Sergey
*************************************************************************/
bool sparsecholeskyskyline(sparsematrix &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Sparse Cholesky decomposition for a matrix  stored  in  any sparse storage,
without rows/cols permutation.

This function is the most convenient (less parameters to specify) although
the less efficient, version of sparse Cholesky.

IMPORTANT: if all you need is to solve a linear system A*x=b with positive
           definite A, we strongly recommend you to use functions provided
           by the DIRECTSPARSESOLVERS  subpackage  of  the SOLVERS package
           instead of this function.

           Functions provided by the TRFAC  subpackage   are  designed  to
           return factorizations, i.e. such L that A=LL' or A=LDL'.  Thus,
           TRFAC unit can not utilize some performance  improvements  that
           allow to solve Ax=b without returning factors L and D.

           In particular, performance backends, dynamic regularization and
           iterative refinement are not supported  by  TRFAC  but  can  be
           included in DIRECTSPARSESOLVERS.

IMPORTANT: the commercial edition of ALGLIB can parallelize this function.
           Specific speed-up due  to  parallelism  heavily  depends  on  a
           sparsity pattern, with the following matrix classes  being  the
           easiest ones to parallelize:
           * large matrices with many nearly-independent sets of rows/cols
           * matrices with large dense blocks on the diagonal
           See the ALGLIB Reference Manual for more information on how  to
           activate parallelism support.

Internally it:
* calls SparseCholeskyAnalyze()  function  to  perform  symbolic  analysis
  phase with no permutation being configured.
* calls SparseCholeskyFactorize() function to perform numerical  phase  of
  the factorization

Following alternatives may result in better performance:
* using SparseCholeskyP(), which selects best  pivoting  available,  which
  almost always results in improved sparsity and cache locality
* using  SparseCholeskyAnalyze() and SparseCholeskyFactorize()   functions
  directly,  which  may  improve  performance of repetitive factorizations
  with same sparsity patterns.

The latter also allows one to perform  LDLT  factorization  of  indefinite
matrix (one with strictly diagonal D, which is known  to  be  stable  only
in few special cases, like quasi-definite matrices).

INPUT PARAMETERS:
    A       -   a square NxN sparse matrix, stored in any storage format.
    IsUpper -   if IsUpper=True, then factorization is performed on  upper
                triangle.  Another triangle is ignored on  input,  dropped
                on output. Similarly, if IsUpper=False, the lower triangle
                is processed.

OUTPUT PARAMETERS:
    A       -   the result of factorization, stored in CRS format:
                * if IsUpper=True, then the upper triangle contains matrix
                  U such  that  A = U^T*U and the lower triangle is empty.
                * similarly, if IsUpper=False, then lower triangular L  is
                  returned and we have A = L*(L^T).
                Note that THIS function does not  perform  permutation  of
                the rows to reduce fill-in.

RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False.  Contents  of  A  is  undefined
    in such case.

NOTE: for  performance  reasons  this  function  does NOT check that input
      matrix  includes  only  finite  values. It is your responsibility to
      make sure that there are no infinite or NAN values in the matrix.

  -- ALGLIB routine --
     16.09.2020
     Bochkanov Sergey
*************************************************************************/
bool sparsecholesky(sparsematrix &a, const bool isupper, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Sparse Cholesky decomposition for a matrix  stored  in  any sparse storage
format, with performance-enhancing permutation of rows/cols.

Present version is configured  to  perform  supernodal  permutation   with
a sparsity reducing ordering.

IMPORTANT: if all you need is to solve a linear system A*x=b with positive
           definite A, we strongly recommend you to use functions provided
           by the DIRECTSPARSESOLVERS  subpackage  of  the SOLVERS package
           instead of this function.

           Functions provided by the TRFAC  subpackage   are  designed  to
           return factorizations, i.e. such L that A=LL' or A=LDL'.  Thus,
           TRFAC unit can not utilize some performance  improvements  that
           allow to solve Ax=b without returning factors L and D.

           In particular, performance backends, dynamic regularization and
           iterative refinement are not supported  by  TRFAC  but  can  be
           included in DIRECTSPARSESOLVERS.

IMPORTANT: the commercial edition of ALGLIB can parallelize this function.
           Specific speed-up due  to  parallelism  heavily  depends  on  a
           sparsity pattern, with the following matrix classes  being  the
           easiest ones to parallelize:
           * large matrices with many nearly-independent sets of rows/cols
           * matrices with large dense blocks on the diagonal
           See the ALGLIB Reference Manual for more information on how  to
           activate parallelism support.

This function is a wrapper around generic sparse  decomposition  functions
that internally:
* calls SparseCholeskyAnalyze()  function  to  perform  symbolic  analysis
  phase with best available permutation being configured.
* calls SparseCholeskyFactorize() function to perform numerical  phase  of
  the factorization.

NOTE: using  SparseCholeskyAnalyze() and SparseCholeskyFactorize() directly
      may improve  performance  of  repetitive  factorizations  with  same
      sparsity patterns. It also allows one to perform  LDLT factorization
      of  indefinite  matrix  -  a factorization with strictly diagonal D,
      which  is  known to be stable only in few special cases, like quasi-
      definite matrices.

INPUT PARAMETERS:
    A       -   a square NxN sparse matrix, stored in any storage format.
    IsUpper -   if IsUpper=True, then factorization is performed on  upper
                triangle.  Another triangle is ignored on  input,  dropped
                on output. Similarly, if IsUpper=False, the lower triangle
                is processed.

OUTPUT PARAMETERS:
    A       -   the result of factorization, stored in CRS format:
                * if IsUpper=True, then the upper triangle contains matrix
                  U such  that  A = U^T*U and the lower triangle is empty.
                * similarly, if IsUpper=False, then lower triangular L  is
                  returned and we have A = L*(L^T).
    P       -   a row/column permutation, a product of P0*P1*...*Pk, k=N-1,
                with Pi being permutation of rows/cols I and P[I]

RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False.  Contents  of  A  is  undefined
    in such case.

NOTE: for  performance  reasons  this  function  does NOT check that input
      matrix  includes  only  finite  values. It is your responsibility to
      make sure that there are no infinite or NAN values in the matrix.

  -- ALGLIB routine --
     16.09.2020
     Bochkanov Sergey
*************************************************************************/
bool sparsecholeskyp(sparsematrix &a, const bool isupper, integer_1d_array &p, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Sparse Cholesky/LDLT decomposition: symbolic analysis phase.

This function is a part of the 'expert' sparse Cholesky API:
* SparseCholeskyAnalyze(), that performs symbolic analysis phase and loads
  matrix to be factorized into internal storage
* SparseCholeskySetModType(), that allows to  use  modified  Cholesky/LDLT
  with lower bounds on pivot magnitudes and additional overflow safeguards
* SparseCholeskyFactorize(),  that performs  numeric  factorization  using
  precomputed symbolic analysis and internally stored matrix - and outputs
  result
* SparseCholeskyReload(), that reloads one more matrix with same  sparsity
  pattern into internal storage so  one  may  reuse  previously  allocated
  temporaries and previously performed symbolic analysis

This specific function performs preliminary analysis of the  Cholesky/LDLT
factorization. It allows to choose  different  permutation  types  and  to
choose between classic Cholesky and  indefinite  LDLT  factorization  (the
latter is computed with strictly diagonal D, i.e.  without  Bunch-Kauffman
pivoting).

IMPORTANT: if all you need is to solve a linear system A*x=b with positive
           definite A, we strongly recommend you to use functions provided
           by the DIRECTSPARSESOLVERS  subpackage  of  the SOLVERS package
           instead of this function.

           Functions provided by the TRFAC  subpackage   are  designed  to
           return factorizations, i.e. such L that A=LL' or A=LDL'.  Thus,
           TRFAC unit can not utilize some performance  improvements  that
           allow to solve Ax=b without returning factors L and D.

           In particular, performance backends, dynamic regularization and
           iterative refinement are not supported  by  TRFAC  but  can  be
           included in DIRECTSPARSESOLVERS.

NOTE: L*D*LT family of factorization may be used to  factorize  indefinite
      matrices. However, numerical stability is guaranteed ONLY for a class
      of quasi-definite matrices.

NOTE: all internal processing is performed with lower triangular  matrices
      stored  in  CRS  format.  Any  other  storage  formats  and/or upper
      triangular storage means  that  one  format  conversion  and/or  one
      transposition will be performed  internally  for  the  analysis  and
      factorization phases. Thus, highest  performance  is  achieved  when
      input is a lower triangular CRS matrix.

INPUT PARAMETERS:
    A           -   sparse square matrix in any sparse storage format.
    IsUpper     -   whether upper or lower  triangle  is  decomposed  (the
                    other one is ignored).
    FactType    -   factorization type:
                    * 0 for traditional Cholesky of SPD matrix
                    * 1 for LDLT decomposition with strictly  diagonal  D,
                        which may have non-positive entries.
    PermType    -   permutation type:
                    *-1 for absence of permutation
                    * 0 for best fill-in reducing  permutation  available,
                        which is 3 in the current version
                    * 1 for supernodal ordering (improves locality and
                      performance, does NOT change fill-in factor)
                    * 2 for original AMD ordering
                    * 3 for  improved  AMD  (approximate  minimum  degree)
                        ordering with better  handling  of  matrices  with
                        dense rows/columns

OUTPUT PARAMETERS:
    Analysis    -   contains:
                    * symbolic analysis of the matrix structure which will
                      be used later to guide numerical factorization.
                    * specific numeric values loaded into internal  memory
                      waiting for the factorization to be performed

This function fails if and only if the matrix A is symbolically degenerate
i.e. has diagonal element which is exactly zero. In  such  case  False  is
returned, contents of Analysis object is undefined.

  -- ALGLIB routine --
     20.09.2020
     Bochkanov Sergey
*************************************************************************/
bool sparsecholeskyanalyze(const sparsematrix &a, const bool isupper, const ae_int_t facttype, const ae_int_t permtype, sparsedecompositionanalysis &analysis, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Sparse Cholesky decomposition: numerical analysis phase.

IMPORTANT: the commercial edition of ALGLIB can parallelize this function.
           Specific speed-up due  to  parallelism  heavily  depends  on  a
           sparsity pattern, with the following matrix classes  being  the
           easiest ones to parallelize:
           * large matrices with many nearly-independent sets of rows/cols
           * matrices with large dense blocks on the diagonal
           See the ALGLIB Reference Manual for more information on how  to
           activate parallelism support.

This function is a part of the 'expert' sparse Cholesky API:
* SparseCholeskyAnalyze(), that performs symbolic analysis phase and loads
  matrix to be factorized into internal storage
* SparseCholeskySetModType(), that allows to  use  modified  Cholesky/LDLT
  with lower bounds on pivot magnitudes and additional overflow safeguards
* SparseCholeskyFactorize(),  that performs  numeric  factorization  using
  precomputed symbolic analysis and internally stored matrix - and outputs
  result
* SparseCholeskyReload(), that reloads one more matrix with same  sparsity
  pattern into internal storage so  one  may  reuse  previously  allocated
  temporaries and previously performed symbolic analysis

Depending on settings specified during SparseCholeskyAnalyze() call it may
produce classic Cholesky or L*D*LT  decomposition  (with strictly diagonal
D), without permutation or with performance-enhancing permutation P.

IMPORTANT: if all you need is to solve a linear system A*x=b with positive
           definite A, we strongly recommend you to use functions provided
           by the DIRECTSPARSESOLVERS  subpackage  of  the SOLVERS package
           instead of this function.

           Functions provided by the TRFAC  subpackage   are  designed  to
           return factorizations, i.e. such L that A=LL' or A=LDL'.  Thus,
           TRFAC unit can not utilize some performance  improvements  that
           allow to solve Ax=b without returning factors L and D.

           In particular, performance backends, dynamic regularization and
           iterative refinement are not supported  by  TRFAC  but  can  be
           included in DIRECTSPARSESOLVERS.

NOTE: all internal processing is performed with lower triangular  matrices
      stored  in  CRS  format.  Any  other  storage  formats  and/or upper
      triangular storage means  that  one  format  conversion  and/or  one
      transposition will be performed  internally  for  the  analysis  and
      factorization phases. Thus, highest  performance  is  achieved  when
      input is a lower triangular CRS matrix, and lower triangular  output
      is requested.

NOTE: L*D*LT family of factorization may be used to  factorize  indefinite
      matrices. However, numerical stability is guaranteed ONLY for a class
      of quasi-definite matrices.

INPUT PARAMETERS:
    Analysis    -   prior analysis with internally stored matrix which will
                    be factorized
    NeedUpper   -   whether upper triangular or lower triangular output is
                    needed

OUTPUT PARAMETERS:
    A           -   Cholesky decomposition of A stored in lower triangular
                    CRS format, i.e. A=L*L' (or upper triangular CRS, with
                    A=U'*U, depending on NeedUpper parameter).
    D           -   array[N], diagonal factor. If no diagonal  factor  was
                    required during analysis  phase,  still  returned  but
                    filled with 1's
    P           -   array[N], pivots. Permutation matrix P is a product of
                    P(0)*P(1)*...*P(N-1), where P(i) is a  permutation  of
                    row/col I and P[I] (with P[I]>=I).
                    If no permutation was requested during analysis phase,
                    still returned but filled with identity permutation.

The function returns True  when  factorization  resulted  in nondegenerate
matrix. False is returned when factorization fails (Cholesky factorization
of indefinite matrix) or LDLT factorization has exactly zero  elements  at
the diagonal. In the latter case contents of A, D and P is undefined.

The analysis object is not changed during  the  factorization.  Subsequent
calls to SparseCholeskyFactorize() will result in same factorization being
performed one more time.

  -- ALGLIB routine --
     20.09.2020
     Bochkanov Sergey
*************************************************************************/
bool sparsecholeskyfactorize(sparsedecompositionanalysis &analysis, const bool needupper, sparsematrix &a, real_1d_array &d, integer_1d_array &p, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Sparse  Cholesky  decomposition:  update  internally  stored  matrix  with
another one with exactly same sparsity pattern.

This function is a part of the 'expert' sparse Cholesky API:
* SparseCholeskyAnalyze(), that performs symbolic analysis phase and loads
  matrix to be factorized into internal storage
* SparseCholeskySetModType(), that allows to  use  modified  Cholesky/LDLT
  with lower bounds on pivot magnitudes and additional overflow safeguards
* SparseCholeskyFactorize(),  that performs  numeric  factorization  using
  precomputed symbolic analysis and internally stored matrix - and outputs
  result
* SparseCholeskyReload(), that reloads one more matrix with same  sparsity
  pattern into internal storage so  one  may  reuse  previously  allocated
  temporaries and previously performed symbolic analysis

This specific function replaces internally stored  numerical  values  with
ones from another sparse matrix (but having exactly same sparsity  pattern
as one that was used for initial SparseCholeskyAnalyze() call).

NOTE: all internal processing is performed with lower triangular  matrices
      stored  in  CRS  format.  Any  other  storage  formats  and/or upper
      triangular storage means  that  one  format  conversion  and/or  one
      transposition will be performed  internally  for  the  analysis  and
      factorization phases. Thus, highest  performance  is  achieved  when
      input is a lower triangular CRS matrix.

INPUT PARAMETERS:
    Analysis    -   analysis object
    A           -   sparse square matrix in any sparse storage format.  It
                    MUST have exactly same sparsity pattern as that of the
                    matrix that was passed to SparseCholeskyAnalyze().
                    Any difference (missing elements or additional elements)
                    may result in unpredictable and undefined  behavior  -
                    an algorithm may fail due to memory access violation.
    IsUpper     -   whether upper or lower  triangle  is  decomposed  (the
                    other one is ignored).

OUTPUT PARAMETERS:
    Analysis    -   contains:
                    * symbolic analysis of the matrix structure which will
                      be used later to guide numerical factorization.
                    * specific numeric values loaded into internal  memory
                      waiting for the factorization to be performed

  -- ALGLIB routine --
     20.09.2020
     Bochkanov Sergey
*************************************************************************/
void sparsecholeskyreload(sparsedecompositionanalysis &analysis, const sparsematrix &a, const bool isupper, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_BDSVD) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Singular value decomposition of a bidiagonal matrix (extended algorithm)

COMMERCIAL EDITION OF ALGLIB:

  ! Commercial version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Hardware vendor library support (Intel MKL  support  on  x64,  other
  !   libraries on other platforms)
  !
  ! Vendor  libraries  give  approximately  constant  with  respect to the
  ! number of worker threads) acceleration factor which depends on the CPU
  ! being  used, problem  size  and  "baseline"  ALGLIB  edition  which is
  ! used for comparison.
  !
  ! Generally, commercial ALGLIB is several times faster than  open-source
  ! generic C edition, and many times faster than open-source C# edition.
  !
  ! Multithreaded acceleration is NOT supported for this function.
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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 -
                THIS PARAMETER IS IGNORED SINCE ALGLIB 3.5.0
                SINGULAR VALUES ARE ALWAYS SEARCHED WITH HIGH ACCURACY.
    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).

NOTE: multiplication U*Q is performed by means of transposition to internal
      buffer, multiplication and backward transposition. It helps to avoid
      costly columnwise operations and speed-up algorithm.

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.
*************************************************************************/
bool rmatrixbdsvd(real_1d_array &d, const real_1d_array &e, const ae_int_t n, const bool isupper, const bool isfractionalaccuracyrequired, real_2d_array &u, const ae_int_t nru, real_2d_array &c, const ae_int_t ncc, real_2d_array &vt, const ae_int_t ncvt, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_SVD) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Singular value decomposition of a rectangular matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions, x64/ARM platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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
*************************************************************************/
bool rmatrixsvd(const real_2d_array &a, const ae_int_t m, const ae_int_t n, const ae_int_t uneeded, const ae_int_t vtneeded, const ae_int_t additionalmemory, real_1d_array &w, real_2d_array &u, real_2d_array &vt, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_RCOND) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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.
*************************************************************************/
double rmatrixrcond1(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Estimate of a matrix condition number (2-norm)

The algorithm calculates exact 2-norm reciprocal condition number using SVD.

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.

Result: 1/cond2(A)

NOTE:
    if k(A) is very large, then the matrix is  assumed to be degenerate,
    k(A)=INF, 0.0 is returned in such cases.
*************************************************************************/
double rmatrixrcond2(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Estimate of a matrix condition number (2-norm) for a rectangular matrix.

The algorithm calculates exact 2-norm reciprocal condition number using SVD.

Input parameters:
    A   -   matrix. Array[M,N]
    M, N-   rows and columns count, >=1

Result: 1/cond2(A)

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double rmatrixrcond2rect(const real_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double rmatrixrcondinf(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double spdmatrixrcond(const real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);


/*************************************************************************
2-norm condition number of a symmetric positive definite matrix using EVD.

Input parameters:
    A       -   symmetric positive definite matrix which is given by its
                upper or lower triangle depending on the value of
                IsUpper. Array[N,N]
    N       -   size of matrix A.
    IsUpper -   storage format.

Result:
    1/cond(A), if matrix A is positive definite,
    0, if matrix A is not positive definite

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double spdmatrixrcond2(const real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double rmatrixtrrcond1(const real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Triangular matrix: reciprocal 2-norm condition number

The algorithm calculates a reciprocal 2-norm condition number using SVD.

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/cond(A)

NOTE:
    if k(A) is very large, then matrix is assumed to be degenerate,
    k(A)=INF, 0.0 is returned in such cases.
*************************************************************************/
double rmatrixtrrcond2(const real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double rmatrixtrrcondinf(const real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double hpdmatrixrcond(const complex_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double cmatrixrcond1(const complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double cmatrixrcondinf(const complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double rmatrixlurcond1(const real_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double rmatrixlurcondinf(const real_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double spdmatrixcholeskyrcond(const real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double hpdmatrixcholeskyrcond(const complex_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double cmatrixlurcond1(const complex_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double cmatrixlurcondinf(const complex_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double cmatrixtrrcond1(const complex_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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.
*************************************************************************/
double cmatrixtrrcondinf(const complex_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_FBLS) || !defined(AE_PARTIAL_BUILD)

#endif

#if defined(AE_COMPILE_NORMESTIMATOR) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
This procedure initializes matrix norm estimator.

USAGE:
1. User initializes algorithm state with NormEstimatorCreate() call
2. User calls NormEstimatorEstimateSparse() (or NormEstimatorIteration())
3. User calls NormEstimatorResults() to get solution.

INPUT PARAMETERS:
    M       -   number of rows in the matrix being estimated, M>0
    N       -   number of columns in the matrix being estimated, N>0
    NStart  -   number of random starting vectors
                recommended value - at least 5.
    NIts    -   number of iterations to do with best starting vector
                recommended value - at least 5.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state


NOTE: this algorithm is effectively deterministic, i.e. it always  returns
same result when repeatedly called for the same matrix. In fact, algorithm
uses randomized starting vectors, but internal  random  numbers  generator
always generates same sequence of the random values (it is a  feature, not
bug).

Algorithm can be made non-deterministic with NormEstimatorSetSeed(0) call.

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorcreate(const ae_int_t m, const ae_int_t n, const ae_int_t nstart, const ae_int_t nits, normestimatorstate &state, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function changes seed value used by algorithm. In some cases we  need
deterministic processing, i.e. subsequent calls must return equal results,
in other cases we need non-deterministic algorithm which returns different
results for the same matrix on every pass.

Setting zero seed will lead to non-deterministic algorithm, while non-zero
value will make our algorithm deterministic.

INPUT PARAMETERS:
    State       -   norm estimator state, must be initialized with a  call
                    to NormEstimatorCreate()
    SeedVal     -   seed value, >=0. Zero value = non-deterministic algo.

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorsetseed(normestimatorstate &state, const ae_int_t seedval, const xparams _xparams = alglib::xdefault);


/*************************************************************************
This function estimates norm of the sparse M*N matrix A.

INPUT PARAMETERS:
    State       -   norm estimator state, must be initialized with a  call
                    to NormEstimatorCreate()
    A           -   sparse M*N matrix, must be converted to CRS format
                    prior to calling this function.

After this function  is  over  you can call NormEstimatorResults() to get
estimate of the norm(A).

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorestimatesparse(normestimatorstate &state, const sparsematrix &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Matrix norm estimation results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    Nrm     -   estimate of the matrix norm, Nrm>=0

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorresults(const normestimatorstate &state, double &nrm, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_MATINV) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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 the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     05.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixluinverse(real_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixluinverse(real_2d_array &a, const integer_1d_array &pivots, matinvreport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Inversion of a general matrix.

INPUT PARAMETERS:
    A       -   matrix.
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixinverse(real_2d_array &a, const ae_int_t n, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixinverse(real_2d_array &a, matinvreport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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 the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     05.02.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixluinverse(complex_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixluinverse(complex_2d_array &a, const integer_1d_array &pivots, matinvreport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Inversion of a general matrix.

Input parameters:
    A       -   matrix
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)

Output parameters:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void cmatrixinverse(complex_2d_array &a, const ae_int_t n, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixinverse(complex_2d_array &a, matinvreport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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 the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)
    IsUpper -   storage type:
                * if True, the symmetric  matrix  A  is given by its upper
                  triangle, and the lower triangle isn't  used/changed  by
                  the function
                * if False, the symmetric matrix  A  is given by its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  the function

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0,   corresponding      triangle
                  contains inverse matrix,   the  other  triangle  is  not
                  modified.
                * for rep.terminationtype<0,  corresponding  triangle   is
                  zero-filled; the other triangle is not modified.
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void spdmatrixcholeskyinverse(real_2d_array &a, const ae_int_t n, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void spdmatrixcholeskyinverse(real_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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[N,N]
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)
    IsUpper -   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

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void spdmatrixinverse(real_2d_array &a, const ae_int_t n, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void spdmatrixinverse(real_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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 the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)
    IsUpper -   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

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void hpdmatrixcholeskyinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void hpdmatrixcholeskyinverse(complex_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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[N,N]
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)
    IsUpper -   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

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void hpdmatrixinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void hpdmatrixinverse(complex_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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 the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a 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:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 05.02.2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixtrinverse(real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixtrinverse(real_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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 the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a 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:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel, ARM) implementations of linear algebra and
  !   other primitives (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 05.02.2010 by Bochkanov Sergey
*************************************************************************/
void cmatrixtrinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixtrinverse(complex_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_INVERSEUPDATE) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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
*************************************************************************/
void rmatrixinvupdatesimple(real_2d_array &inva, const ae_int_t n, const ae_int_t updrow, const ae_int_t updcolumn, const double updval, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixinvupdaterow(real_2d_array &inva, const ae_int_t n, const ae_int_t updrow, const real_1d_array &v, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixinvupdatecolumn(real_2d_array &inva, const ae_int_t n, const ae_int_t updcolumn, const real_1d_array &u, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
void rmatrixinvupdateuv(real_2d_array &inva, const ae_int_t n, const real_1d_array &u, const real_1d_array &v, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_SCHUR) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Subroutine performing the Schur decomposition of a general matrix by using
the QR algorithm with multiple shifts.

COMMERCIAL EDITION OF ALGLIB:

  ! Commercial version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Hardware vendor library support (lightweight Intel  MKL  is  shipped
  !   with ALGLIB for x64, other libs for other platforms)
  !
  ! Vendor libs give approximately constant  (with  respect  to  number of
  ! worker threads) acceleration factor which depends on CPU  being  used,
  ! problem  size  and  "baseline"  ALGLIB  edition  which  is  used   for
  ! comparison.
  !
  ! Multithreaded acceleration is NOT supported for this function.
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

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).
*************************************************************************/
bool rmatrixschur(real_2d_array &a, const ae_int_t n, real_2d_array &s, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_SPDGEVD) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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
*************************************************************************/
bool smatrixgevd(const real_2d_array &a, const ae_int_t n, const bool isuppera, const real_2d_array &b, const bool isupperb, const ae_int_t zneeded, const ae_int_t problemtype, real_1d_array &d, real_2d_array &z, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
bool smatrixgevdreduce(real_2d_array &a, const ae_int_t n, const bool isuppera, const real_2d_array &b, const bool isupperb, const ae_int_t problemtype, real_2d_array &r, bool &isupperr, const xparams _xparams = alglib::xdefault);
#endif

#if defined(AE_COMPILE_MATDET) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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
*************************************************************************/
double rmatrixludet(const real_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double rmatrixludet(const real_2d_array &a, const integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
double rmatrixdet(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double rmatrixdet(const real_2d_array &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
alglib::complex cmatrixludet(const complex_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, const xparams _xparams = alglib::xdefault);
alglib::complex cmatrixludet(const complex_2d_array &a, const integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
alglib::complex cmatrixdet(const complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
alglib::complex cmatrixdet(const complex_2d_array &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
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
*************************************************************************/
double spdmatrixcholeskydet(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double spdmatrixcholeskydet(const real_2d_array &a, const xparams _xparams = alglib::xdefault);


/*************************************************************************
Determinant calculation of the symmetric positive definite matrix.

Input parameters:
    A       -   matrix, array[N,N]
    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 -   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

Result:
    determinant of matrix A.
    If matrix A is not positive definite, an exception is generated.

  -- ALGLIB --
     Copyright 2005-2008 by Bochkanov Sergey
*************************************************************************/
double spdmatrixdet(const real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
double spdmatrixdet(const real_2d_array &a, const bool isupper, const xparams _xparams = alglib::xdefault);
#endif
}

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (FUNCTIONS)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
#if defined(AE_COMPILE_ABLAS) || !defined(AE_PARTIAL_BUILD)
void ablassplitlength(/* Real    */ const ae_matrix* a,
     ae_int_t n,
     ae_int_t* n1,
     ae_int_t* n2,
     ae_state *_state);
void ablascomplexsplitlength(/* Complex */ const ae_matrix* a,
     ae_int_t n,
     ae_int_t* n1,
     ae_int_t* n2,
     ae_state *_state);
ae_int_t gemmparallelsize(ae_state *_state);
ae_int_t ablasblocksize(/* Real    */ const ae_matrix* a,
     ae_state *_state);
ae_int_t ablascomplexblocksize(/* Complex */ const ae_matrix* a,
     ae_state *_state);
ae_int_t ablasmicroblocksize(ae_state *_state);
void generatereflection(/* Real    */ ae_vector* x,
     ae_int_t n,
     double* tau,
     ae_state *_state);
void applyreflectionfromtheleft(/* Real    */ ae_matrix* c,
     double tau,
     /* Real    */ const ae_vector* v,
     ae_int_t m1,
     ae_int_t m2,
     ae_int_t n1,
     ae_int_t n2,
     /* Real    */ ae_vector* work,
     ae_state *_state);
void applyreflectionfromtheright(/* Real    */ ae_matrix* c,
     double tau,
     /* Real    */ const ae_vector* v,
     ae_int_t m1,
     ae_int_t m2,
     ae_int_t n1,
     ae_int_t n2,
     /* Real    */ ae_vector* work,
     ae_state *_state);
void cmatrixtranspose(ae_int_t m,
     ae_int_t n,
     /* Complex */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     /* Complex */ ae_matrix* b,
     ae_int_t ib,
     ae_int_t jb,
     ae_state *_state);
void rmatrixtranspose(ae_int_t m,
     ae_int_t n,
     /* Real    */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     /* Real    */ ae_matrix* b,
     ae_int_t ib,
     ae_int_t jb,
     ae_state *_state);
void rmatrixenforcesymmetricity(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
void cmatrixcopy(ae_int_t m,
     ae_int_t n,
     /* Complex */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     /* Complex */ ae_matrix* b,
     ae_int_t ib,
     ae_int_t jb,
     ae_state *_state);
void rvectorcopy(ae_int_t n,
     /* Real    */ const ae_vector* a,
     ae_int_t ia,
     /* Real    */ ae_vector* b,
     ae_int_t ib,
     ae_state *_state);
void rmatrixcopy(ae_int_t m,
     ae_int_t n,
     /* Real    */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     /* Real    */ ae_matrix* b,
     ae_int_t ib,
     ae_int_t jb,
     ae_state *_state);
void rmatrixgencopy(ae_int_t m,
     ae_int_t n,
     double alpha,
     /* Real    */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     double beta,
     /* Real    */ ae_matrix* b,
     ae_int_t ib,
     ae_int_t jb,
     ae_state *_state);
void rmatrixger(ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     double alpha,
     /* Real    */ const ae_vector* u,
     ae_int_t iu,
     /* Real    */ const ae_vector* v,
     ae_int_t iv,
     ae_state *_state);
void cmatrixrank1(ae_int_t m,
     ae_int_t n,
     /* Complex */ ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     /* Complex */ const ae_vector* u,
     ae_int_t iu,
     /* Complex */ const ae_vector* v,
     ae_int_t iv,
     ae_state *_state);
void rmatrixrank1(ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     /* Real    */ const ae_vector* u,
     ae_int_t iu,
     /* Real    */ const ae_vector* v,
     ae_int_t iv,
     ae_state *_state);
void rmatrixgemv(ae_int_t m,
     ae_int_t n,
     double alpha,
     /* Real    */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t opa,
     /* Real    */ const ae_vector* x,
     ae_int_t ix,
     double beta,
     /* Real    */ ae_vector* y,
     ae_int_t iy,
     ae_state *_state);
void cmatrixmv(ae_int_t m,
     ae_int_t n,
     /* Complex */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t opa,
     /* Complex */ const ae_vector* x,
     ae_int_t ix,
     /* Complex */ ae_vector* y,
     ae_int_t iy,
     ae_state *_state);
void rmatrixmv(ae_int_t m,
     ae_int_t n,
     /* Real    */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t opa,
     /* Real    */ const ae_vector* x,
     ae_int_t ix,
     /* Real    */ ae_vector* y,
     ae_int_t iy,
     ae_state *_state);
void rmatrixsymv(ae_int_t n,
     double alpha,
     /* Real    */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_bool isupper,
     /* Real    */ const ae_vector* x,
     ae_int_t ix,
     double beta,
     /* Real    */ ae_vector* y,
     ae_int_t iy,
     ae_state *_state);
double rmatrixsyvmv(ae_int_t n,
     /* Real    */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_bool isupper,
     /* Real    */ const ae_vector* x,
     ae_int_t ix,
     /* Real    */ ae_vector* tmp,
     ae_state *_state);
void rmatrixtrsv(ae_int_t n,
     /* Real    */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_bool isupper,
     ae_bool isunit,
     ae_int_t optype,
     /* Real    */ ae_vector* x,
     ae_int_t ix,
     ae_state *_state);
void cmatrixrighttrsm(ae_int_t m,
     ae_int_t n,
     /* Complex */ const ae_matrix* a,
     ae_int_t i1,
     ae_int_t j1,
     ae_bool isupper,
     ae_bool isunit,
     ae_int_t optype,
     /* Complex */ ae_matrix* x,
     ae_int_t i2,
     ae_int_t j2,
     ae_state *_state);
ae_bool _trypexec_cmatrixrighttrsm(ae_int_t m,
    ae_int_t n,
    /* Complex */ const ae_matrix* a,
    ae_int_t i1,
    ae_int_t j1,
    ae_bool isupper,
    ae_bool isunit,
    ae_int_t optype,
    /* Complex */ ae_matrix* x,
    ae_int_t i2,
    ae_int_t j2, ae_state *_state);
void cmatrixlefttrsm(ae_int_t m,
     ae_int_t n,
     /* Complex */ const ae_matrix* a,
     ae_int_t i1,
     ae_int_t j1,
     ae_bool isupper,
     ae_bool isunit,
     ae_int_t optype,
     /* Complex */ ae_matrix* x,
     ae_int_t i2,
     ae_int_t j2,
     ae_state *_state);
ae_bool _trypexec_cmatrixlefttrsm(ae_int_t m,
    ae_int_t n,
    /* Complex */ const ae_matrix* a,
    ae_int_t i1,
    ae_int_t j1,
    ae_bool isupper,
    ae_bool isunit,
    ae_int_t optype,
    /* Complex */ ae_matrix* x,
    ae_int_t i2,
    ae_int_t j2, ae_state *_state);
void rmatrixrighttrsm(ae_int_t m,
     ae_int_t n,
     /* Real    */ const ae_matrix* a,
     ae_int_t i1,
     ae_int_t j1,
     ae_bool isupper,
     ae_bool isunit,
     ae_int_t optype,
     /* Real    */ ae_matrix* x,
     ae_int_t i2,
     ae_int_t j2,
     ae_state *_state);
ae_bool _trypexec_rmatrixrighttrsm(ae_int_t m,
    ae_int_t n,
    /* Real    */ const ae_matrix* a,
    ae_int_t i1,
    ae_int_t j1,
    ae_bool isupper,
    ae_bool isunit,
    ae_int_t optype,
    /* Real    */ ae_matrix* x,
    ae_int_t i2,
    ae_int_t j2, ae_state *_state);
void rmatrixlefttrsm(ae_int_t m,
     ae_int_t n,
     /* Real    */ const ae_matrix* a,
     ae_int_t i1,
     ae_int_t j1,
     ae_bool isupper,
     ae_bool isunit,
     ae_int_t optype,
     /* Real    */ ae_matrix* x,
     ae_int_t i2,
     ae_int_t j2,
     ae_state *_state);
ae_bool _trypexec_rmatrixlefttrsm(ae_int_t m,
    ae_int_t n,
    /* Real    */ const ae_matrix* a,
    ae_int_t i1,
    ae_int_t j1,
    ae_bool isupper,
    ae_bool isunit,
    ae_int_t optype,
    /* Real    */ ae_matrix* x,
    ae_int_t i2,
    ae_int_t j2, ae_state *_state);
void cmatrixherk(ae_int_t n,
     ae_int_t k,
     double alpha,
     /* Complex */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t optypea,
     double beta,
     /* Complex */ ae_matrix* c,
     ae_int_t ic,
     ae_int_t jc,
     ae_bool isupper,
     ae_state *_state);
ae_bool _trypexec_cmatrixherk(ae_int_t n,
    ae_int_t k,
    double alpha,
    /* Complex */ const ae_matrix* a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t optypea,
    double beta,
    /* Complex */ ae_matrix* c,
    ae_int_t ic,
    ae_int_t jc,
    ae_bool isupper, ae_state *_state);
void rmatrixsyrk(ae_int_t n,
     ae_int_t k,
     double alpha,
     /* Real    */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t optypea,
     double beta,
     /* Real    */ ae_matrix* c,
     ae_int_t ic,
     ae_int_t jc,
     ae_bool isupper,
     ae_state *_state);
ae_bool _trypexec_rmatrixsyrk(ae_int_t n,
    ae_int_t k,
    double alpha,
    /* Real    */ const ae_matrix* a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t optypea,
    double beta,
    /* Real    */ ae_matrix* c,
    ae_int_t ic,
    ae_int_t jc,
    ae_bool isupper, ae_state *_state);
void cmatrixgemm(ae_int_t m,
     ae_int_t n,
     ae_int_t k,
     ae_complex alpha,
     /* Complex */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t optypea,
     /* Complex */ const ae_matrix* b,
     ae_int_t ib,
     ae_int_t jb,
     ae_int_t optypeb,
     ae_complex beta,
     /* Complex */ ae_matrix* c,
     ae_int_t ic,
     ae_int_t jc,
     ae_state *_state);
ae_bool _trypexec_cmatrixgemm(ae_int_t m,
    ae_int_t n,
    ae_int_t k,
    ae_complex alpha,
    /* Complex */ const ae_matrix* a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t optypea,
    /* Complex */ const ae_matrix* b,
    ae_int_t ib,
    ae_int_t jb,
    ae_int_t optypeb,
    ae_complex beta,
    /* Complex */ ae_matrix* c,
    ae_int_t ic,
    ae_int_t jc, ae_state *_state);
void rmatrixgemm(ae_int_t m,
     ae_int_t n,
     ae_int_t k,
     double alpha,
     /* Real    */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t optypea,
     /* Real    */ const ae_matrix* b,
     ae_int_t ib,
     ae_int_t jb,
     ae_int_t optypeb,
     double beta,
     /* Real    */ ae_matrix* c,
     ae_int_t ic,
     ae_int_t jc,
     ae_state *_state);
ae_bool _trypexec_rmatrixgemm(ae_int_t m,
    ae_int_t n,
    ae_int_t k,
    double alpha,
    /* Real    */ const ae_matrix* a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t optypea,
    /* Real    */ const ae_matrix* b,
    ae_int_t ib,
    ae_int_t jb,
    ae_int_t optypeb,
    double beta,
    /* Real    */ ae_matrix* c,
    ae_int_t ic,
    ae_int_t jc, ae_state *_state);
void cmatrixsyrk(ae_int_t n,
     ae_int_t k,
     double alpha,
     /* Complex */ const ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t optypea,
     double beta,
     /* Complex */ ae_matrix* c,
     ae_int_t ic,
     ae_int_t jc,
     ae_bool isupper,
     ae_state *_state);
void rowwisegramschmidt(/* Real    */ const ae_matrix* q,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* qx,
     ae_bool needqx,
     ae_state *_state);
#endif
#if defined(AE_COMPILE_ORTFAC) || !defined(AE_PARTIAL_BUILD)
void rmatrixqr(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* tau,
     ae_state *_state);
void rmatrixlq(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* tau,
     ae_state *_state);
void cmatrixqr(/* Complex */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Complex */ ae_vector* tau,
     ae_state *_state);
void cmatrixlq(/* Complex */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Complex */ ae_vector* tau,
     ae_state *_state);
void rmatrixqrunpackq(/* Real    */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ const ae_vector* tau,
     ae_int_t qcolumns,
     /* Real    */ ae_matrix* q,
     ae_state *_state);
void rmatrixqrunpackr(/* Real    */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_matrix* r,
     ae_state *_state);
void rmatrixlqunpackq(/* Real    */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ const ae_vector* tau,
     ae_int_t qrows,
     /* Real    */ ae_matrix* q,
     ae_state *_state);
void rmatrixlqunpackl(/* Real    */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_matrix* l,
     ae_state *_state);
void cmatrixqrunpackq(/* Complex */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Complex */ const ae_vector* tau,
     ae_int_t qcolumns,
     /* Complex */ ae_matrix* q,
     ae_state *_state);
void cmatrixqrunpackr(/* Complex */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Complex */ ae_matrix* r,
     ae_state *_state);
void cmatrixlqunpackq(/* Complex */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Complex */ const ae_vector* tau,
     ae_int_t qrows,
     /* Complex */ ae_matrix* q,
     ae_state *_state);
void cmatrixlqunpackl(/* Complex */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Complex */ ae_matrix* l,
     ae_state *_state);
void rmatrixqrbasecase(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* work,
     /* Real    */ ae_vector* t,
     /* Real    */ ae_vector* tau,
     ae_state *_state);
void rmatrixlqbasecase(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* work,
     /* Real    */ ae_vector* t,
     /* Real    */ ae_vector* tau,
     ae_state *_state);
void rmatrixbd(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* tauq,
     /* Real    */ ae_vector* taup,
     ae_state *_state);
void rmatrixbdunpackq(/* Real    */ const ae_matrix* qp,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ const ae_vector* tauq,
     ae_int_t qcolumns,
     /* Real    */ ae_matrix* q,
     ae_state *_state);
void rmatrixbdmultiplybyq(/* Real    */ const ae_matrix* qp,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ const ae_vector* tauq,
     /* Real    */ ae_matrix* z,
     ae_int_t zrows,
     ae_int_t zcolumns,
     ae_bool fromtheright,
     ae_bool dotranspose,
     ae_state *_state);
void rmatrixbdunpackpt(/* Real    */ const ae_matrix* qp,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ const ae_vector* taup,
     ae_int_t ptrows,
     /* Real    */ ae_matrix* pt,
     ae_state *_state);
void rmatrixbdmultiplybyp(/* Real    */ const ae_matrix* qp,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ const ae_vector* taup,
     /* Real    */ ae_matrix* z,
     ae_int_t zrows,
     ae_int_t zcolumns,
     ae_bool fromtheright,
     ae_bool dotranspose,
     ae_state *_state);
void rmatrixbdunpackdiagonals(/* Real    */ const ae_matrix* b,
     ae_int_t m,
     ae_int_t n,
     ae_bool* isupper,
     /* Real    */ ae_vector* d,
     /* Real    */ ae_vector* e,
     ae_state *_state);
void rmatrixhessenberg(/* Real    */ ae_matrix* a,
     ae_int_t n,
     /* Real    */ ae_vector* tau,
     ae_state *_state);
void rmatrixhessenbergunpackq(/* Real    */ const ae_matrix* a,
     ae_int_t n,
     /* Real    */ const ae_vector* tau,
     /* Real    */ ae_matrix* q,
     ae_state *_state);
void rmatrixhessenbergunpackh(/* Real    */ const ae_matrix* a,
     ae_int_t n,
     /* Real    */ ae_matrix* h,
     ae_state *_state);
void smatrixtd(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ ae_vector* tau,
     /* Real    */ ae_vector* d,
     /* Real    */ ae_vector* e,
     ae_state *_state);
void smatrixtdunpackq(/* Real    */ const ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ const ae_vector* tau,
     /* Real    */ ae_matrix* q,
     ae_state *_state);
void hmatrixtd(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Complex */ ae_vector* tau,
     /* Real    */ ae_vector* d,
     /* Real    */ ae_vector* e,
     ae_state *_state);
void hmatrixtdunpackq(/* Complex */ const ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Complex */ const ae_vector* tau,
     /* Complex */ ae_matrix* q,
     ae_state *_state);
#endif
#if defined(AE_COMPILE_MATGEN) || !defined(AE_PARTIAL_BUILD)
void rmatrixrndorthogonal(ae_int_t n,
     /* Real    */ ae_matrix* a,
     ae_state *_state);
void rmatrixrndcond(ae_int_t n,
     double c,
     /* Real    */ ae_matrix* a,
     ae_state *_state);
void cmatrixrndorthogonal(ae_int_t n,
     /* Complex */ ae_matrix* a,
     ae_state *_state);
void cmatrixrndcond(ae_int_t n,
     double c,
     /* Complex */ ae_matrix* a,
     ae_state *_state);
void smatrixrndcond(ae_int_t n,
     double c,
     /* Real    */ ae_matrix* a,
     ae_state *_state);
void spdmatrixrndcond(ae_int_t n,
     double c,
     /* Real    */ ae_matrix* a,
     ae_state *_state);
void hmatrixrndcond(ae_int_t n,
     double c,
     /* Complex */ ae_matrix* a,
     ae_state *_state);
void hpdmatrixrndcond(ae_int_t n,
     double c,
     /* Complex */ ae_matrix* a,
     ae_state *_state);
void rmatrixrndorthogonalfromtheright(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     ae_state *_state);
void rmatrixrndorthogonalfromtheleft(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     ae_state *_state);
void cmatrixrndorthogonalfromtheright(/* Complex */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     ae_state *_state);
void cmatrixrndorthogonalfromtheleft(/* Complex */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     ae_state *_state);
void smatrixrndmultiply(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_state *_state);
void hmatrixrndmultiply(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_state *_state);
#endif
#if defined(AE_COMPILE_SPARSE) || !defined(AE_PARTIAL_BUILD)
void sparsecreate(ae_int_t m,
     ae_int_t n,
     ae_int_t k,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatebuf(ae_int_t m,
     ae_int_t n,
     ae_int_t k,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatecrs(ae_int_t m,
     ae_int_t n,
     /* Integer */ const ae_vector* ner,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatecrsbuf(ae_int_t m,
     ae_int_t n,
     /* Integer */ const ae_vector* ner,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatecrsfromdense(/* Real    */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatecrsfromdensebuf(/* Real    */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatecrsfromdensev(/* Real    */ const ae_vector* a,
     ae_int_t m,
     ae_int_t n,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatecrsfromdensevbuf(/* Real    */ const ae_vector* a,
     ae_int_t m,
     ae_int_t n,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatesks(ae_int_t m,
     ae_int_t n,
     /* Integer */ const ae_vector* d,
     /* Integer */ const ae_vector* u,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatesksbuf(ae_int_t m,
     ae_int_t n,
     /* Integer */ const ae_vector* d,
     /* Integer */ const ae_vector* u,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatesksband(ae_int_t m,
     ae_int_t n,
     ae_int_t bw,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatesksbandbuf(ae_int_t m,
     ae_int_t n,
     ae_int_t bw,
     sparsematrix* s,
     ae_state *_state);
void sparsecopy(const sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparseswap(sparsematrix* s0, sparsematrix* s1, ae_state *_state);
void sparseadd(sparsematrix* s,
     ae_int_t i,
     ae_int_t j,
     double v,
     ae_state *_state);
void sparseset(sparsematrix* s,
     ae_int_t i,
     ae_int_t j,
     double v,
     ae_state *_state);
double sparseget(const sparsematrix* s,
     ae_int_t i,
     ae_int_t j,
     ae_state *_state);
ae_bool sparseexists(const sparsematrix* s,
     ae_int_t i,
     ae_int_t j,
     ae_state *_state);
double sparsegetdiagonal(const sparsematrix* s,
     ae_int_t i,
     ae_state *_state);
void sparsemv(const sparsematrix* s,
     /* Real    */ const ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_state *_state);
void sparsemtv(const sparsematrix* s,
     /* Real    */ const ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_state *_state);
void sparsegemv(const sparsematrix* s,
     double alpha,
     ae_int_t ops,
     /* Real    */ const ae_vector* x,
     ae_int_t ix,
     double beta,
     /* Real    */ ae_vector* y,
     ae_int_t iy,
     ae_state *_state);
void sparsemv2(const sparsematrix* s,
     /* Real    */ const ae_vector* x,
     /* Real    */ ae_vector* y0,
     /* Real    */ ae_vector* y1,
     ae_state *_state);
void sparsesmv(const sparsematrix* s,
     ae_bool isupper,
     /* Real    */ const ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_state *_state);
void sparsemultiplycolsby(sparsematrix* s,
     /* Real    */ const ae_vector* x,
     ae_state *_state);
void sparsemultiplyrowsby(sparsematrix* s,
     /* Real    */ const ae_vector* x,
     ae_state *_state);
void sparsemultiplyrowscolsby(sparsematrix* s,
     /* Real    */ const ae_vector* x,
     /* Real    */ const ae_vector* y,
     ae_state *_state);
double sparsevsmv(const sparsematrix* s,
     ae_bool isupper,
     /* Real    */ const ae_vector* x,
     ae_state *_state);
void sparsemm(const sparsematrix* s,
     /* Real    */ const ae_matrix* a,
     ae_int_t k,
     /* Real    */ ae_matrix* b,
     ae_state *_state);
void sparsemtm(const sparsematrix* s,
     /* Real    */ const ae_matrix* a,
     ae_int_t k,
     /* Real    */ ae_matrix* b,
     ae_state *_state);
void sparsemm2(const sparsematrix* s,
     /* Real    */ const ae_matrix* a,
     ae_int_t k,
     /* Real    */ ae_matrix* b0,
     /* Real    */ ae_matrix* b1,
     ae_state *_state);
void sparsesmm(const sparsematrix* s,
     ae_bool isupper,
     /* Real    */ const ae_matrix* a,
     ae_int_t k,
     /* Real    */ ae_matrix* b,
     ae_state *_state);
void sparsetrmv(const sparsematrix* s,
     ae_bool isupper,
     ae_bool isunit,
     ae_int_t optype,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_state *_state);
void sparsetrsv(const sparsematrix* s,
     ae_bool isupper,
     ae_bool isunit,
     ae_int_t optype,
     /* Real    */ ae_vector* x,
     ae_state *_state);
void sparsesymmpermtbl(const sparsematrix* a,
     ae_bool isupper,
     /* Integer */ const ae_vector* p,
     sparsematrix* b,
     ae_state *_state);
void sparsesymmpermtbltranspose(const sparsematrix* a,
     ae_bool isupper,
     /* Integer */ const ae_vector* p,
     sparsematrix* b,
     ae_state *_state);
void sparsesymmpermtblbuf(const sparsematrix* a,
     ae_bool isupper,
     /* Integer */ const ae_vector* p,
     sparsematrix* b,
     ae_state *_state);
void sparsesymmpermtbltransposebuf(const sparsematrix* a,
     ae_bool isupper,
     /* Integer */ const ae_vector* p,
     sparsematrix* b,
     ae_state *_state);
void sparseresizematrix(sparsematrix* s, ae_state *_state);
double sparsegetaveragelengthofchain(const sparsematrix* s,
     ae_state *_state);
ae_bool sparseenumerate(const sparsematrix* s,
     ae_int_t* t0,
     ae_int_t* t1,
     ae_int_t* i,
     ae_int_t* j,
     double* v,
     ae_state *_state);
ae_bool sparserewriteexisting(sparsematrix* s,
     ae_int_t i,
     ae_int_t j,
     double v,
     ae_state *_state);
void sparsegetrow(const sparsematrix* s,
     ae_int_t i,
     /* Real    */ ae_vector* irow,
     ae_state *_state);
void sparsegetcompressedrow(const sparsematrix* s,
     ae_int_t i,
     /* Integer */ ae_vector* colidx,
     /* Real    */ ae_vector* vals,
     ae_int_t* nzcnt,
     ae_state *_state);
void sparseappendcompressedrow(sparsematrix* s,
     /* Integer */ const ae_vector* colidx,
     /* Real    */ const ae_vector* vals,
     ae_int_t nz,
     ae_state *_state);
void sparseappendemptyrow(sparsematrix* s, ae_state *_state);
void sparseappendelement(sparsematrix* s,
     ae_int_t k,
     double v,
     ae_state *_state);
void sparsetransposesks(sparsematrix* s, ae_state *_state);
void sparsetransposecrs(sparsematrix* s, ae_state *_state);
void sparsecopytransposecrs(const sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparsecopytransposecrsbuf(const sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparsecopytransposecrsxbuf(const sparsematrix* s0,
     ae_int_t triangle,
     sparsematrix* s1,
     ae_state *_state);
void sparseconvertto(sparsematrix* s0, ae_int_t fmt, ae_state *_state);
void sparsecopytobuf(const sparsematrix* s0,
     ae_int_t fmt,
     sparsematrix* s1,
     ae_state *_state);
void sparseconverttohash(sparsematrix* s, ae_state *_state);
void sparsecopytohash(const sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparsecopytohashbuf(const sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparsecopytocrs(const sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparseconverttosks(sparsematrix* s, ae_state *_state);
void sparsecopytosks(const sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparsecopytosksbuf(const sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparsecreatecrsinplace(sparsematrix* s, ae_state *_state);
ae_int_t sparsegetmatrixtype(const sparsematrix* s, ae_state *_state);
ae_bool sparseishash(const sparsematrix* s, ae_state *_state);
ae_bool sparseissks(const sparsematrix* s, ae_state *_state);
void sparsefree(sparsematrix* s, ae_state *_state);
ae_int_t sparsegetuppercount(const sparsematrix* s, ae_state *_state);
ae_int_t sparsegetlowercount(const sparsematrix* s, ae_state *_state);
void sparsescale(sparsematrix* s,
     ae_int_t scltype,
     ae_bool scalerows,
     ae_bool scalecols,
     ae_bool colsfirst,
     /* Real    */ ae_vector* r,
     /* Real    */ ae_vector* c,
     ae_state *_state);
void sparseappendcoltocrs(sparsematrix* s, ae_state *_state);
ae_bool sparsetrygatherclear(sparsematrix* dst,
     const sparsematrix* src,
     ae_state *_state);
void sparsemergepatterns(const sparsematrix* s1,
     const sparsematrix* s2,
     sparsematrix* dst,
     ae_state *_state);
void sparsealloc(ae_serializer* s,
     const sparsematrix* a,
     ae_state *_state);
void sparseserialize(ae_serializer* s,
     const sparsematrix* a,
     ae_state *_state);
void sparseunserialize(ae_serializer* s,
     sparsematrix* a,
     ae_state *_state);
double spdotvr(/* Real    */ const ae_vector* x,
     const sparsematrix* s,
     ae_int_t r,
     ae_state *_state);
double spdotr2(const sparsematrix* s, ae_int_t r, ae_state *_state);
void _sparsebuffers_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _sparsebuffers_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _sparsebuffers_clear(void* _p);
void _sparsebuffers_destroy(void* _p);
#endif
#if defined(AE_COMPILE_HSSCHUR) || !defined(AE_PARTIAL_BUILD)
void rmatrixinternalschurdecomposition(/* Real    */ ae_matrix* h,
     ae_int_t n,
     ae_int_t tneeded,
     ae_int_t zneeded,
     /* Real    */ ae_vector* wr,
     /* Real    */ ae_vector* wi,
     /* Real    */ ae_matrix* z,
     ae_int_t* info,
     ae_state *_state);
ae_bool upperhessenbergschurdecomposition(/* Real    */ ae_matrix* h,
     ae_int_t n,
     /* Real    */ ae_matrix* s,
     ae_state *_state);
void internalschurdecomposition(/* Real    */ ae_matrix* h,
     ae_int_t n,
     ae_int_t tneeded,
     ae_int_t zneeded,
     /* Real    */ ae_vector* wr,
     /* Real    */ ae_vector* wi,
     /* Real    */ ae_matrix* z,
     ae_int_t* info,
     ae_state *_state);
#endif
#if defined(AE_COMPILE_EVD) || !defined(AE_PARTIAL_BUILD)
void eigsubspacecreate(ae_int_t n,
     ae_int_t k,
     eigsubspacestate* state,
     ae_state *_state);
void eigsubspacecreatebuf(ae_int_t n,
     ae_int_t k,
     eigsubspacestate* state,
     ae_state *_state);
void eigsubspacesetcond(eigsubspacestate* state,
     double eps,
     ae_int_t maxits,
     ae_state *_state);
void eigsubspacesetwarmstart(eigsubspacestate* state,
     ae_bool usewarmstart,
     ae_state *_state);
void eigsubspaceoocstart(eigsubspacestate* state,
     ae_int_t mtype,
     ae_state *_state);
ae_bool eigsubspaceooccontinue(eigsubspacestate* state, ae_state *_state);
void eigsubspaceoocgetrequestinfo(eigsubspacestate* state,
     ae_int_t* requesttype,
     ae_int_t* requestsize,
     ae_state *_state);
void eigsubspaceoocgetrequestdata(eigsubspacestate* state,
     /* Real    */ ae_matrix* x,
     ae_state *_state);
void eigsubspaceoocsendresult(eigsubspacestate* state,
     /* Real    */ const ae_matrix* ax,
     ae_state *_state);
void eigsubspaceoocstop(eigsubspacestate* state,
     /* Real    */ ae_vector* w,
     /* Real    */ ae_matrix* z,
     eigsubspacereport* rep,
     ae_state *_state);
void eigsubspacesolvedenses(eigsubspacestate* state,
     /* Real    */ const ae_matrix* a,
     ae_bool isupper,
     /* Real    */ ae_vector* w,
     /* Real    */ ae_matrix* z,
     eigsubspacereport* rep,
     ae_state *_state);
void eigsubspacesolvesparses(eigsubspacestate* state,
     const sparsematrix* a,
     ae_bool isupper,
     /* Real    */ ae_vector* w,
     /* Real    */ ae_matrix* z,
     eigsubspacereport* rep,
     ae_state *_state);
ae_bool eigsubspaceiteration(eigsubspacestate* state, ae_state *_state);
ae_bool smatrixevd(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_int_t zneeded,
     ae_bool isupper,
     /* Real    */ ae_vector* d,
     /* Real    */ ae_matrix* z,
     ae_state *_state);
ae_bool smatrixevdr(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_int_t zneeded,
     ae_bool isupper,
     double b1,
     double b2,
     ae_int_t* m,
     /* Real    */ ae_vector* w,
     /* Real    */ ae_matrix* z,
     ae_state *_state);
ae_bool smatrixevdi(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_int_t zneeded,
     ae_bool isupper,
     ae_int_t i1,
     ae_int_t i2,
     /* Real    */ ae_vector* w,
     /* Real    */ ae_matrix* z,
     ae_state *_state);
ae_bool hmatrixevd(/* Complex */ const ae_matrix* _a,
     ae_int_t n,
     ae_int_t zneeded,
     ae_bool isupper,
     /* Real    */ ae_vector* d,
     /* Complex */ ae_matrix* z,
     ae_state *_state);
ae_bool hmatrixevdr(/* Complex */ const ae_matrix* _a,
     ae_int_t n,
     ae_int_t zneeded,
     ae_bool isupper,
     double b1,
     double b2,
     ae_int_t* m,
     /* Real    */ ae_vector* w,
     /* Complex */ ae_matrix* z,
     ae_state *_state);
ae_bool hmatrixevdi(/* Complex */ const ae_matrix* _a,
     ae_int_t n,
     ae_int_t zneeded,
     ae_bool isupper,
     ae_int_t i1,
     ae_int_t i2,
     /* Real    */ ae_vector* w,
     /* Complex */ ae_matrix* z,
     ae_state *_state);
ae_bool smatrixtdevd(/* Real    */ ae_vector* d,
     /* Real    */ const ae_vector* _e,
     ae_int_t n,
     ae_int_t zneeded,
     /* Real    */ ae_matrix* z,
     ae_state *_state);
ae_bool smatrixtdevdr(/* Real    */ ae_vector* d,
     /* Real    */ const ae_vector* e,
     ae_int_t n,
     ae_int_t zneeded,
     double a,
     double b,
     ae_int_t* m,
     /* Real    */ ae_matrix* z,
     ae_state *_state);
ae_bool smatrixtdevdi(/* Real    */ ae_vector* d,
     /* Real    */ const ae_vector* e,
     ae_int_t n,
     ae_int_t zneeded,
     ae_int_t i1,
     ae_int_t i2,
     /* Real    */ ae_matrix* z,
     ae_state *_state);
ae_bool rmatrixevd(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_int_t vneeded,
     /* Real    */ ae_vector* wr,
     /* Real    */ ae_vector* wi,
     /* Real    */ ae_matrix* vl,
     /* Real    */ ae_matrix* vr,
     ae_state *_state);
void _eigsubspacestate_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _eigsubspacestate_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _eigsubspacestate_clear(void* _p);
void _eigsubspacestate_destroy(void* _p);
void _eigsubspacereport_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _eigsubspacereport_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _eigsubspacereport_clear(void* _p);
void _eigsubspacereport_destroy(void* _p);
#endif
#if defined(AE_COMPILE_DLU) || !defined(AE_PARTIAL_BUILD)
void cmatrixluprec(/* Complex */ ae_matrix* a,
     ae_int_t offs,
     ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* pivots,
     /* Complex */ ae_vector* tmp,
     ae_state *_state);
void rmatrixluprec(/* Real    */ ae_matrix* a,
     ae_int_t offs,
     ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* pivots,
     /* Real    */ ae_vector* tmp,
     ae_state *_state);
void cmatrixplurec(/* Complex */ ae_matrix* a,
     ae_int_t offs,
     ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* pivots,
     /* Complex */ ae_vector* tmp,
     ae_state *_state);
void rmatrixplurec(/* Real    */ ae_matrix* a,
     ae_int_t offs,
     ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* pivots,
     /* Real    */ ae_vector* tmp,
     ae_state *_state);
#endif
#if defined(AE_COMPILE_SPTRF) || !defined(AE_PARTIAL_BUILD)
ae_bool sptrflu(sparsematrix* a,
     ae_int_t pivottype,
     /* Integer */ ae_vector* pr,
     /* Integer */ ae_vector* pc,
     sluv2buffer* buf,
     ae_state *_state);
void _sluv2list1matrix_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _sluv2list1matrix_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _sluv2list1matrix_clear(void* _p);
void _sluv2list1matrix_destroy(void* _p);
void _sluv2sparsetrail_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _sluv2sparsetrail_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _sluv2sparsetrail_clear(void* _p);
void _sluv2sparsetrail_destroy(void* _p);
void _sluv2densetrail_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _sluv2densetrail_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _sluv2densetrail_clear(void* _p);
void _sluv2densetrail_destroy(void* _p);
void _sluv2buffer_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _sluv2buffer_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _sluv2buffer_clear(void* _p);
void _sluv2buffer_destroy(void* _p);
#endif
#if defined(AE_COMPILE_AMDORDERING) || !defined(AE_PARTIAL_BUILD)
void generateamdpermutation(const sparsematrix* a,
     ae_int_t n,
     /* Integer */ ae_vector* perm,
     /* Integer */ ae_vector* invperm,
     amdbuffer* buf,
     ae_state *_state);
ae_int_t generateamdpermutationx(const sparsematrix* a,
     /* Boolean */ const ae_vector* eligible,
     ae_int_t n,
     double promoteabove,
     /* Integer */ ae_vector* perm,
     /* Integer */ ae_vector* invperm,
     ae_int_t amdtype,
     amdbuffer* buf,
     ae_state *_state);
void _amdknset_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _amdknset_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _amdknset_clear(void* _p);
void _amdknset_destroy(void* _p);
void _amdvertexset_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _amdvertexset_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _amdvertexset_clear(void* _p);
void _amdvertexset_destroy(void* _p);
void _amdllmatrix_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _amdllmatrix_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _amdllmatrix_clear(void* _p);
void _amdllmatrix_destroy(void* _p);
void _amdbuffer_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _amdbuffer_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _amdbuffer_clear(void* _p);
void _amdbuffer_destroy(void* _p);
#endif
#if defined(AE_COMPILE_SPCHOL) || !defined(AE_PARTIAL_BUILD)
ae_int_t spsymmgetmaxfastkernel(ae_state *_state);
ae_bool spsymmanalyze(const sparsematrix* a,
     /* Integer */ const ae_vector* priorities,
     double promoteabove,
     ae_int_t promoteto,
     ae_int_t facttype,
     ae_int_t permtype,
     ae_int_t memreuse,
     spcholanalysis* analysis,
     ae_state *_state);
void spsymmsetmodificationstrategy(spcholanalysis* analysis,
     ae_int_t modstrategy,
     double p0,
     double p1,
     double p2,
     double p3,
     ae_state *_state);
void spsymmcontrolsign(spcholanalysis* analysis,
     /* Boolean */ const ae_vector* ispositive,
     double eps,
     ae_state *_state);
void spsymmreload(spcholanalysis* analysis,
     const sparsematrix* a,
     ae_state *_state);
void spsymmreloaddiagonal(spcholanalysis* analysis,
     /* Real    */ const ae_vector* d,
     ae_state *_state);
ae_bool spsymmfactorize(spcholanalysis* analysis, ae_state *_state);
void spsymmenforceparallelism(spcholanalysis* analysis, ae_state *_state);
void spsymmextract(spcholanalysis* analysis,
     sparsematrix* a,
     /* Real    */ ae_vector* d,
     /* Integer */ ae_vector* p,
     ae_state *_state);
void spsymmsolve(spcholanalysis* analysis,
     /* Real    */ ae_vector* b,
     ae_state *_state);
void spsymmdiagerr(spcholanalysis* analysis,
     double* sumsq,
     double* errsq,
     ae_state *_state);
void _priorityamdbuffers_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _priorityamdbuffers_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _priorityamdbuffers_clear(void* _p);
void _priorityamdbuffers_destroy(void* _p);
void _spcholadj_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _spcholadj_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _spcholadj_clear(void* _p);
void _spcholadj_destroy(void* _p);
void _spcholanalysis_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _spcholanalysis_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _spcholanalysis_clear(void* _p);
void _spcholanalysis_destroy(void* _p);
#endif
#if defined(AE_COMPILE_TRFAC) || !defined(AE_PARTIAL_BUILD)
void rmatrixlu(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* pivots,
     ae_state *_state);
void cmatrixlu(/* Complex */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* pivots,
     ae_state *_state);
ae_bool hpdmatrixcholesky(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
ae_bool spdmatrixcholesky(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
void spdmatrixcholeskyupdateadd1(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ const ae_vector* u,
     ae_state *_state);
void spdmatrixcholeskyupdatefix(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Boolean */ const ae_vector* fix,
     ae_state *_state);
void spdmatrixcholeskyupdateadd1buf(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ const ae_vector* u,
     /* Real    */ ae_vector* bufr,
     ae_state *_state);
void spdmatrixcholeskyupdatefixbuf(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Boolean */ const ae_vector* fix,
     /* Real    */ ae_vector* bufr,
     ae_state *_state);
ae_bool sparselu(sparsematrix* a,
     ae_int_t pivottype,
     /* Integer */ ae_vector* p,
     /* Integer */ ae_vector* q,
     ae_state *_state);
ae_bool sparsecholeskyskyline(sparsematrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
ae_bool sparsecholesky(sparsematrix* a, ae_bool isupper, ae_state *_state);
ae_bool sparsecholeskyp(sparsematrix* a,
     ae_bool isupper,
     /* Integer */ ae_vector* p,
     ae_state *_state);
ae_bool sparsecholeskyanalyze(const sparsematrix* a,
     ae_bool isupper,
     ae_int_t facttype,
     ae_int_t permtype,
     sparsedecompositionanalysis* analysis,
     ae_state *_state);
void sparsecholeskysetmodtype(sparsedecompositionanalysis* analysis,
     ae_int_t modstrategy,
     double p0,
     double p1,
     double p2,
     double p3,
     ae_state *_state);
ae_bool sparsecholeskyfactorize(sparsedecompositionanalysis* analysis,
     ae_bool needupper,
     sparsematrix* a,
     /* Real    */ ae_vector* d,
     /* Integer */ ae_vector* p,
     ae_state *_state);
void sparsecholeskyreload(sparsedecompositionanalysis* analysis,
     const sparsematrix* a,
     ae_bool isupper,
     ae_state *_state);
void rmatrixlup(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* pivots,
     ae_state *_state);
void cmatrixlup(/* Complex */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* pivots,
     ae_state *_state);
void rmatrixplu(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* pivots,
     ae_state *_state);
void cmatrixplu(/* Complex */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* pivots,
     ae_state *_state);
ae_bool spdmatrixcholeskyrec(/* Real    */ ae_matrix* a,
     ae_int_t offs,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ ae_vector* tmp,
     ae_state *_state);
void _sparsedecompositionanalysis_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _sparsedecompositionanalysis_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _sparsedecompositionanalysis_clear(void* _p);
void _sparsedecompositionanalysis_destroy(void* _p);
#endif
#if defined(AE_COMPILE_BDSVD) || !defined(AE_PARTIAL_BUILD)
ae_bool rmatrixbdsvd(/* Real    */ ae_vector* d,
     /* Real    */ const ae_vector* _e,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isfractionalaccuracyrequired,
     /* Real    */ ae_matrix* u,
     ae_int_t nru,
     /* Real    */ ae_matrix* c,
     ae_int_t ncc,
     /* Real    */ ae_matrix* vt,
     ae_int_t ncvt,
     ae_state *_state);
ae_bool bidiagonalsvddecomposition(/* Real    */ ae_vector* d,
     /* Real    */ const ae_vector* _e,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isfractionalaccuracyrequired,
     /* Real    */ ae_matrix* u,
     ae_int_t nru,
     /* Real    */ ae_matrix* c,
     ae_int_t ncc,
     /* Real    */ ae_matrix* vt,
     ae_int_t ncvt,
     ae_state *_state);
#endif
#if defined(AE_COMPILE_SVD) || !defined(AE_PARTIAL_BUILD)
ae_bool rmatrixsvd(/* Real    */ const ae_matrix* _a,
     ae_int_t m,
     ae_int_t n,
     ae_int_t uneeded,
     ae_int_t vtneeded,
     ae_int_t additionalmemory,
     /* Real    */ ae_vector* w,
     /* Real    */ ae_matrix* u,
     /* Real    */ ae_matrix* vt,
     ae_state *_state);
#endif
#if defined(AE_COMPILE_RCOND) || !defined(AE_PARTIAL_BUILD)
double rmatrixrcond1(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_state *_state);
double rmatrixrcond2(/* Real    */ const ae_matrix* a,
     ae_int_t n,
     ae_state *_state);
double rmatrixrcond2rect(/* Real    */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     ae_state *_state);
double rmatrixrcondinf(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_state *_state);
double spdmatrixrcond(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
double spdmatrixrcond2(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
double rmatrixtrrcond1(/* Real    */ const ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     ae_state *_state);
double rmatrixtrrcond2(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     ae_state *_state);
double rmatrixtrrcondinf(/* Real    */ const ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     ae_state *_state);
double hpdmatrixrcond(/* Complex */ const ae_matrix* _a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
double cmatrixrcond1(/* Complex */ const ae_matrix* _a,
     ae_int_t n,
     ae_state *_state);
double cmatrixrcondinf(/* Complex */ const ae_matrix* _a,
     ae_int_t n,
     ae_state *_state);
double rmatrixlurcond1(/* Real    */ const ae_matrix* lua,
     ae_int_t n,
     ae_state *_state);
double rmatrixlurcondinf(/* Real    */ const ae_matrix* lua,
     ae_int_t n,
     ae_state *_state);
double spdmatrixcholeskyrcond(/* Real    */ const ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
double hpdmatrixcholeskyrcond(/* Complex */ const ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
double cmatrixlurcond1(/* Complex */ const ae_matrix* lua,
     ae_int_t n,
     ae_state *_state);
double cmatrixlurcondinf(/* Complex */ const ae_matrix* lua,
     ae_int_t n,
     ae_state *_state);
double cmatrixtrrcond1(/* Complex */ const ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     ae_state *_state);
double cmatrixtrrcondinf(/* Complex */ const ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     ae_state *_state);
double rcondthreshold(ae_state *_state);
#endif
#if defined(AE_COMPILE_FBLS) || !defined(AE_PARTIAL_BUILD)
void fblscholeskysolve(/* Real    */ const ae_matrix* cha,
     double sqrtscalea,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ ae_vector* xb,
     /* Real    */ ae_vector* tmp,
     ae_state *_state);
void fblssolvecgx(/* Real    */ const ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     double alpha,
     /* Real    */ const ae_vector* b,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* buf,
     ae_state *_state);
void fblscgcreate(/* Real    */ const ae_vector* x,
     /* Real    */ const ae_vector* b,
     ae_int_t n,
     fblslincgstate* state,
     ae_state *_state);
ae_bool fblscgiteration(fblslincgstate* state, ae_state *_state);
void fblsgmrescreate(/* Real    */ const ae_vector* b,
     ae_int_t n,
     ae_int_t k,
     fblsgmresstate* state,
     ae_state *_state);
ae_bool fblsgmresiteration(fblsgmresstate* state, ae_state *_state);
void fblssolvels(/* Real    */ ae_matrix* a,
     /* Real    */ ae_vector* b,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* tmp0,
     /* Real    */ ae_vector* tmp1,
     /* Real    */ ae_vector* tmp2,
     ae_state *_state);
void _fblslincgstate_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _fblslincgstate_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _fblslincgstate_clear(void* _p);
void _fblslincgstate_destroy(void* _p);
void _fblsgmresstate_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _fblsgmresstate_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _fblsgmresstate_clear(void* _p);
void _fblsgmresstate_destroy(void* _p);
#endif
#if defined(AE_COMPILE_NORMESTIMATOR) || !defined(AE_PARTIAL_BUILD)
void normestimatorcreate(ae_int_t m,
     ae_int_t n,
     ae_int_t nstart,
     ae_int_t nits,
     normestimatorstate* state,
     ae_state *_state);
void normestimatorsetseed(normestimatorstate* state,
     ae_int_t seedval,
     ae_state *_state);
ae_bool normestimatoriteration(normestimatorstate* state,
     ae_state *_state);
void normestimatorestimatesparse(normestimatorstate* state,
     const sparsematrix* a,
     ae_state *_state);
void normestimatorresults(const normestimatorstate* state,
     double* nrm,
     ae_state *_state);
void normestimatorrestart(normestimatorstate* state, ae_state *_state);
void _normestimatorstate_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _normestimatorstate_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _normestimatorstate_clear(void* _p);
void _normestimatorstate_destroy(void* _p);
#endif
#if defined(AE_COMPILE_MATINV) || !defined(AE_PARTIAL_BUILD)
void rmatrixluinverse(/* Real    */ ae_matrix* a,
     /* Integer */ const ae_vector* pivots,
     ae_int_t n,
     matinvreport* rep,
     ae_state *_state);
void rmatrixinverse(/* Real    */ ae_matrix* a,
     ae_int_t n,
     matinvreport* rep,
     ae_state *_state);
void cmatrixluinverse(/* Complex */ ae_matrix* a,
     /* Integer */ const ae_vector* pivots,
     ae_int_t n,
     matinvreport* rep,
     ae_state *_state);
void cmatrixinverse(/* Complex */ ae_matrix* a,
     ae_int_t n,
     matinvreport* rep,
     ae_state *_state);
void spdmatrixcholeskyinverse(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     matinvreport* rep,
     ae_state *_state);
void spdmatrixinverse(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     matinvreport* rep,
     ae_state *_state);
void hpdmatrixcholeskyinverse(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     matinvreport* rep,
     ae_state *_state);
void hpdmatrixinverse(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     matinvreport* rep,
     ae_state *_state);
void rmatrixtrinverse(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     matinvreport* rep,
     ae_state *_state);
void cmatrixtrinverse(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     matinvreport* rep,
     ae_state *_state);
void spdmatrixcholeskyinverserec(/* Real    */ ae_matrix* a,
     ae_int_t offs,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ ae_vector* tmp,
     matinvreport* rep,
     ae_state *_state);
ae_bool _trypexec_spdmatrixcholeskyinverserec(/* Real    */ ae_matrix* a,
    ae_int_t offs,
    ae_int_t n,
    ae_bool isupper,
    /* Real    */ ae_vector* tmp,
    matinvreport* rep, ae_state *_state);
void _matinvreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _matinvreport_init_copy(void* _dst, const void* _src, ae_state *_state, ae_bool make_automatic);
void _matinvreport_clear(void* _p);
void _matinvreport_destroy(void* _p);
#endif
#if defined(AE_COMPILE_INVERSEUPDATE) || !defined(AE_PARTIAL_BUILD)
void rmatrixinvupdatesimple(/* Real    */ ae_matrix* inva,
     ae_int_t n,
     ae_int_t updrow,
     ae_int_t updcolumn,
     double updval,
     ae_state *_state);
void rmatrixinvupdaterow(/* Real    */ ae_matrix* inva,
     ae_int_t n,
     ae_int_t updrow,
     /* Real    */ const ae_vector* v,
     ae_state *_state);
void rmatrixinvupdatecolumn(/* Real    */ ae_matrix* inva,
     ae_int_t n,
     ae_int_t updcolumn,
     /* Real    */ const ae_vector* u,
     ae_state *_state);
void rmatrixinvupdateuv(/* Real    */ ae_matrix* inva,
     ae_int_t n,
     /* Real    */ const ae_vector* u,
     /* Real    */ const ae_vector* v,
     ae_state *_state);
#endif
#if defined(AE_COMPILE_SCHUR) || !defined(AE_PARTIAL_BUILD)
ae_bool rmatrixschur(/* Real    */ ae_matrix* a,
     ae_int_t n,
     /* Real    */ ae_matrix* s,
     ae_state *_state);
#endif
#if defined(AE_COMPILE_SPDGEVD) || !defined(AE_PARTIAL_BUILD)
ae_bool smatrixgevd(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_bool isuppera,
     /* Real    */ const ae_matrix* b,
     ae_bool isupperb,
     ae_int_t zneeded,
     ae_int_t problemtype,
     /* Real    */ ae_vector* d,
     /* Real    */ ae_matrix* z,
     ae_state *_state);
ae_bool smatrixgevdreduce(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isuppera,
     /* Real    */ const ae_matrix* b,
     ae_bool isupperb,
     ae_int_t problemtype,
     /* Real    */ ae_matrix* r,
     ae_bool* isupperr,
     ae_state *_state);
#endif
#if defined(AE_COMPILE_MATDET) || !defined(AE_PARTIAL_BUILD)
double rmatrixludet(/* Real    */ const ae_matrix* a,
     /* Integer */ const ae_vector* pivots,
     ae_int_t n,
     ae_state *_state);
double rmatrixdet(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_state *_state);
ae_complex cmatrixludet(/* Complex */ const ae_matrix* a,
     /* Integer */ const ae_vector* pivots,
     ae_int_t n,
     ae_state *_state);
ae_complex cmatrixdet(/* Complex */ const ae_matrix* _a,
     ae_int_t n,
     ae_state *_state);
double spdmatrixcholeskydet(/* Real    */ const ae_matrix* a,
     ae_int_t n,
     ae_state *_state);
double spdmatrixdet(/* Real    */ const ae_matrix* _a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
#endif

}
#endif

