linalg.h

/*************************************************************************
ALGLIB 3.11.0 (source code generated 2017-05-11)
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
{
typedef struct
{
    ae_vector vals;
    ae_vector idx;
    ae_vector ridx;
    ae_vector didx;
    ae_vector uidx;
    ae_int_t matrixtype;
    ae_int_t m;
    ae_int_t n;
    ae_int_t nfree;
    ae_int_t ninitialized;
    ae_int_t tablesize;
} sparsematrix;
typedef struct
{
    ae_vector d;
    ae_vector u;
    sparsematrix s;
} sparsebuffers;
typedef struct
{
    double r1;
    double rinf;
} matinvreport;
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_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;
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 matrixtype;
    hqrndstate rs;
    ae_bool running;
    ae_vector tau;
    ae_matrix qcur;
    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 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;
 
}
 
//
// THIS SECTION CONTAINS C++ INTERFACE
//
namespace alglib
{
 
/*************************************************************************
Sparse matrix structure.
 
You should use ALGLIB functions to work with sparse matrix. Never  try  to
access its fields directly!
 
NOTES ON THE SPARSE STORAGE FORMATS
 
Sparse matrices can be stored using several formats:
* Hash-Table representation
* Compressed Row Storage (CRS)
* Skyline matrix storage (SKS)
 
Each of the formats has benefits and drawbacks:
* Hash-table is good for dynamic operations (insertion of new elements),
  but does not support linear algebra operations
* CRS is good for operations like matrix-vector or matrix-matrix products,
  but its initialization is less convenient - you have to tell row   sizes
  at the initialization, and you have to fill  matrix  only  row  by  row,
  from left to right.
* SKS is a special format which is used to store triangular  factors  from
  Cholesky factorization. It does not support  dynamic  modification,  and
  support for linear algebra operations is very limited.
 
Tables below outline information about these two formats:
 
    OPERATIONS WITH MATRIX      HASH        CRS         SKS
    creation                    +           +           +
    SparseGet                   +           +           +
    SparseRewriteExisting       +           +           +
    SparseSet                   +
    SparseAdd                   +
    SparseGetRow                            +           +
    SparseGetCompressedRow                  +           +
    sparse-dense linear algebra             +           +
*************************************************************************/
class _sparsematrix_owner
{
public:
    _sparsematrix_owner();
    _sparsematrix_owner(const _sparsematrix_owner &rhs);
    _sparsematrix_owner& operator=(const _sparsematrix_owner &rhs);
    virtual ~_sparsematrix_owner();
    alglib_impl::sparsematrix* c_ptr();
    alglib_impl::sparsematrix* c_ptr() const;
protected:
    alglib_impl::sparsematrix *p_struct;
};
class sparsematrix : public _sparsematrix_owner
{
public:
    sparsematrix();
    sparsematrix(const sparsematrix &rhs);
    sparsematrix& operator=(const sparsematrix &rhs);
    virtual ~sparsematrix();
 
};
 
 
/*************************************************************************
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(const _sparsebuffers_owner &rhs);
    _sparsebuffers_owner& operator=(const _sparsebuffers_owner &rhs);
    virtual ~_sparsebuffers_owner();
    alglib_impl::sparsebuffers* c_ptr();
    alglib_impl::sparsebuffers* c_ptr() const;
protected:
    alglib_impl::sparsebuffers *p_struct;
};
class sparsebuffers : public _sparsebuffers_owner
{
public:
    sparsebuffers();
    sparsebuffers(const sparsebuffers &rhs);
    sparsebuffers& operator=(const sparsebuffers &rhs);
    virtual ~sparsebuffers();
 
};
 
 
 
 
 
 
 
 
 
/*************************************************************************
Matrix inverse report:
* R1    reciprocal of condition number in 1-norm
* RInf  reciprocal of condition number in inf-norm
*************************************************************************/
class _matinvreport_owner
{
public:
    _matinvreport_owner();
    _matinvreport_owner(const _matinvreport_owner &rhs);
    _matinvreport_owner& operator=(const _matinvreport_owner &rhs);
    virtual ~_matinvreport_owner();
    alglib_impl::matinvreport* c_ptr();
    alglib_impl::matinvreport* c_ptr() const;
protected:
    alglib_impl::matinvreport *p_struct;
};
class matinvreport : public _matinvreport_owner
{
public:
    matinvreport();
    matinvreport(const matinvreport &rhs);
    matinvreport& operator=(const matinvreport &rhs);
    virtual ~matinvreport();
    double &r1;
    double &rinf;
 
};
 
 
 
 
 
 
 
 
 
/*************************************************************************
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(const _normestimatorstate_owner &rhs);
    _normestimatorstate_owner& operator=(const _normestimatorstate_owner &rhs);
    virtual ~_normestimatorstate_owner();
    alglib_impl::normestimatorstate* c_ptr();
    alglib_impl::normestimatorstate* c_ptr() const;
protected:
    alglib_impl::normestimatorstate *p_struct;
};
class normestimatorstate : public _normestimatorstate_owner
{
public:
    normestimatorstate();
    normestimatorstate(const normestimatorstate &rhs);
    normestimatorstate& operator=(const normestimatorstate &rhs);
    virtual ~normestimatorstate();
 
};
 
/*************************************************************************
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(const _eigsubspacestate_owner &rhs);
    _eigsubspacestate_owner& operator=(const _eigsubspacestate_owner &rhs);
    virtual ~_eigsubspacestate_owner();
    alglib_impl::eigsubspacestate* c_ptr();
    alglib_impl::eigsubspacestate* c_ptr() const;
protected:
    alglib_impl::eigsubspacestate *p_struct;
};
class eigsubspacestate : public _eigsubspacestate_owner
{
public:
    eigsubspacestate();
    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(const _eigsubspacereport_owner &rhs);
    _eigsubspacereport_owner& operator=(const _eigsubspacereport_owner &rhs);
    virtual ~_eigsubspacereport_owner();
    alglib_impl::eigsubspacereport* c_ptr();
    alglib_impl::eigsubspacereport* c_ptr() const;
protected:
    alglib_impl::eigsubspacereport *p_struct;
};
class eigsubspacereport : public _eigsubspacereport_owner
{
public:
    eigsubspacereport();
    eigsubspacereport(const eigsubspacereport &rhs);
    eigsubspacereport& operator=(const eigsubspacereport &rhs);
    virtual ~eigsubspacereport();
    ae_int_t &iterationscount;
 
};
 
/*************************************************************************
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);
void sparsecreate(const ae_int_t m, const ae_int_t n, sparsematrix &s);
 
 
/*************************************************************************
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, const sparsematrix &s);
void sparsecreatebuf(const ae_int_t m, const ae_int_t n, const sparsematrix &s);
 
 
/*************************************************************************
This function creates sparse matrix in a CRS format (expert function for
situations when you are running out of memory).
 
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
 
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);
 
 
/*************************************************************************
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, const sparsematrix &s);
 
 
/*************************************************************************
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 SparseRewriteExisting() 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);
 
 
/*************************************************************************
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()/SparseAdd() 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, const sparsematrix &s);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
This function copies S0 to S1.
Memory already allocated in S1 is reused as much as possible.
 
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 sparsecopybuf(const sparsematrix &s0, const sparsematrix &s1);
 
 
/*************************************************************************
This function efficiently swaps contents of S0 and S1.
 
  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparseswap(const sparsematrix &s0, const sparsematrix &s1);
 
 
/*************************************************************************
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(const sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v);
 
 
/*************************************************************************
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
* new value MUST be non-zero. Exception will be thrown for zero V.
* elements must be initialized in correct order -  from top row to bottom,
  within row - from left to right.
 
For SKS storage: NOT SUPPORTED! Use SparseRewriteExisting() to  work  with
SKS matrices.
 
INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table or CRS representation.
    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(const sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v);
 
 
/*************************************************************************
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 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
 
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);
 
 
/*************************************************************************
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 modify, 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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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, const real_1d_array &x, real_1d_array &y);
 
 
/*************************************************************************
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, const real_1d_array &x);
 
 
/*************************************************************************
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(const sparsematrix &s);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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(const sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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(const sparsematrix &s);
 
 
/*************************************************************************
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(const sparsematrix &s0, const ae_int_t fmt);
 
 
/*************************************************************************
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, const sparsematrix &s1);
 
 
/*************************************************************************
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(const sparsematrix &s);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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, const sparsematrix &s1);
 
 
/*************************************************************************
This function converts matrix to CRS format.
 
Some  algorithms  (linear  algebra ones, for example) require matrices in
CRS format. This function allows to perform in-place conversion.
 
INPUT PARAMETERS
    S           -   sparse M*N matrix in any format
 
OUTPUT PARAMETERS
    S           -   matrix in CRS format
 
NOTE: this   function  has  no  effect  when  called with matrix which is
      already in CRS mode.
 
NOTE: this function allocates temporary memory to store a   copy  of  the
      matrix. If you perform a lot of repeated conversions, we  recommend
      you  to  use  SparseCopyToCRSBuf()  function,   which   can   reuse
      previously allocated memory.
 
  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparseconverttocrs(const sparsematrix &s);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
This  function  performs  out-of-place  conversion  to  CRS 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.
    S1          -   matrix which may contain some pre-allocated memory, or
                    can be just uninitialized structure.
 
OUTPUT PARAMETERS
    S1          -   sparse matrix in CRS format.
 
NOTE: if S0 is stored as CRS, it is just copied without conversion.
 
  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytocrsbuf(const sparsematrix &s0, const sparsematrix &s1);
 
 
/*************************************************************************
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(const sparsematrix &s);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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, const sparsematrix &s1);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using CRS representation.
 
INPUT PARAMETERS:
    S   -   sparse matrix.
 
RESULT:
    True if matrix type is CRS
    False if matrix type is not CRS
 
  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
bool sparseiscrs(const sparsematrix &s);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
The function returns number of rows of a sparse matrix.
 
RESULT: number of rows of a sparse matrix.
 
  -- ALGLIB PROJECT --
     Copyright 23.08.2012 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetnrows(const sparsematrix &s);
 
 
/*************************************************************************
The function returns number of columns of a sparse matrix.
 
RESULT: number of columns of a sparse matrix.
 
  -- ALGLIB PROJECT --
     Copyright 23.08.2012 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetncols(const sparsematrix &s);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
/*************************************************************************
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);
 
 
/*************************************************************************
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, const real_2d_array &b, const ae_int_t ib, const ae_int_t jb);
 
 
/*************************************************************************
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(const real_2d_array &a, const ae_int_t n, const bool isupper);
 
 
/*************************************************************************
Copy
 
Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied and transposed
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, 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);
 
 
/*************************************************************************
Copy
 
Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied and transposed
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, 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);
 
 
/*************************************************************************
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, complex_1d_array &u, const ae_int_t iu, complex_1d_array &v, const ae_int_t iv);
 
 
/*************************************************************************
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, real_1d_array &u, const ae_int_t iu, real_1d_array &v, const ae_int_t iv);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
This subroutine calculates X*op(A^-1) where:
* X is MxN general matrix
* A is NxN upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition, conjugate transposition
 
Multiplication result replaces X.
Cache-oblivious algorithm is used.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  Because  starting/stopping  worker  thread always
  ! involves some overhead, parallelism starts to be  profitable  for  N's
  ! larger than 128.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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
    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
 
  -- ALGLIB routine --
     15.12.2009
     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, const complex_2d_array &x, const ae_int_t i2, const ae_int_t j2);
void smp_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, const complex_2d_array &x, const ae_int_t i2, const ae_int_t j2);
 
 
/*************************************************************************
This subroutine calculates op(A^-1)*X where:
* X is MxN general matrix
* A is MxM upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition, conjugate transposition
 
Multiplication result replaces X.
Cache-oblivious algorithm is used.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  Because  starting/stopping  worker  thread always
  ! involves some overhead, parallelism starts to be  profitable  for  N's
  ! larger than 128.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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
    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
 
  -- ALGLIB routine --
     15.12.2009
     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, const complex_2d_array &x, const ae_int_t i2, const ae_int_t j2);
void smp_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, const complex_2d_array &x, const ae_int_t i2, const ae_int_t j2);
 
 
/*************************************************************************
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.
Cache-oblivious algorithm is used.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  Because  starting/stopping  worker  thread always
  ! involves some overhead, parallelism starts to be  profitable  for  N's
  ! larger than 128.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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
    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
 
  -- ALGLIB routine --
     15.12.2009
     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, const real_2d_array &x, const ae_int_t i2, const ae_int_t j2);
void smp_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, const real_2d_array &x, const ae_int_t i2, const ae_int_t j2);
 
 
/*************************************************************************
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.
Cache-oblivious algorithm is used.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  Because  starting/stopping  worker  thread always
  ! involves some overhead, parallelism starts to be  profitable  for  N's
  ! larger than 128.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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
    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
 
  -- ALGLIB routine --
     15.12.2009
     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, const real_2d_array &x, const ae_int_t i2, const ae_int_t j2);
void smp_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, const real_2d_array &x, const ae_int_t i2, const ae_int_t j2);
 
 
/*************************************************************************
This subroutine calculates  C=alpha*A*A^H+beta*C  or  C=alpha*A^H*A+beta*C
where:
* C is NxN Hermitian matrix given by its upper/lower triangle
* A is NxK matrix when A*A^H is calculated, KxN matrix otherwise
 
Additional info:
* cache-oblivious algorithm is used.
* multiplication result replaces C. If Beta=0, C elements are not used in
  calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  Because  starting/stopping  worker  thread always
  ! involves some overhead, parallelism starts to be  profitable  for  N's
  ! larger than 128.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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
    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).
 
  -- ALGLIB routine --
     16.12.2009
     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, const complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper);
void smp_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, const complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper);
 
 
/*************************************************************************
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:
* 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.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  Because  starting/stopping  worker  thread always
  ! involves some overhead, parallelism starts to be  profitable  for  N's
  ! larger than 128.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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
    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
 
  -- ALGLIB routine --
     16.12.2009
     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, const real_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper);
void smp_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, const real_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper);
 
 
/*************************************************************************
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.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  Because  starting/stopping  worker  thread always
  ! involves some overhead, parallelism starts to be  profitable  for  N's
  ! larger than 128.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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.
 
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
 
  -- ALGLIB routine --
     16.12.2009
     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, const complex_2d_array &c, const ae_int_t ic, const ae_int_t jc);
void smp_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, const complex_2d_array &c, const ae_int_t ic, const ae_int_t jc);
 
 
/*************************************************************************
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.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  Because  starting/stopping  worker  thread always
  ! involves some overhead, parallelism starts to be  profitable  for  N's
  ! larger than 128.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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.
 
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
 
  -- ALGLIB routine --
     2009-2013
     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, const real_2d_array &c, const ae_int_t ic, const ae_int_t jc);
void smp_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, const real_2d_array &c, const ae_int_t ic, const ae_int_t jc);
 
 
/*************************************************************************
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, const complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper);
void smp_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, const complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper);
 
/*************************************************************************
LU decomposition of a general real matrix with row pivoting
 
A is represented as A = P*L*U, where:
* L is lower unitriangular matrix
* U is upper triangular matrix
* P = P0*P1*...*PK, K=min(M,N)-1,
  Pi - permutation matrix for I and Pivots[I]
 
This is cache-oblivous implementation of LU decomposition.
It is optimized for square matrices. As for rectangular matrices:
* best case - M>>N
* worst case - N>>M, small M, large N, matrix does not fit in CPU cache
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that LU decomposition  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=1024,  achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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       -   array[0..M-1, 0..N-1].
    M       -   number of rows in matrix A.
    N       -   number of columns in matrix A.
 
 
OUTPUT PARAMETERS:
    A       -   matrices L and U in compact form:
                * L is stored under main diagonal
                * U is stored on and above main diagonal
    Pivots  -   permutation matrix in compact form.
                array[0..Min(M-1,N-1)].
 
  -- ALGLIB routine --
     10.01.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixlu(real_2d_array &a, const ae_int_t m, const ae_int_t n, integer_1d_array &pivots);
void smp_rmatrixlu(real_2d_array &a, const ae_int_t m, const ae_int_t n, integer_1d_array &pivots);
 
 
/*************************************************************************
LU decomposition of a general complex matrix with row pivoting
 
A is represented as A = P*L*U, where:
* L is lower unitriangular matrix
* U is upper triangular matrix
* P = P0*P1*...*PK, K=min(M,N)-1,
  Pi - permutation matrix for I and Pivots[I]
 
This is cache-oblivous implementation of LU decomposition. It is optimized
for square matrices. As for rectangular matrices:
* best case - M>>N
* worst case - N>>M, small M, large N, matrix does not fit in CPU cache
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that LU decomposition  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=1024,  achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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       -   array[0..M-1, 0..N-1].
    M       -   number of rows in matrix A.
    N       -   number of columns in matrix A.
 
 
OUTPUT PARAMETERS:
    A       -   matrices L and U in compact form:
                * L is stored under main diagonal
                * U is stored on and above main diagonal
    Pivots  -   permutation matrix in compact form.
                array[0..Min(M-1,N-1)].
 
  -- ALGLIB routine --
     10.01.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixlu(complex_2d_array &a, const ae_int_t m, const ae_int_t n, integer_1d_array &pivots);
void smp_cmatrixlu(complex_2d_array &a, const ae_int_t m, const ae_int_t n, integer_1d_array &pivots);
 
 
/*************************************************************************
Cache-oblivious Cholesky decomposition
 
The algorithm computes Cholesky decomposition  of  a  Hermitian  positive-
definite matrix. The result of an algorithm is a representation  of  A  as
A=U'*U  or A=L*L' (here X' detones conj(X^T)).
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that Cholesky decomposition is harder
  ! to parallelize than, say, matrix-matrix product - this  algorithm  has
  ! several synchronization points which  can  not  be  avoided.  However,
  ! parallelism starts to be profitable starting from N=500.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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       -   upper or lower triangle of a factorized matrix.
                array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   if IsUpper=True, then A contains an upper triangle of
                a symmetric matrix, otherwise A contains a lower one.
 
OUTPUT PARAMETERS:
    A       -   the result of factorization. If IsUpper=True, then
                the upper triangle contains matrix U, so that A = U'*U,
                and the elements below the main diagonal are not modified.
                Similarly, if IsUpper = False.
 
RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False. Contents of A is not determined
    in such case.
 
  -- ALGLIB routine --
     15.12.2009
     Bochkanov Sergey
*************************************************************************/
bool hpdmatrixcholesky(complex_2d_array &a, const ae_int_t n, const bool isupper);
bool smp_hpdmatrixcholesky(complex_2d_array &a, const ae_int_t n, const bool isupper);
 
 
/*************************************************************************
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
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that Cholesky decomposition is harder
  ! to parallelize than, say, matrix-matrix product - this  algorithm  has
  ! several synchronization points which  can  not  be  avoided.  However,
  ! parallelism starts to be profitable starting from N=500.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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       -   upper or lower triangle of a factorized matrix.
                array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   if IsUpper=True, then A contains an upper triangle of
                a symmetric matrix, otherwise A contains a lower one.
 
OUTPUT PARAMETERS:
    A       -   the result of factorization. If IsUpper=True, then
                the upper triangle contains matrix U, so that A = U^T*U,
                and the elements below the main diagonal are not modified.
                Similarly, if IsUpper = False.
 
RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False. Contents of A is not determined
    in such case.
 
  -- ALGLIB routine --
     15.12.2009
     Bochkanov Sergey
*************************************************************************/
bool spdmatrixcholesky(real_2d_array &a, const ae_int_t n, const bool isupper);
bool smp_spdmatrixcholesky(real_2d_array &a, const ae_int_t n, const bool isupper);
 
 
/*************************************************************************
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.
    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.
 
  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdateadd1(const real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &u);
 
 
/*************************************************************************
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(const real_2d_array &a, const ae_int_t n, const bool isupper, const boolean_1d_array &fix);
 
 
/*************************************************************************
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(const real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &u, real_1d_array &bufr);
 
 
/*************************************************************************
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(const real_2d_array &a, const ae_int_t n, const bool isupper, const boolean_1d_array &fix, real_1d_array &bufr);
 
 
/*************************************************************************
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  is  a  more  efficient alternative to general, but  slower
SparseCholeskyX(), because it does not  create  temporary  copies  of  the
target. It performs factorization in-place, which gives  best  performance
on low-profile matrices. Its drawback, however, is that it can not perform
profile-reducing permutation of input matrix.
 
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(const sparsematrix &a, const ae_int_t n, const bool isupper);
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
/*************************************************************************
Inversion of a matrix given by its LU decomposition.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that matrix inversion  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=1024,  achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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       -   LU decomposition of the matrix
                (output of RMatrixLU subroutine).
    Pivots  -   table of permutations
                (the output of RMatrixLU subroutine).
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
 
OUTPUT PARAMETERS:
    Info    -   return code:
                * -3    A is singular, or VERY close to singular.
                        it is filled by zeros in such cases.
                *  1    task is solved (but matrix A may be ill-conditioned,
                        check R1/RInf parameters for condition numbers).
    Rep     -   solver report, see below for more info
    A       -   inverse of matrix A.
                Array whose indexes range within [0..N-1, 0..N-1].
 
SOLVER REPORT
 
Subroutine sets following fields of the Rep structure:
* R1        reciprocal of condition number: 1/cond(A), 1-norm.
* RInf      reciprocal of condition number: 1/cond(A), inf-norm.
 
  -- ALGLIB routine --
     05.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixluinverse(real_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, ae_int_t &info, matinvreport &rep);
void smp_rmatrixluinverse(real_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, ae_int_t &info, matinvreport &rep);
void rmatrixluinverse(real_2d_array &a, const integer_1d_array &pivots, ae_int_t &info, matinvreport &rep);
void smp_rmatrixluinverse(real_2d_array &a, const integer_1d_array &pivots, ae_int_t &info, matinvreport &rep);
 
 
/*************************************************************************
Inversion of a general matrix.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that matrix inversion  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=1024,  achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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.
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
 
Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse
 
Result:
    True, if the matrix is not singular.
    False, if the matrix is singular.
 
  -- ALGLIB --
     Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixinverse(real_2d_array &a, const ae_int_t n, ae_int_t &info, matinvreport &rep);
void smp_rmatrixinverse(real_2d_array &a, const ae_int_t n, ae_int_t &info, matinvreport &rep);
void rmatrixinverse(real_2d_array &a, ae_int_t &info, matinvreport &rep);
void smp_rmatrixinverse(real_2d_array &a, ae_int_t &info, matinvreport &rep);
 
 
/*************************************************************************
Inversion of a matrix given by its LU decomposition.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that matrix inversion  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=1024,  achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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       -   LU decomposition of the matrix
                (output of CMatrixLU subroutine).
    Pivots  -   table of permutations
                (the output of CMatrixLU subroutine).
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
 
OUTPUT PARAMETERS:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse
 
  -- ALGLIB routine --
     05.02.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixluinverse(complex_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, ae_int_t &info, matinvreport &rep);
void smp_cmatrixluinverse(complex_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, ae_int_t &info, matinvreport &rep);
void cmatrixluinverse(complex_2d_array &a, const integer_1d_array &pivots, ae_int_t &info, matinvreport &rep);
void smp_cmatrixluinverse(complex_2d_array &a, const integer_1d_array &pivots, ae_int_t &info, matinvreport &rep);
 
 
/*************************************************************************
Inversion of a general matrix.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that matrix inversion  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=1024,  achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
 
Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse
 
  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void cmatrixinverse(complex_2d_array &a, const ae_int_t n, ae_int_t &info, matinvreport &rep);
void smp_cmatrixinverse(complex_2d_array &a, const ae_int_t n, ae_int_t &info, matinvreport &rep);
void cmatrixinverse(complex_2d_array &a, ae_int_t &info, matinvreport &rep);
void smp_cmatrixinverse(complex_2d_array &a, ae_int_t &info, matinvreport &rep);
 
 
/*************************************************************************
Inversion of a symmetric positive definite matrix which is given
by Cholesky decomposition.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  However,  Cholesky  inversion  is  a  "difficult"
  ! algorithm  -  it  has  lots  of  internal synchronization points which
  ! prevents efficient  parallelization  of  algorithm.  Only  very  large
  ! problems (N=thousands) can be efficiently parallelized.
  !
  ! 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       -   Cholesky decomposition of the matrix to be inverted:
                A=U'*U or A = L*L'.
                Output of  SPDMatrixCholesky subroutine.
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   storage type (optional):
                * if True, symmetric  matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't  used/changed  by
                  function
                * if False,  symmetric matrix  A  is  given  by  its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  function
                * if not given, lower half is used.
 
Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse
 
  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void spdmatrixcholeskyinverse(real_2d_array &a, const ae_int_t n, const bool isupper, ae_int_t &info, matinvreport &rep);
void smp_spdmatrixcholeskyinverse(real_2d_array &a, const ae_int_t n, const bool isupper, ae_int_t &info, matinvreport &rep);
void spdmatrixcholeskyinverse(real_2d_array &a, ae_int_t &info, matinvreport &rep);
void smp_spdmatrixcholeskyinverse(real_2d_array &a, ae_int_t &info, matinvreport &rep);
 
 
/*************************************************************************
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.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  However,  Cholesky  inversion  is  a  "difficult"
  ! algorithm  -  it  has  lots  of  internal synchronization points which
  ! prevents efficient  parallelization  of  algorithm.  Only  very  large
  ! problems (N=thousands) can be efficiently parallelized.
  !
  ! 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 inverted (upper or lower triangle).
                Array with elements [0..N-1,0..N-1].
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   storage type (optional):
                * if True, symmetric  matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't  used/changed  by
                  function
                * if False,  symmetric matrix  A  is  given  by  its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  function
                * if not given,  both lower and upper  triangles  must  be
                  filled.
 
Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse
 
  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void spdmatrixinverse(real_2d_array &a, const ae_int_t n, const bool isupper, ae_int_t &info, matinvreport &rep);
void smp_spdmatrixinverse(real_2d_array &a, const ae_int_t n, const bool isupper, ae_int_t &info, matinvreport &rep);
void spdmatrixinverse(real_2d_array &a, ae_int_t &info, matinvreport &rep);
void smp_spdmatrixinverse(real_2d_array &a, ae_int_t &info, matinvreport &rep);
 
 
/*************************************************************************
Inversion of a Hermitian positive definite matrix which is given
by Cholesky decomposition.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  However,  Cholesky  inversion  is  a  "difficult"
  ! algorithm  -  it  has  lots  of  internal synchronization points which
  ! prevents efficient  parallelization  of  algorithm.  Only  very  large
  ! problems (N=thousands) can be efficiently parallelized.
  !
  ! 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       -   Cholesky decomposition of the matrix to be inverted:
                A=U'*U or A = L*L'.
                Output of  HPDMatrixCholesky subroutine.
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   storage type (optional):
                * if True, symmetric  matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't  used/changed  by
                  function
                * if False,  symmetric matrix  A  is  given  by  its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  function
                * if not given, lower half is used.
 
Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse
 
  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void hpdmatrixcholeskyinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, ae_int_t &info, matinvreport &rep);
void smp_hpdmatrixcholeskyinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, ae_int_t &info, matinvreport &rep);
void hpdmatrixcholeskyinverse(complex_2d_array &a, ae_int_t &info, matinvreport &rep);
void smp_hpdmatrixcholeskyinverse(complex_2d_array &a, ae_int_t &info, matinvreport &rep);
 
 
/*************************************************************************
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.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of  this  function.  However,  Cholesky  inversion  is  a  "difficult"
  ! algorithm  -  it  has  lots  of  internal synchronization points which
  ! prevents efficient  parallelization  of  algorithm.  Only  very  large
  ! problems (N=thousands) can be efficiently parallelized.
  !
  ! 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 inverted (upper or lower triangle).
                Array with elements [0..N-1,0..N-1].
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   storage type (optional):
                * if True, symmetric  matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't  used/changed  by
                  function
                * if False,  symmetric matrix  A  is  given  by  its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  function
                * if not given,  both lower and upper  triangles  must  be
                  filled.
 
Output parameters:
    Info    -   return code, same as in RMatrixLUInverse
    Rep     -   solver report, same as in RMatrixLUInverse
    A       -   inverse of matrix A, same as in RMatrixLUInverse
 
  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void hpdmatrixinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, ae_int_t &info, matinvreport &rep);
void smp_hpdmatrixinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, ae_int_t &info, matinvreport &rep);
void hpdmatrixinverse(complex_2d_array &a, ae_int_t &info, matinvreport &rep);
void smp_hpdmatrixinverse(complex_2d_array &a, ae_int_t &info, matinvreport &rep);
 
 
/*************************************************************************
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.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that triangular inverse is harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=1024,  achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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, array[0..N-1, 0..N-1].
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   diagonal type (optional):
                * if True, matrix has unit diagonal (a[i,i] are NOT used)
                * if False, matrix diagonal is arbitrary
                * if not given, False is assumed
 
Output parameters:
    Info    -   same as for RMatrixLUInverse
    Rep     -   same as for RMatrixLUInverse
    A       -   same as for RMatrixLUInverse.
 
  -- ALGLIB --
     Copyright 05.02.2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixtrinverse(real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, ae_int_t &info, matinvreport &rep);
void smp_rmatrixtrinverse(real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, ae_int_t &info, matinvreport &rep);
void rmatrixtrinverse(real_2d_array &a, const bool isupper, ae_int_t &info, matinvreport &rep);
void smp_rmatrixtrinverse(real_2d_array &a, const bool isupper, ae_int_t &info, matinvreport &rep);
 
 
/*************************************************************************
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.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that triangular inverse is harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=1024,  achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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, array[0..N-1, 0..N-1].
    N       -   size of matrix A (optional) :
                * if given, only principal NxN submatrix is processed  and
                  overwritten. other elements are unchanged.
                * if not given,  size  is  automatically  determined  from
                  matrix size (A must be square matrix)
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   diagonal type (optional):
                * if True, matrix has unit diagonal (a[i,i] are NOT used)
                * if False, matrix diagonal is arbitrary
                * if not given, False is assumed
 
Output parameters:
    Info    -   same as for RMatrixLUInverse
    Rep     -   same as for RMatrixLUInverse
    A       -   same as for RMatrixLUInverse.
 
  -- ALGLIB --
     Copyright 05.02.2010 by Bochkanov Sergey
*************************************************************************/
void cmatrixtrinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, ae_int_t &info, matinvreport &rep);
void smp_cmatrixtrinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, ae_int_t &info, matinvreport &rep);
void cmatrixtrinverse(complex_2d_array &a, const bool isupper, ae_int_t &info, matinvreport &rep);
void smp_cmatrixtrinverse(complex_2d_array &a, const bool isupper, ae_int_t &info, matinvreport &rep);
 
/*************************************************************************
QR decomposition of a rectangular matrix of size MxN
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that QP decomposition  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=512,   achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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 whose indexes range within [0..M-1, 0..N-1].
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.
 
Output parameters:
    A   -   matrices Q and R in compact form (see below).
    Tau -   array of scalar factors which are used to form
            matrix Q. Array whose index ranges within [0.. Min(M-1,N-1)].
 
Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
MxM, R - upper triangular (or upper trapezoid) matrix of size M x N.
 
The elements of matrix R are located on and above the main diagonal of
matrix A. The elements which are located in Tau array and below the main
diagonal of matrix A are used to form matrix Q as follows:
 
Matrix Q is represented as a product of elementary reflections
 
Q = H(0)*H(2)*...*H(k-1),
 
where k = min(m,n), and each H(i) is in the form
 
H(i) = 1 - tau * v * (v^T)
 
where tau is a scalar stored in Tau[I]; v - real vector,
so that v(0:i-1) = 0, v(i) = 1, v(i+1:m-1) stored in A(i+1:m-1,i).
 
  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixqr(real_2d_array &a, const ae_int_t m, const ae_int_t n, real_1d_array &tau);
void smp_rmatrixqr(real_2d_array &a, const ae_int_t m, const ae_int_t n, real_1d_array &tau);
 
 
/*************************************************************************
LQ decomposition of a rectangular matrix of size MxN
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that QP decomposition  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=512,   achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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 whose indexes range within [0..M-1, 0..N-1].
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.
 
Output parameters:
    A   -   matrices L and Q in compact form (see below)
    Tau -   array of scalar factors which are used to form
            matrix Q. Array whose index ranges within [0..Min(M,N)-1].
 
Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size
MxM, L - lower triangular (or lower trapezoid) matrix of size M x N.
 
The elements of matrix L are located on and below  the  main  diagonal  of
matrix A. The elements which are located in Tau array and above  the  main
diagonal of matrix A are used to form matrix Q as follows:
 
Matrix Q is represented as a product of elementary reflections
 
Q = H(k-1)*H(k-2)*...*H(1)*H(0),
 
where k = min(m,n), and each H(i) is of the form
 
H(i) = 1 - tau * v * (v^T)
 
where tau is a scalar stored in Tau[I]; v - real vector, so that v(0:i-1)=0,
v(i) = 1, v(i+1:n-1) stored in A(i,i+1:n-1).
 
  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixlq(real_2d_array &a, const ae_int_t m, const ae_int_t n, real_1d_array &tau);
void smp_rmatrixlq(real_2d_array &a, const ae_int_t m, const ae_int_t n, real_1d_array &tau);
 
 
/*************************************************************************
QR decomposition of a rectangular complex matrix of size MxN
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that QP decomposition  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=512,   achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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 whose indexes range within [0..M-1, 0..N-1]
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.
 
Output parameters:
    A   -   matrices Q and R in compact form
    Tau -   array of scalar factors which are used to form matrix Q. Array
            whose indexes range within [0.. Min(M,N)-1]
 
Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
MxM, R - upper triangular (or upper trapezoid) matrix of size MxN.
 
  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     September 30, 1994
*************************************************************************/
void cmatrixqr(complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_1d_array &tau);
void smp_cmatrixqr(complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_1d_array &tau);
 
 
/*************************************************************************
LQ decomposition of a rectangular complex matrix of size MxN
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that QP decomposition  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=512,   achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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 whose indexes range within [0..M-1, 0..N-1]
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.
 
Output parameters:
    A   -   matrices Q and L in compact form
    Tau -   array of scalar factors which are used to form matrix Q. Array
            whose indexes range within [0.. Min(M,N)-1]
 
Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size
MxM, L - lower triangular (or lower trapezoid) matrix of size MxN.
 
  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     September 30, 1994
*************************************************************************/
void cmatrixlq(complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_1d_array &tau);
void smp_cmatrixlq(complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_1d_array &tau);
 
 
/*************************************************************************
Partial unpacking of matrix Q from the QR decomposition of a matrix A
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that QP decomposition  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=512,   achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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       -   matrices Q and R in compact form.
                Output of RMatrixQR subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.
    Tau     -   scalar factors which are used to form Q.
                Output of the RMatrixQR subroutine.
    QColumns -  required number of columns of matrix Q. M>=QColumns>=0.
 
Output parameters:
    Q       -   first QColumns columns of matrix Q.
                Array whose indexes range within [0..M-1, 0..QColumns-1].
                If QColumns=0, the array remains unchanged.
 
  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
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);
void smp_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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
Partial unpacking of matrix Q from the LQ decomposition of a matrix A
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that QP decomposition  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=512,   achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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       -   matrices L and Q in compact form.
                Output of RMatrixLQ subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.
    Tau     -   scalar factors which are used to form Q.
                Output of the RMatrixLQ subroutine.
    QRows   -   required number of rows in matrix Q. N>=QRows>=0.
 
Output parameters:
    Q       -   first QRows rows of matrix Q. Array whose indexes range
                within [0..QRows-1, 0..N-1]. If QRows=0, the array remains
                unchanged.
 
  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
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);
void smp_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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
Partial unpacking of matrix Q from QR decomposition of a complex matrix A.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that QP decomposition  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=512,   achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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           -   matrices Q and R in compact form.
                    Output of CMatrixQR subroutine .
    M           -   number of rows in matrix A. M>=0.
    N           -   number of columns in matrix A. N>=0.
    Tau         -   scalar factors which are used to form Q.
                    Output of CMatrixQR subroutine .
    QColumns    -   required number of columns in matrix Q. M>=QColumns>=0.
 
Output parameters:
    Q           -   first QColumns columns of matrix Q.
                    Array whose index ranges within [0..M-1, 0..QColumns-1].
                    If QColumns=0, array isn't changed.
 
  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
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);
void smp_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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
Partial unpacking of matrix Q from LQ decomposition of a complex matrix A.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  ! * multicore support
  !
  ! Intel MKL gives 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.
  !
  ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be:
  ! * about 2-3x faster than ALGLIB for C++ without MKL
  ! * about 7-10x faster than "pure C#" edition of ALGLIB
  ! Difference in performance will be more striking  on  newer  CPU's with
  ! support for newer SIMD instructions. Generally,  MKL  accelerates  any
  ! problem whose size is at least 128, with best  efficiency achieved for
  ! N's larger than 512.
  !
  ! Commercial edition of ALGLIB also supports multithreaded  acceleration
  ! of this function. We should note that QP decomposition  is  harder  to
  ! parallelize than, say, matrix-matrix  product  -  this  algorithm  has
  ! many internal synchronization points which can not be avoided. However
  ! parallelism starts to be profitable starting  from  N=512,   achieving
  ! near-linear speedup for N=4096 or higher.
  !
  ! In order to use multicore features you have to:
  ! * use commercial version of ALGLIB
  ! * call  this  function  with  "smp_"  prefix,  which  indicates  that
  !   multicore code will be used (for multicore support)
  !
  ! 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           -   matrices Q and R in compact form.
                    Output of CMatrixLQ subroutine .
    M           -   number of rows in matrix A. M>=0.
    N           -   number of columns in matrix A. N>=0.
    Tau         -   scalar factors which are used to form Q.
                    Output of CMatrixLQ subroutine .
    QRows       -   required number of rows in matrix Q. N>=QColumns>=0.
 
Output parameters:
    Q           -   first QRows rows of matrix Q.
                    Array whose index ranges within [0..QRows-1, 0..N-1].
                    If QRows=0, array isn't changed.
 
  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
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);
void smp_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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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 version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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  because
  ! bidiagonal decompostion is inherently sequential in nature.
  !
  ! 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);
 
 
/*************************************************************************
Unpacking matrix Q which reduces a matrix to bidiagonal form.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
 
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);
 
 
/*************************************************************************
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 version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
 
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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 version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
  !
  ! 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.
 
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);
 
 
/*************************************************************************
Unpacking matrix Q which reduces matrix A to upper Hessenberg form
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
  !
  ! 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.
 
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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 version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
  !
  ! 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.
 
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);
 
 
/*************************************************************************
Unpacking matrix Q which reduces symmetric matrix to a tridiagonal
form.
 
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
  !
  ! 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.
 
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);
 
 
/*************************************************************************
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 version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
  !
  ! 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.
 
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);
 
 
/*************************************************************************
Unpacking matrix Q which reduces a Hermitian matrix to a real  tridiagonal
form.
 
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
  !
  ! 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.
 
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);
 
 
 
/*************************************************************************
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#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
  !
  ! 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);
 
/*************************************************************************
Singular value decomposition of a rectangular matrix.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
  !
  ! 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 only partially supported (some parts are
  ! optimized, but most - are not).
  !
  ! 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);
bool smp_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);
 
/*************************************************************************
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);
 
 
/*************************************************************************
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(const normestimatorstate &state, const ae_int_t seedval);
 
 
/*************************************************************************
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(const normestimatorstate &state, const sparsematrix &a);
 
 
/*************************************************************************
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);
 
/*************************************************************************
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
 
  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacecreate(const ae_int_t n, const ae_int_t k, eigsubspacestate &state);
 
 
/*************************************************************************
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, const eigsubspacestate &state);
 
 
/*************************************************************************
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(const eigsubspacestate &state, const double eps, const ae_int_t maxits);
 
 
/*************************************************************************
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:
                    * 0 for real  symmetric  matrix  (solver  assumes that
                      matrix  being   processed  is  symmetric;  symmetric
                      direct eigensolver is used for  smaller  subproblems
                      arising during solution of larger "full" task)
                    Future versions of ALGLIB may  introduce  support  for
                    other  matrix   types;   for   now,   only   symmetric
                    eigenproblems are supported.
 
 
  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocstart(const eigsubspacestate &state, const ae_int_t mtype);
 
 
/*************************************************************************
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(const eigsubspacestate &state);
 
 
/*************************************************************************
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(const eigsubspacestate &state, ae_int_t &requesttype, ae_int_t &requestsize);
 
 
/*************************************************************************
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(const eigsubspacestate &state, real_2d_array &x);
 
 
/*************************************************************************
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(const eigsubspacestate &state, const real_2d_array &ax);
 
 
/*************************************************************************
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(const eigsubspacestate &state, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep);
 
 
/*************************************************************************
This  function runs eigensolver for dense NxN symmetric matrix A, given by
upper or lower triangle.
 
This function can not process nonsymmetric matrices.
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes two  important  improvements  of
  ! this function, which can be used from C++ and C#:
  ! * multithreading support
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! For a situation  when  you  need  just  a  few  eigenvectors  (~1-10),
  ! multithreading typically gives sublinear (wrt to cores count) speedup.
  ! For larger  problems  it  may  give  you  nearly  linear  increase  in
  ! performance.
  !
  ! Intel MKL gives 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. Best results are achieved  for  high-dimensional  problems
  ! (NVars is at least 256).
  !
  ! 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:
    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.
 
  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesolvedenses(const eigsubspacestate &state, const real_2d_array &a, const bool isupper, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep);
void smp_eigsubspacesolvedenses(const eigsubspacestate &state, const real_2d_array &a, const bool isupper, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep);
 
 
/*************************************************************************
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(const eigsubspacestate &state, const sparsematrix &a, const bool isupper, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep);
 
 
/*************************************************************************
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 version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
  !
  ! 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.
 
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);
 
 
/*************************************************************************
Subroutine for finding the eigenvalues (and eigenvectors) of  a  symmetric
matrix  in  a  given half open interval (A, B] by using  a  bisection  and
inverse iteration
 
Input parameters:
    A       -   symmetric matrix which is given by its upper or lower
                triangular part. Array [0..N-1, 0..N-1].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.
    IsUpperA -  storage format of matrix A.
    B1, B2 -    half open interval (B1, B2] to search eigenvalues in.
 
Output parameters:
    M       -   number of eigenvalues found in a given half-interval (M>=0).
    W       -   array of the eigenvalues found.
                Array whose index ranges within [0..M-1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains eigenvectors.
                Array whose indexes range within [0..N-1, 0..M-1].
                The eigenvectors are stored in the matrix columns.
 
Result:
    True, if successful. M contains the number of eigenvalues in the given
    half-interval (could be equal to 0), W contains the eigenvalues,
    Z contains the eigenvectors (if needed).
 
    False, if the bisection method subroutine wasn't able to find the
    eigenvalues in the given interval or if the inverse iteration subroutine
    wasn't able to find all the corresponding eigenvectors.
    In that case, the eigenvalues and eigenvectors are not returned,
    M is equal to 0.
 
  -- ALGLIB --
     Copyright 07.01.2006 by Bochkanov Sergey
*************************************************************************/
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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 version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
  !
  ! 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.
 
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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 version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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.
  !
  ! 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.
 
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
Finding eigenvalues and eigenvectors of a general (unsymmetric) matrix
 
COMMERCIAL EDITION OF ALGLIB:
 
  ! Commercial version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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. Speed-up provided by MKL for this particular problem (EVD)
  ! is really high, because  MKL  uses combination of (a) better low-level
  ! optimizations, and (b) better EVD algorithms.
  !
  ! On one particular SSE-capable  machine  for  N=1024,  commercial  MKL-
  ! -capable ALGLIB was:
  ! * 7-10 times faster than open source "generic C" version
  ! * 15-18 times faster than "pure C#" version
  !
  ! 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 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);
 
/*************************************************************************
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#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives 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);
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
 
/*************************************************************************
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);
 
/*************************************************************************
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);
double rmatrixludet(const real_2d_array &a, const integer_1d_array &pivots);
 
 
/*************************************************************************
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);
double rmatrixdet(const real_2d_array &a);
 
 
/*************************************************************************
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);
alglib::complex cmatrixludet(const complex_2d_array &a, const integer_1d_array &pivots);
 
 
/*************************************************************************
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);
alglib::complex cmatrixdet(const complex_2d_array &a);
 
 
/*************************************************************************
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);
double spdmatrixcholeskydet(const real_2d_array &a);
 
 
/*************************************************************************
Determinant calculation of the symmetric positive definite matrix.
 
Input parameters:
    A       -   matrix. Array with elements [0..N-1, 0..N-1].
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)
    IsUpper -   (optional) storage type:
                * if True, symmetric matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't used/changed  by
                  function
                * if False, symmetric matrix  A  is  given  by  its lower
                  triangle, and the upper triangle isn't used/changed  by
                  function
                * if not given, both lower and upper  triangles  must  be
                  filled.
 
Result:
    determinant of matrix A.
    If matrix A is not positive definite, exception is thrown.
 
  -- ALGLIB --
     Copyright 2005-2008 by Bochkanov Sergey
*************************************************************************/
double spdmatrixdet(const real_2d_array &a, const ae_int_t n, const bool isupper);
double spdmatrixdet(const real_2d_array &a);
}
 
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (FUNCTIONS)
//
namespace alglib_impl
{
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 */ ae_vector* ner,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatecrsbuf(ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* ner,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatesks(ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* d,
     /* Integer */ ae_vector* u,
     sparsematrix* s,
     ae_state *_state);
void sparsecreatesksbuf(ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* d,
     /* Integer */ ae_vector* u,
     sparsematrix* s,
     ae_state *_state);
void sparsecopy(sparsematrix* s0, sparsematrix* s1, ae_state *_state);
void sparsecopybuf(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(sparsematrix* s,
     ae_int_t i,
     ae_int_t j,
     ae_state *_state);
double sparsegetdiagonal(sparsematrix* s, ae_int_t i, ae_state *_state);
void sparsemv(sparsematrix* s,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_state *_state);
void sparsemtv(sparsematrix* s,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_state *_state);
void sparsemv2(sparsematrix* s,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y0,
     /* Real    */ ae_vector* y1,
     ae_state *_state);
void sparsesmv(sparsematrix* s,
     ae_bool isupper,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_state *_state);
double sparsevsmv(sparsematrix* s,
     ae_bool isupper,
     /* Real    */ ae_vector* x,
     ae_state *_state);
void sparsemm(sparsematrix* s,
     /* Real    */ ae_matrix* a,
     ae_int_t k,
     /* Real    */ ae_matrix* b,
     ae_state *_state);
void sparsemtm(sparsematrix* s,
     /* Real    */ ae_matrix* a,
     ae_int_t k,
     /* Real    */ ae_matrix* b,
     ae_state *_state);
void sparsemm2(sparsematrix* s,
     /* Real    */ ae_matrix* a,
     ae_int_t k,
     /* Real    */ ae_matrix* b0,
     /* Real    */ ae_matrix* b1,
     ae_state *_state);
void sparsesmm(sparsematrix* s,
     ae_bool isupper,
     /* Real    */ ae_matrix* a,
     ae_int_t k,
     /* Real    */ ae_matrix* b,
     ae_state *_state);
void sparsetrmv(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(sparsematrix* s,
     ae_bool isupper,
     ae_bool isunit,
     ae_int_t optype,
     /* Real    */ ae_vector* x,
     ae_state *_state);
void sparseresizematrix(sparsematrix* s, ae_state *_state);
double sparsegetaveragelengthofchain(sparsematrix* s, ae_state *_state);
ae_bool sparseenumerate(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(sparsematrix* s,
     ae_int_t i,
     /* Real    */ ae_vector* irow,
     ae_state *_state);
void sparsegetcompressedrow(sparsematrix* s,
     ae_int_t i,
     /* Integer */ ae_vector* colidx,
     /* Real    */ ae_vector* vals,
     ae_int_t* nzcnt,
     ae_state *_state);
void sparsetransposesks(sparsematrix* s, ae_state *_state);
void sparseconvertto(sparsematrix* s0, ae_int_t fmt, ae_state *_state);
void sparsecopytobuf(sparsematrix* s0,
     ae_int_t fmt,
     sparsematrix* s1,
     ae_state *_state);
void sparseconverttohash(sparsematrix* s, ae_state *_state);
void sparsecopytohash(sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparsecopytohashbuf(sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparseconverttocrs(sparsematrix* s, ae_state *_state);
void sparsecopytocrs(sparsematrix* s0, sparsematrix* s1, ae_state *_state);
void sparsecopytocrsbuf(sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparseconverttosks(sparsematrix* s, ae_state *_state);
void sparsecopytosks(sparsematrix* s0, sparsematrix* s1, ae_state *_state);
void sparsecopytosksbuf(sparsematrix* s0,
     sparsematrix* s1,
     ae_state *_state);
void sparsecreatecrsinplace(sparsematrix* s, ae_state *_state);
ae_int_t sparsegetmatrixtype(sparsematrix* s, ae_state *_state);
ae_bool sparseishash(sparsematrix* s, ae_state *_state);
ae_bool sparseiscrs(sparsematrix* s, ae_state *_state);
ae_bool sparseissks(sparsematrix* s, ae_state *_state);
void sparsefree(sparsematrix* s, ae_state *_state);
ae_int_t sparsegetnrows(sparsematrix* s, ae_state *_state);
ae_int_t sparsegetncols(sparsematrix* s, ae_state *_state);
ae_int_t sparsegetuppercount(sparsematrix* s, ae_state *_state);
ae_int_t sparsegetlowercount(sparsematrix* s, ae_state *_state);
void _sparsematrix_init(void* _p, ae_state *_state);
void _sparsematrix_init_copy(void* _dst, void* _src, ae_state *_state);
void _sparsematrix_clear(void* _p);
void _sparsematrix_destroy(void* _p);
void _sparsebuffers_init(void* _p, ae_state *_state);
void _sparsebuffers_init_copy(void* _dst, void* _src, ae_state *_state);
void _sparsebuffers_clear(void* _p);
void _sparsebuffers_destroy(void* _p);
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);
void ablassplitlength(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_int_t* n1,
     ae_int_t* n2,
     ae_state *_state);
void ablascomplexsplitlength(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_int_t* n1,
     ae_int_t* n2,
     ae_state *_state);
ae_int_t ablasblocksize(/* Real    */ ae_matrix* a, ae_state *_state);
ae_int_t ablascomplexblocksize(/* Complex */ ae_matrix* a,
     ae_state *_state);
ae_int_t ablasmicroblocksize(ae_state *_state);
void cmatrixtranspose(ae_int_t m,
     ae_int_t n,
     /* Complex */ 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    */ 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 */ 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 rmatrixcopy(ae_int_t m,
     ae_int_t n,
     /* Real    */ 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 cmatrixrank1(ae_int_t m,
     ae_int_t n,
     /* Complex */ ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     /* Complex */ ae_vector* u,
     ae_int_t iu,
     /* Complex */ 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    */ ae_vector* u,
     ae_int_t iu,
     /* Real    */ ae_vector* v,
     ae_int_t iv,
     ae_state *_state);
void cmatrixmv(ae_int_t m,
     ae_int_t n,
     /* Complex */ ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t opa,
     /* Complex */ 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    */ ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t opa,
     /* Real    */ ae_vector* x,
     ae_int_t ix,
     /* Real    */ ae_vector* y,
     ae_int_t iy,
     ae_state *_state);
void cmatrixrighttrsm(ae_int_t m,
     ae_int_t n,
     /* Complex */ 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 _pexec_cmatrixrighttrsm(ae_int_t m,
    ae_int_t n,
    /* Complex */ 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 */ 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 _pexec_cmatrixlefttrsm(ae_int_t m,
    ae_int_t n,
    /* Complex */ 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    */ 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 _pexec_rmatrixrighttrsm(ae_int_t m,
    ae_int_t n,
    /* Real    */ 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    */ 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 _pexec_rmatrixlefttrsm(ae_int_t m,
    ae_int_t n,
    /* Real    */ 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 */ 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 _pexec_cmatrixherk(ae_int_t n,
    ae_int_t k,
    double alpha,
    /* Complex */ 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    */ 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 _pexec_rmatrixsyrk(ae_int_t n,
    ae_int_t k,
    double alpha,
    /* Real    */ 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 */ ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t optypea,
     /* Complex */ 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 _pexec_cmatrixgemm(ae_int_t m,
    ae_int_t n,
    ae_int_t k,
    ae_complex alpha,
    /* Complex */ ae_matrix* a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t optypea,
    /* Complex */ 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    */ ae_matrix* a,
     ae_int_t ia,
     ae_int_t ja,
     ae_int_t optypea,
     /* Real    */ 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 _pexec_rmatrixgemm(ae_int_t m,
    ae_int_t n,
    ae_int_t k,
    double alpha,
    /* Real    */ ae_matrix* a,
    ae_int_t ia,
    ae_int_t ja,
    ae_int_t optypea,
    /* Real    */ 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 */ 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 _pexec_cmatrixsyrk(ae_int_t n,
    ae_int_t k,
    double alpha,
    /* Complex */ 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 rmatrixlu(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Integer */ ae_vector* pivots,
     ae_state *_state);
void _pexec_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);
void _pexec_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 _pexec_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);
ae_bool _pexec_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    */ ae_vector* u,
     ae_state *_state);
void spdmatrixcholeskyupdatefix(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Boolean */ ae_vector* fix,
     ae_state *_state);
void spdmatrixcholeskyupdateadd1buf(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ ae_vector* u,
     /* Real    */ ae_vector* bufr,
     ae_state *_state);
void spdmatrixcholeskyupdatefixbuf(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Boolean */ ae_vector* fix,
     /* Real    */ ae_vector* bufr,
     ae_state *_state);
ae_bool sparsecholeskyskyline(sparsematrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
ae_bool sparsecholeskyx(sparsematrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Integer */ ae_vector* p0,
     /* Integer */ ae_vector* p1,
     ae_int_t ordering,
     ae_int_t algo,
     ae_int_t fmt,
     sparsebuffers* buf,
     sparsematrix* c,
     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);
double rmatrixrcond1(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_state *_state);
double rmatrixrcondinf(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_state *_state);
double spdmatrixrcond(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
double rmatrixtrrcond1(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     ae_state *_state);
double rmatrixtrrcondinf(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     ae_state *_state);
double hpdmatrixrcond(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
double cmatrixrcond1(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_state *_state);
double cmatrixrcondinf(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_state *_state);
double rmatrixlurcond1(/* Real    */ ae_matrix* lua,
     ae_int_t n,
     ae_state *_state);
double rmatrixlurcondinf(/* Real    */ ae_matrix* lua,
     ae_int_t n,
     ae_state *_state);
double spdmatrixcholeskyrcond(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
double hpdmatrixcholeskyrcond(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
double cmatrixlurcond1(/* Complex */ ae_matrix* lua,
     ae_int_t n,
     ae_state *_state);
double cmatrixlurcondinf(/* Complex */ ae_matrix* lua,
     ae_int_t n,
     ae_state *_state);
double cmatrixtrrcond1(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     ae_state *_state);
double cmatrixtrrcondinf(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     ae_state *_state);
double rcondthreshold(ae_state *_state);
void rmatrixluinverse(/* Real    */ ae_matrix* a,
     /* Integer */ ae_vector* pivots,
     ae_int_t n,
     ae_int_t* info,
     matinvreport* rep,
     ae_state *_state);
void _pexec_rmatrixluinverse(/* Real    */ ae_matrix* a,
    /* Integer */ ae_vector* pivots,
    ae_int_t n,
    ae_int_t* info,
    matinvreport* rep, ae_state *_state);
void rmatrixinverse(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_int_t* info,
     matinvreport* rep,
     ae_state *_state);
void _pexec_rmatrixinverse(/* Real    */ ae_matrix* a,
    ae_int_t n,
    ae_int_t* info,
    matinvreport* rep, ae_state *_state);
void cmatrixluinverse(/* Complex */ ae_matrix* a,
     /* Integer */ ae_vector* pivots,
     ae_int_t n,
     ae_int_t* info,
     matinvreport* rep,
     ae_state *_state);
void _pexec_cmatrixluinverse(/* Complex */ ae_matrix* a,
    /* Integer */ ae_vector* pivots,
    ae_int_t n,
    ae_int_t* info,
    matinvreport* rep, ae_state *_state);
void cmatrixinverse(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_int_t* info,
     matinvreport* rep,
     ae_state *_state);
void _pexec_cmatrixinverse(/* Complex */ ae_matrix* a,
    ae_int_t n,
    ae_int_t* info,
    matinvreport* rep, ae_state *_state);
void spdmatrixcholeskyinverse(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_int_t* info,
     matinvreport* rep,
     ae_state *_state);
void _pexec_spdmatrixcholeskyinverse(/* Real    */ ae_matrix* a,
    ae_int_t n,
    ae_bool isupper,
    ae_int_t* info,
    matinvreport* rep, ae_state *_state);
void spdmatrixinverse(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_int_t* info,
     matinvreport* rep,
     ae_state *_state);
void _pexec_spdmatrixinverse(/* Real    */ ae_matrix* a,
    ae_int_t n,
    ae_bool isupper,
    ae_int_t* info,
    matinvreport* rep, ae_state *_state);
void hpdmatrixcholeskyinverse(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_int_t* info,
     matinvreport* rep,
     ae_state *_state);
void _pexec_hpdmatrixcholeskyinverse(/* Complex */ ae_matrix* a,
    ae_int_t n,
    ae_bool isupper,
    ae_int_t* info,
    matinvreport* rep, ae_state *_state);
void hpdmatrixinverse(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_int_t* info,
     matinvreport* rep,
     ae_state *_state);
void _pexec_hpdmatrixinverse(/* Complex */ ae_matrix* a,
    ae_int_t n,
    ae_bool isupper,
    ae_int_t* info,
    matinvreport* rep, ae_state *_state);
void rmatrixtrinverse(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     ae_int_t* info,
     matinvreport* rep,
     ae_state *_state);
void _pexec_rmatrixtrinverse(/* Real    */ ae_matrix* a,
    ae_int_t n,
    ae_bool isupper,
    ae_bool isunit,
    ae_int_t* info,
    matinvreport* rep, ae_state *_state);
void cmatrixtrinverse(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_bool isunit,
     ae_int_t* info,
     matinvreport* rep,
     ae_state *_state);
void _pexec_cmatrixtrinverse(/* Complex */ ae_matrix* a,
    ae_int_t n,
    ae_bool isupper,
    ae_bool isunit,
    ae_int_t* info,
    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,
     ae_state *_state);
void _matinvreport_init(void* _p, ae_state *_state);
void _matinvreport_init_copy(void* _dst, void* _src, ae_state *_state);
void _matinvreport_clear(void* _p);
void _matinvreport_destroy(void* _p);
void rmatrixqr(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* tau,
     ae_state *_state);
void _pexec_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 _pexec_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 _pexec_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 _pexec_cmatrixlq(/* Complex */ ae_matrix* a,
    ae_int_t m,
    ae_int_t n,
    /* Complex */ ae_vector* tau, ae_state *_state);
void rmatrixqrunpackq(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* tau,
     ae_int_t qcolumns,
     /* Real    */ ae_matrix* q,
     ae_state *_state);
void _pexec_rmatrixqrunpackq(/* Real    */ ae_matrix* a,
    ae_int_t m,
    ae_int_t n,
    /* Real    */ ae_vector* tau,
    ae_int_t qcolumns,
    /* Real    */ ae_matrix* q, ae_state *_state);
void rmatrixqrunpackr(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_matrix* r,
     ae_state *_state);
void rmatrixlqunpackq(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* tau,
     ae_int_t qrows,
     /* Real    */ ae_matrix* q,
     ae_state *_state);
void _pexec_rmatrixlqunpackq(/* Real    */ ae_matrix* a,
    ae_int_t m,
    ae_int_t n,
    /* Real    */ ae_vector* tau,
    ae_int_t qrows,
    /* Real    */ ae_matrix* q, ae_state *_state);
void rmatrixlqunpackl(/* Real    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_matrix* l,
     ae_state *_state);
void cmatrixqrunpackq(/* Complex */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Complex */ ae_vector* tau,
     ae_int_t qcolumns,
     /* Complex */ ae_matrix* q,
     ae_state *_state);
void _pexec_cmatrixqrunpackq(/* Complex */ ae_matrix* a,
    ae_int_t m,
    ae_int_t n,
    /* Complex */ ae_vector* tau,
    ae_int_t qcolumns,
    /* Complex */ ae_matrix* q, ae_state *_state);
void cmatrixqrunpackr(/* Complex */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Complex */ ae_matrix* r,
     ae_state *_state);
void cmatrixlqunpackq(/* Complex */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     /* Complex */ ae_vector* tau,
     ae_int_t qrows,
     /* Complex */ ae_matrix* q,
     ae_state *_state);
void _pexec_cmatrixlqunpackq(/* Complex */ ae_matrix* a,
    ae_int_t m,
    ae_int_t n,
    /* Complex */ ae_vector* tau,
    ae_int_t qrows,
    /* Complex */ ae_matrix* q, ae_state *_state);
void cmatrixlqunpackl(/* Complex */ 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    */ ae_matrix* qp,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* tauq,
     ae_int_t qcolumns,
     /* Real    */ ae_matrix* q,
     ae_state *_state);
void rmatrixbdmultiplybyq(/* Real    */ ae_matrix* qp,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ 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    */ ae_matrix* qp,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ ae_vector* taup,
     ae_int_t ptrows,
     /* Real    */ ae_matrix* pt,
     ae_state *_state);
void rmatrixbdmultiplybyp(/* Real    */ ae_matrix* qp,
     ae_int_t m,
     ae_int_t n,
     /* Real    */ 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    */ 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    */ ae_matrix* a,
     ae_int_t n,
     /* Real    */ ae_vector* tau,
     /* Real    */ ae_matrix* q,
     ae_state *_state);
void rmatrixhessenbergunpackh(/* Real    */ 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    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Real    */ 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 */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     /* Complex */ ae_vector* tau,
     /* Complex */ ae_matrix* q,
     ae_state *_state);
void fblscholeskysolve(/* Real    */ 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    */ ae_matrix* a,
     ae_int_t m,
     ae_int_t n,
     double alpha,
     /* Real    */ ae_vector* b,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* buf,
     ae_state *_state);
void fblscgcreate(/* Real    */ ae_vector* x,
     /* Real    */ ae_vector* b,
     ae_int_t n,
     fblslincgstate* state,
     ae_state *_state);
ae_bool fblscgiteration(fblslincgstate* 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);
void _fblslincgstate_init_copy(void* _dst, void* _src, ae_state *_state);
void _fblslincgstate_clear(void* _p);
void _fblslincgstate_destroy(void* _p);
ae_bool rmatrixbdsvd(/* Real    */ ae_vector* d,
     /* Real    */ 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    */ 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 rmatrixsvd(/* Real    */ 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);
ae_bool _pexec_rmatrixsvd(/* Real    */ 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);
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,
     sparsematrix* a,
     ae_state *_state);
void normestimatorresults(normestimatorstate* state,
     double* nrm,
     ae_state *_state);
void normestimatorrestart(normestimatorstate* state, ae_state *_state);
void _normestimatorstate_init(void* _p, ae_state *_state);
void _normestimatorstate_init_copy(void* _dst, void* _src, ae_state *_state);
void _normestimatorstate_clear(void* _p);
void _normestimatorstate_destroy(void* _p);
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 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    */ 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    */ ae_matrix* a,
     ae_bool isupper,
     /* Real    */ ae_vector* w,
     /* Real    */ ae_matrix* z,
     eigsubspacereport* rep,
     ae_state *_state);
void _pexec_eigsubspacesolvedenses(eigsubspacestate* state,
    /* Real    */ ae_matrix* a,
    ae_bool isupper,
    /* Real    */ ae_vector* w,
    /* Real    */ ae_matrix* z,
    eigsubspacereport* rep, ae_state *_state);
void eigsubspacesolvesparses(eigsubspacestate* state,
     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    */ 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    */ 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    */ 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 */ 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 */ 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 */ 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    */ 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    */ 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    */ 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    */ 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);
void _eigsubspacestate_init_copy(void* _dst, void* _src, ae_state *_state);
void _eigsubspacestate_clear(void* _p);
void _eigsubspacestate_destroy(void* _p);
void _eigsubspacereport_init(void* _p, ae_state *_state);
void _eigsubspacereport_init_copy(void* _dst, void* _src, ae_state *_state);
void _eigsubspacereport_clear(void* _p);
void _eigsubspacereport_destroy(void* _p);
ae_bool rmatrixschur(/* Real    */ ae_matrix* a,
     ae_int_t n,
     /* Real    */ ae_matrix* s,
     ae_state *_state);
ae_bool smatrixgevd(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isuppera,
     /* Real    */ 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    */ ae_matrix* b,
     ae_bool isupperb,
     ae_int_t problemtype,
     /* Real    */ ae_matrix* r,
     ae_bool* isupperr,
     ae_state *_state);
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    */ ae_vector* v,
     ae_state *_state);
void rmatrixinvupdatecolumn(/* Real    */ ae_matrix* inva,
     ae_int_t n,
     ae_int_t updcolumn,
     /* Real    */ ae_vector* u,
     ae_state *_state);
void rmatrixinvupdateuv(/* Real    */ ae_matrix* inva,
     ae_int_t n,
     /* Real    */ ae_vector* u,
     /* Real    */ ae_vector* v,
     ae_state *_state);
double rmatrixludet(/* Real    */ ae_matrix* a,
     /* Integer */ ae_vector* pivots,
     ae_int_t n,
     ae_state *_state);
double rmatrixdet(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_state *_state);
ae_complex cmatrixludet(/* Complex */ ae_matrix* a,
     /* Integer */ ae_vector* pivots,
     ae_int_t n,
     ae_state *_state);
ae_complex cmatrixdet(/* Complex */ ae_matrix* a,
     ae_int_t n,
     ae_state *_state);
double spdmatrixcholeskydet(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_state *_state);
double spdmatrixdet(/* Real    */ ae_matrix* a,
     ae_int_t n,
     ae_bool isupper,
     ae_state *_state);
 
}
#endif