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/*************************************************************************
ALGLIB 3.16.0 (source code generated 2019-12-19)
Copyright (c) Sergey Bochkanov (ALGLIB project).
>>> SOURCE LICENSE >>>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation (www.fsf.org); either version 2 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
A copy of the GNU General Public License is available at
http://www.fsf.org/licensing/licenses
>>> END OF LICENSE >>>
*************************************************************************/
#ifndef _linalg_pkg_h
#define _linalg_pkg_h
#include "ap.h"
#include "alglibinternal.h"
#include "alglibmisc.h"
/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (DATATYPES)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
#if defined(AE_COMPILE_SPARSE) || !defined(AE_PARTIAL_BUILD)
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;
#endif
#if defined(AE_COMPILE_ABLAS) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_DLU) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SPTRF) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
ae_int_t nfixed;
ae_int_t ndynamic;
ae_vector idxfirst;
ae_vector strgidx;
ae_vector strgval;
ae_int_t nallocated;
ae_int_t nused;
} sluv2list1matrix;
typedef struct
{
ae_int_t n;
ae_int_t k;
ae_vector nzc;
ae_int_t maxwrkcnt;
ae_int_t maxwrknz;
ae_int_t wrkcnt;
ae_vector wrkset;
ae_vector colid;
ae_vector isdensified;
ae_vector slscolptr;
ae_vector slsrowptr;
ae_vector slsidx;
ae_vector slsval;
ae_int_t slsused;
ae_vector tmp0;
} sluv2sparsetrail;
typedef struct
{
ae_int_t n;
ae_int_t ndense;
ae_matrix d;
ae_vector did;
} sluv2densetrail;
typedef struct
{
ae_int_t n;
sparsematrix sparsel;
sparsematrix sparseut;
sluv2list1matrix bleft;
sluv2list1matrix bupper;
sluv2sparsetrail strail;
sluv2densetrail dtrail;
ae_vector rowpermrawidx;
ae_matrix dbuf;
ae_vector v0i;
ae_vector v1i;
ae_vector v0r;
ae_vector v1r;
ae_vector tmp0;
ae_vector tmpi;
ae_vector tmpp;
} sluv2buffer;
#endif
#if defined(AE_COMPILE_MATGEN) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_TRFAC) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_RCOND) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_MATINV) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
double r1;
double rinf;
} matinvreport;
#endif
#if defined(AE_COMPILE_ORTFAC) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_FBLS) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
double e1;
double e2;
ae_vector x;
ae_vector ax;
double xax;
ae_int_t n;
ae_vector rk;
ae_vector rk1;
ae_vector xk;
ae_vector xk1;
ae_vector pk;
ae_vector pk1;
ae_vector b;
rcommstate rstate;
ae_vector tmp2;
} fblslincgstate;
#endif
#if defined(AE_COMPILE_BDSVD) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SVD) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_NORMESTIMATOR) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
ae_int_t n;
ae_int_t m;
ae_int_t nstart;
ae_int_t nits;
ae_int_t seedval;
ae_vector x0;
ae_vector x1;
ae_vector t;
ae_vector xbest;
hqrndstate r;
ae_vector x;
ae_vector mv;
ae_vector mtv;
ae_bool needmv;
ae_bool needmtv;
double repnorm;
rcommstate rstate;
} normestimatorstate;
#endif
#if defined(AE_COMPILE_HSSCHUR) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_EVD) || !defined(AE_PARTIAL_BUILD)
typedef struct
{
ae_int_t n;
ae_int_t k;
ae_int_t nwork;
ae_int_t maxits;
double eps;
ae_int_t eigenvectorsneeded;
ae_int_t matrixtype;
ae_bool usewarmstart;
ae_bool firstcall;
hqrndstate rs;
ae_bool running;
ae_vector tau;
ae_matrix q0;
ae_matrix qcur;
ae_matrix qnew;
ae_matrix znew;
ae_matrix r;
ae_matrix rz;
ae_matrix tz;
ae_matrix rq;
ae_matrix dummy;
ae_vector rw;
ae_vector tw;
ae_vector wcur;
ae_vector wprev;
ae_vector wrank;
apbuffers buf;
ae_matrix x;
ae_matrix ax;
ae_int_t requesttype;
ae_int_t requestsize;
ae_int_t repiterationscount;
rcommstate rstate;
} eigsubspacestate;
typedef struct
{
ae_int_t iterationscount;
} eigsubspacereport;
#endif
#if defined(AE_COMPILE_SCHUR) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SPDGEVD) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_INVERSEUPDATE) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_MATDET) || !defined(AE_PARTIAL_BUILD)
#endif
}
/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS C++ INTERFACE
//
/////////////////////////////////////////////////////////////////////////
namespace alglib
{
#if defined(AE_COMPILE_SPARSE) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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();
};
#endif
#if defined(AE_COMPILE_ABLAS) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_DLU) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SPTRF) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_MATGEN) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_TRFAC) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_RCOND) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_MATINV) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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;
};
#endif
#if defined(AE_COMPILE_ORTFAC) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_FBLS) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_BDSVD) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SVD) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_NORMESTIMATOR) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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();
};
#endif
#if defined(AE_COMPILE_HSSCHUR) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_EVD) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
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;
};
#endif
#if defined(AE_COMPILE_SCHUR) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SPDGEVD) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_INVERSEUPDATE) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_MATDET) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SPARSE) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
This function creates sparse matrix in a Hash-Table format.
This function creates Hast-Table matrix, which can be converted to CRS
format after its initialization is over. Typical usage scenario for a
sparse matrix is:
1. creation in a Hash-Table format
2. insertion of the matrix elements
3. conversion to the CRS representation
4. matrix is passed to some linear algebra algorithm
Some information about different matrix formats can be found below, in
the "NOTES" section.
INPUT PARAMETERS
M - number of rows in a matrix, M>=1
N - number of columns in a matrix, N>=1
K - K>=0, expected number of non-zero elements in a matrix.
K can be inexact approximation, can be less than actual
number of elements (table will grow when needed) or
even zero).
It is important to understand that although hash-table
may grow automatically, it is better to provide good
estimate of data size.
OUTPUT PARAMETERS
S - sparse M*N matrix in Hash-Table representation.
All elements of the matrix are zero.
NOTE 1
Hash-tables use memory inefficiently, and they have to keep some amount
of the "spare memory" in order to have good performance. Hash table for
matrix with K non-zero elements will need C*K*(8+2*sizeof(int)) bytes,
where C is a small constant, about 1.5-2 in magnitude.
CRS storage, from the other side, is more memory-efficient, and needs
just K*(8+sizeof(int))+M*sizeof(int) bytes, where M is a number of rows
in a matrix.
When you convert from the Hash-Table to CRS representation, all unneeded
memory will be freed.
NOTE 2
Comments of SparseMatrix structure outline information about different
sparse storage formats. We recommend you to read them before starting to
use ALGLIB sparse matrices.
NOTE 3
This function completely overwrites S with new sparse matrix. Previously
allocated storage is NOT reused. If you want to reuse already allocated
memory, call SparseCreateBuf function.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecreate(const ae_int_t m, const ae_int_t n, const ae_int_t k, sparsematrix &s, const xparams _xparams = alglib::xdefault);
void sparsecreate(const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This version of SparseCreate function creates sparse matrix in Hash-Table
format, reusing previously allocated storage as much as possible. Read
comments for SparseCreate() for more information.
INPUT PARAMETERS
M - number of rows in a matrix, M>=1
N - number of columns in a matrix, N>=1
K - K>=0, expected number of non-zero elements in a matrix.
K can be inexact approximation, can be less than actual
number of elements (table will grow when needed) or
even zero).
It is important to understand that although hash-table
may grow automatically, it is better to provide good
estimate of data size.
S - SparseMatrix structure which MAY contain some already
allocated storage.
OUTPUT PARAMETERS
S - sparse M*N matrix in Hash-Table representation.
All elements of the matrix are zero.
Previously allocated storage is reused, if its size
is compatible with expected number of non-zeros K.
-- ALGLIB PROJECT --
Copyright 14.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatebuf(const ae_int_t m, const ae_int_t n, const ae_int_t k, const sparsematrix &s, const xparams _xparams = alglib::xdefault);
void sparsecreatebuf(const ae_int_t m, const ae_int_t n, const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function creates sparse matrix in a CRS format (expert function for
situations when you are running out of memory).
This 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function creates sparse matrix in a CRS format (expert function for
situations when you are running out of memory). This version of CRS
matrix creation function may reuse memory already allocated in S.
This function creates CRS matrix. Typical usage scenario for a CRS matrix
is:
1. creation (you have to tell number of non-zero elements at each row at
this moment)
2. insertion of the matrix elements (row by row, from left to right)
3. matrix is passed to some linear algebra algorithm
This function is a memory-efficient alternative to SparseCreate(), but it
is more complex because it requires you to know in advance how large your
matrix is. Some information about different matrix formats can be found
in comments on SparseMatrix structure. We recommend you to read them
before starting to use ALGLIB sparse matrices..
INPUT PARAMETERS
M - number of rows in a matrix, M>=1
N - number of columns in a matrix, N>=1
NER - number of elements at each row, array[M], NER[I]>=0
S - sparse matrix structure with possibly preallocated
memory.
OUTPUT PARAMETERS
S - sparse M*N matrix in CRS representation.
You have to fill ALL non-zero elements by calling
SparseSet() BEFORE you try to use this matrix.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsbuf(const ae_int_t m, const ae_int_t n, const integer_1d_array &ner, const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function creates sparse matrix in a SKS format (skyline storage
format). In most cases you do not need this function - CRS format better
suits most use cases.
INPUT PARAMETERS
M, N - number of rows(M) and columns (N) in a matrix:
* M=N (as for now, ALGLIB supports only square SKS)
* N>=1
* M>=1
D - "bottom" bandwidths, array[M], D[I]>=0.
I-th element stores number of non-zeros at I-th row,
below the diagonal (diagonal itself is not included)
U - "top" bandwidths, array[N], U[I]>=0.
I-th element stores number of non-zeros at I-th row,
above the diagonal (diagonal itself is not included)
OUTPUT PARAMETERS
S - sparse M*N matrix in SKS representation.
All elements are filled by zeros.
You may use sparseset() to change their values.
NOTE: this function completely overwrites S with new sparse matrix.
Previously allocated storage is NOT reused. If you want to reuse
already allocated memory, call SparseCreateSKSBuf function.
-- ALGLIB PROJECT --
Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesks(const ae_int_t m, const ae_int_t n, const integer_1d_array &d, const integer_1d_array &u, sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This is "buffered" version of SparseCreateSKS() which reuses memory
previously allocated in S (of course, memory is reallocated if needed).
This function creates sparse matrix in a SKS format (skyline storage
format). In most cases you do not need this function - CRS format better
suits most use cases.
INPUT PARAMETERS
M, N - number of rows(M) and columns (N) in a matrix:
* M=N (as for now, ALGLIB supports only square SKS)
* N>=1
* M>=1
D - "bottom" bandwidths, array[M], 0<=D[I]<=I.
I-th element stores number of non-zeros at I-th row,
below the diagonal (diagonal itself is not included)
U - "top" bandwidths, array[N], 0<=U[I]<=I.
I-th element stores number of non-zeros at I-th row,
above the diagonal (diagonal itself is not included)
OUTPUT PARAMETERS
S - sparse M*N matrix in SKS representation.
All elements are filled by zeros.
You may use sparseset() to change their values.
-- ALGLIB PROJECT --
Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesksbuf(const ae_int_t m, const ae_int_t n, const integer_1d_array &d, const integer_1d_array &u, const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function creates sparse matrix in a SKS format (skyline storage
format). Unlike more general sparsecreatesks(), this function creates
sparse matrix with constant bandwidth.
You may want to use this function instead of sparsecreatesks() when your
matrix has constant or nearly-constant bandwidth, and you want to
simplify source code.
INPUT PARAMETERS
M, N - number of rows(M) and columns (N) in a matrix:
* M=N (as for now, ALGLIB supports only square SKS)
* N>=1
* M>=1
BW - matrix bandwidth, BW>=0
OUTPUT PARAMETERS
S - sparse M*N matrix in SKS representation.
All elements are filled by zeros.
You may use sparseset() to change their values.
NOTE: this function completely overwrites S with new sparse matrix.
Previously allocated storage is NOT reused. If you want to reuse
already allocated memory, call sparsecreatesksbandbuf function.
-- ALGLIB PROJECT --
Copyright 25.12.2017 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesksband(const ae_int_t m, const ae_int_t n, const ae_int_t bw, sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This is "buffered" version of sparsecreatesksband() which reuses memory
previously allocated in S (of course, memory is reallocated if needed).
You may want to use this function instead of sparsecreatesksbuf() when
your matrix has constant or nearly-constant bandwidth, and you want to
simplify source code.
INPUT PARAMETERS
M, N - number of rows(M) and columns (N) in a matrix:
* M=N (as for now, ALGLIB supports only square SKS)
* N>=1
* M>=1
BW - bandwidth, BW>=0
OUTPUT PARAMETERS
S - sparse M*N matrix in SKS representation.
All elements are filled by zeros.
You may use sparseset() to change their values.
-- ALGLIB PROJECT --
Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesksbandbuf(const ae_int_t m, const ae_int_t n, const ae_int_t bw, const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function copies S0 to S1.
This function completely deallocates memory owned by S1 before creating a
copy of S0. If you want to reuse memory, use SparseCopyBuf.
NOTE: this function does not verify its arguments, it just copies all
fields of the structure.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecopy(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function efficiently swaps contents of S0 and S1.
-- ALGLIB PROJECT --
Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparseswap(const sparsematrix &s0, const sparsematrix &s1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function adds value to S[i,j] - element of the sparse matrix. Matrix
must be in a Hash-Table mode.
In case S[i,j] already exists in the table, V i added to its value. In
case S[i,j] is non-existent, it is inserted in the table. Table
automatically grows when necessary.
INPUT PARAMETERS
S - sparse M*N matrix in Hash-Table representation.
Exception will be thrown for CRS matrix.
I - row index of the element to modify, 0<=I<M
J - column index of the element to modify, 0<=J<N
V - value to add, must be finite number
OUTPUT PARAMETERS
S - modified matrix
NOTE 1: when S[i,j] is exactly zero after modification, it is deleted
from the table.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparseadd(const sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function modifies S[i,j] - element of the sparse matrix.
For Hash-based storage format:
* this function can be called at any moment - during matrix initialization
or later
* new value can be zero or non-zero. In case new value of S[i,j] is zero,
this element is deleted from the table.
* this function has no effect when called with zero V for non-existent
element.
For CRS-bases storage format:
* this function can be called ONLY DURING MATRIX INITIALIZATION
* zero values are stored in the matrix similarly to non-zero ones
* elements must be initialized in correct order - from top row to bottom,
within row - from left to right.
For SKS storage:
* this function can be called at any moment - during matrix initialization
or later
* zero values are stored in the matrix similarly to non-zero ones
* this function CAN NOT be called for non-existent (outside of the band
specified during SKS matrix creation) elements. Say, if you created SKS
matrix with bandwidth=2 and tried to call sparseset(s,0,10,VAL), an
exception will be generated.
INPUT PARAMETERS
S - sparse M*N matrix in Hash-Table, SKS or CRS format.
I - row index of the element to modify, 0<=I<M
J - column index of the element to modify, 0<=J<N
V - value to set, must be finite number, can be zero
OUTPUT PARAMETERS
S - modified matrix
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparseset(const sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function returns S[i,j] - element of the sparse matrix. Matrix can
be in any mode (Hash-Table, CRS, SKS), but this function is less efficient
for CRS matrices. Hash-Table and SKS matrices can find element in O(1)
time, while CRS matrices need O(log(RS)) time, where RS is an number of
non-zero elements in a row.
INPUT PARAMETERS
S - sparse M*N matrix 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function returns I-th diagonal element of the sparse matrix.
Matrix can be in any mode (Hash-Table or CRS storage), but this function
is most efficient for CRS matrices - it requires less than 50 CPU cycles
to extract diagonal element. For Hash-Table matrices we still have O(1)
query time, but function is many times slower.
INPUT PARAMETERS
S - sparse M*N matrix in Hash-Table representation.
Exception will be thrown for CRS matrix.
I - index of the element to 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function calculates matrix-vector product S*x. Matrix S must be
stored in CRS or SKS format (exception will be thrown otherwise).
INPUT PARAMETERS
S - sparse M*N matrix in CRS or SKS format.
X - array[N], input vector. For performance reasons we
make only quick checks - we check that array size is
at least N, but we do not check for NAN's or INF's.
Y - output buffer, possibly preallocated. In case buffer
size is too small to store result, this buffer is
automatically resized.
OUTPUT PARAMETERS
Y - array[M], S*x
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before using
this function.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemv(const sparsematrix &s, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function calculates matrix-vector product S^T*x. Matrix S must be
stored in CRS or SKS format (exception will be thrown otherwise).
INPUT PARAMETERS
S - sparse M*N matrix in CRS or SKS format.
X - array[M], input vector. For performance reasons we
make only quick checks - we check that array size is
at least M, but we do not check for NAN's or INF's.
Y - output buffer, possibly preallocated. In case buffer
size is too small to store result, this buffer is
automatically resized.
OUTPUT PARAMETERS
Y - array[N], S^T*x
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before using
this function.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemtv(const sparsematrix &s, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function calculates generalized sparse matrix-vector product
y := alpha*op(S)*x + beta*y
Matrix S must be stored in CRS or SKS format (exception will be thrown
otherwise). op(S) can be either S or S^T.
NOTE: this function expects Y to be large enough to store result. No
automatic preallocation happens for smaller arrays.
INPUT PARAMETERS
S - sparse matrix in CRS or SKS format.
Alpha - source coefficient
OpS - operation type:
* OpS=0 => op(S) = S
* OpS=1 => op(S) = S^T
X - input vector, must have at least Cols(op(S))+IX elements
IX - subvector offset
Beta - destination coefficient
Y - preallocated output array, must have at least Rows(op(S))+IY elements
IY - subvector offset
OUTPUT PARAMETERS
Y - elements [IY...IY+Rows(op(S))-1] are replaced by result,
other elements are not modified
HANDLING OF SPECIAL CASES:
* below M=Rows(op(S)) and N=Cols(op(S)). Although current ALGLIB version
does not allow you to create zero-sized sparse matrices, internally
ALGLIB can deal with such matrices. So, comments for M or N equal to
zero are for internal use only.
* if M=0, then subroutine does nothing. It does not even touch arrays.
* if N=0 or Alpha=0.0, then:
* if Beta=0, then Y is filled by zeros. S and X are not referenced at
all. Initial values of Y are ignored (we do not multiply Y by zero,
we just rewrite it by zeros)
* if Beta<>0, then Y is replaced by Beta*Y
* if M>0, N>0, Alpha<>0, but Beta=0, then Y is replaced by alpha*op(S)*x
initial state of Y is ignored (rewritten without initial multiplication
by zeros).
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before using
this function.
-- ALGLIB PROJECT --
Copyright 10.12.2019 by Bochkanov Sergey
*************************************************************************/
void sparsegemv(const sparsematrix &s, const double alpha, const ae_int_t ops, const real_1d_array &x, const ae_int_t ix, const double beta, const real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function simultaneously calculates two matrix-vector products:
S*x and S^T*x.
S must be square (non-rectangular) matrix stored in CRS or SKS format
(exception will be thrown otherwise).
INPUT PARAMETERS
S - sparse N*N matrix in CRS or SKS format.
X - array[N], input vector. For performance reasons we
make only quick checks - we check that array size is
at least N, but we do not check for NAN's or INF's.
Y0 - output buffer, possibly preallocated. In case buffer
size is too small to store result, this buffer is
automatically resized.
Y1 - output buffer, possibly preallocated. In case buffer
size is too small to store result, this buffer is
automatically resized.
OUTPUT PARAMETERS
Y0 - array[N], S*x
Y1 - array[N], S^T*x
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before using
this function.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemv2(const sparsematrix &s, const real_1d_array &x, real_1d_array &y0, real_1d_array &y1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function calculates matrix-vector product S*x, when S is symmetric
matrix. Matrix S must be stored in CRS or SKS format (exception will be
thrown otherwise).
INPUT PARAMETERS
S - sparse M*M matrix in CRS or SKS format.
IsUpper - whether upper or lower triangle of S is given:
* if upper triangle is given, only S[i,j] for j>=i
are used, and lower triangle is ignored (it can be
empty - these elements are not referenced at all).
* if lower triangle is given, only S[i,j] for j<=i
are used, and upper triangle is ignored.
X - array[N], input vector. For performance reasons we
make only quick checks - we check that array size is
at least N, but we do not check for NAN's or INF's.
Y - output buffer, possibly preallocated. In case buffer
size is too small to store result, this buffer is
automatically resized.
OUTPUT PARAMETERS
Y - array[M], S*x
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before using
this function.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsesmv(const sparsematrix &s, const bool isupper, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function calculates vector-matrix-vector product x'*S*x, where S is
symmetric matrix. Matrix S must be stored in CRS or SKS format (exception
will be thrown otherwise).
INPUT PARAMETERS
S - sparse M*M matrix in CRS or SKS format.
IsUpper - whether upper or lower triangle of S is given:
* if upper triangle is given, only S[i,j] for j>=i
are used, and lower triangle is ignored (it can be
empty - these elements are not referenced at all).
* if lower triangle is given, only S[i,j] for j<=i
are used, and upper triangle is ignored.
X - array[N], input vector. For performance reasons we
make only quick checks - we check that array size is
at least N, but we do not check for NAN's or INF's.
RESULT
x'*S*x
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before using
this function.
-- ALGLIB PROJECT --
Copyright 27.01.2014 by Bochkanov Sergey
*************************************************************************/
double sparsevsmv(const sparsematrix &s, const bool isupper, const real_1d_array &x, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function calculates matrix-matrix product S*A. Matrix S must be
stored in CRS or SKS format (exception will be thrown otherwise).
INPUT PARAMETERS
S - sparse M*N matrix in CRS or SKS format.
A - array[N][K], input dense matrix. For performance reasons
we make only quick checks - we check that array size
is at least N, but we do not check for NAN's or INF's.
K - number of columns of matrix (A).
B - output buffer, possibly preallocated. In case buffer
size is too small to store result, this buffer is
automatically resized.
OUTPUT PARAMETERS
B - array[M][K], S*A
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before using
this function.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemm(const sparsematrix &s, const real_2d_array &a, const ae_int_t k, real_2d_array &b, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function calculates matrix-matrix product S^T*A. Matrix S must be
stored in CRS or SKS format (exception will be thrown otherwise).
INPUT PARAMETERS
S - sparse M*N matrix in CRS or SKS format.
A - array[M][K], input dense matrix. For performance reasons
we make only quick checks - we check that array size is
at least M, but we do not check for NAN's or INF's.
K - number of columns of matrix (A).
B - output buffer, possibly preallocated. In case buffer
size is too small to store result, this buffer is
automatically resized.
OUTPUT PARAMETERS
B - array[N][K], S^T*A
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before using
this function.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemtm(const sparsematrix &s, const real_2d_array &a, const ae_int_t k, real_2d_array &b, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function simultaneously calculates two matrix-matrix products:
S*A and S^T*A.
S must be square (non-rectangular) matrix stored in CRS or SKS format
(exception will be thrown otherwise).
INPUT PARAMETERS
S - sparse N*N matrix in CRS or SKS format.
A - array[N][K], input dense matrix. For performance reasons
we make only quick checks - we check that array size is
at least N, but we do not check for NAN's or INF's.
K - number of columns of matrix (A).
B0 - output buffer, possibly preallocated. In case buffer
size is too small to store result, this buffer is
automatically resized.
B1 - output buffer, possibly preallocated. In case buffer
size is too small to store result, this buffer is
automatically resized.
OUTPUT PARAMETERS
B0 - array[N][K], S*A
B1 - array[N][K], S^T*A
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before using
this function.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemm2(const sparsematrix &s, const real_2d_array &a, const ae_int_t k, real_2d_array &b0, real_2d_array &b1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function calculates matrix-matrix product S*A, when S is symmetric
matrix. Matrix S must be stored in CRS or SKS format (exception will be
thrown otherwise).
INPUT PARAMETERS
S - sparse M*M matrix in CRS or SKS format.
IsUpper - whether upper or lower triangle of S is given:
* if upper triangle is given, only S[i,j] for j>=i
are used, and lower triangle is ignored (it can be
empty - these elements are not referenced at all).
* if lower triangle is given, only S[i,j] for j<=i
are used, and upper triangle is ignored.
A - array[N][K], input dense matrix. For performance reasons
we make only quick checks - we check that array size is
at least N, but we do not check for NAN's or INF's.
K - number of columns of matrix (A).
B - output buffer, possibly preallocated. In case buffer
size is too small to store result, this buffer is
automatically resized.
OUTPUT PARAMETERS
B - array[M][K], S*A
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before using
this function.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsesmm(const sparsematrix &s, const bool isupper, const real_2d_array &a, const ae_int_t k, real_2d_array &b, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function calculates matrix-vector product op(S)*x, when x is vector,
S is symmetric triangular matrix, op(S) is transposition or no operation.
Matrix S must be stored in CRS or SKS format (exception will be thrown
otherwise).
INPUT PARAMETERS
S - sparse square matrix in CRS or SKS format.
IsUpper - whether upper or lower triangle of S is used:
* if upper triangle is given, only S[i,j] for j>=i
are used, and lower triangle is ignored (it can be
empty - these elements are not referenced at all).
* if lower triangle is given, only S[i,j] for j<=i
are used, and upper triangle is ignored.
IsUnit - unit or non-unit diagonal:
* if True, diagonal elements of triangular matrix are
considered equal to 1.0. Actual elements stored in
S are not referenced at all.
* if False, diagonal stored in S is used
OpType - operation type:
* if 0, S*x is calculated
* if 1, (S^T)*x is calculated (transposition)
X - array[N] which stores input vector. For performance
reasons we make only quick checks - we check that
array size is at least N, but we do not check for
NAN's or INF's.
Y - possibly preallocated input buffer. Automatically
resized if its size is too small.
OUTPUT PARAMETERS
Y - array[N], op(S)*x
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before using
this function.
-- ALGLIB PROJECT --
Copyright 20.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsetrmv(const sparsematrix &s, const bool isupper, const bool isunit, const ae_int_t optype, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function solves linear system op(S)*y=x where x is vector, S is
symmetric triangular matrix, op(S) is transposition or no operation.
Matrix S must be stored in CRS or SKS format (exception will be thrown
otherwise).
INPUT PARAMETERS
S - sparse square matrix in CRS or SKS format.
IsUpper - whether upper or lower triangle of S is used:
* if upper triangle is given, only S[i,j] for j>=i
are used, and lower triangle is ignored (it can be
empty - these elements are not referenced at all).
* if lower triangle is given, only S[i,j] for j<=i
are used, and upper triangle is ignored.
IsUnit - unit or non-unit diagonal:
* if True, diagonal elements of triangular matrix are
considered equal to 1.0. Actual elements stored in
S are not referenced at all.
* if False, diagonal stored in S is used. It is your
responsibility to make sure that diagonal is
non-zero.
OpType - operation type:
* if 0, S*x is calculated
* if 1, (S^T)*x is calculated (transposition)
X - array[N] which stores input vector. For performance
reasons we make only quick checks - we check that
array size is at least N, but we do not check for
NAN's or INF's.
OUTPUT PARAMETERS
X - array[N], inv(op(S))*x
NOTE: this function throws exception when called for non-CRS/SKS matrix.
You must convert your matrix with SparseConvertToCRS/SKS() before
using this function.
NOTE: no assertion or tests are done during algorithm operation. It is
your responsibility to provide invertible matrix to algorithm.
-- ALGLIB PROJECT --
Copyright 20.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsetrsv(const sparsematrix &s, const bool isupper, const bool isunit, const ae_int_t optype, const real_1d_array &x, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This procedure resizes Hash-Table matrix. It can be called when you have
deleted too many elements from the matrix, and you want to free unneeded
memory.
-- ALGLIB PROJECT --
Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparseresizematrix(const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function is used to enumerate all elements of the sparse matrix.
Before first call user initializes T0 and T1 counters by zero. These
counters are used to remember current position in a matrix; after each
call they are updated by the function.
Subsequent calls to this function return non-zero elements of the sparse
matrix, one by one. If you enumerate CRS matrix, matrix is traversed from
left to right, from top to bottom. In case you enumerate matrix stored as
Hash table, elements are returned in random order.
EXAMPLE
> T0=0
> T1=0
> while SparseEnumerate(S,T0,T1,I,J,V) do
> ....do something with I,J,V
INPUT PARAMETERS
S - sparse M*N matrix in Hash-Table or CRS representation.
T0 - internal counter
T1 - internal counter
OUTPUT PARAMETERS
T0 - new value of the internal counter
T1 - new value of the internal counter
I - row index of non-zero element, 0<=I<M.
J - column index of non-zero element, 0<=J<N
V - value of the T-th element
RESULT
True in case of success (next non-zero element was retrieved)
False in case all non-zero elements were enumerated
NOTE: you may call SparseRewriteExisting() during enumeration, but it is
THE ONLY matrix modification function you can call!!! Other
matrix modification functions should not be called during enumeration!
-- ALGLIB PROJECT --
Copyright 14.03.2012 by Bochkanov Sergey
*************************************************************************/
bool sparseenumerate(const sparsematrix &s, ae_int_t &t0, ae_int_t &t1, ae_int_t &i, ae_int_t &j, double &v, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function rewrites existing (non-zero) element. It returns True if
element exists or False, when it is called for non-existing (zero)
element.
This function works with any kind of the matrix.
The purpose of this function is to provide convenient thread-safe way to
modify sparse matrix. Such modification (already existing element is
rewritten) is guaranteed to be thread-safe without any synchronization, as
long as different threads modify different elements.
INPUT PARAMETERS
S - sparse M*N matrix in any kind of representation
(Hash, SKS, CRS).
I - row index of non-zero element to modify, 0<=I<M
J - column index of non-zero element to modify, 0<=J<N
V - value to rewrite, must be finite number
OUTPUT PARAMETERS
S - modified matrix
RESULT
True in case when element exists
False in case when element doesn't exist or it is zero
-- ALGLIB PROJECT --
Copyright 14.03.2012 by Bochkanov Sergey
*************************************************************************/
bool sparserewriteexisting(const sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function returns I-th row of the sparse matrix. Matrix must be stored
in CRS or SKS format.
INPUT PARAMETERS:
S - sparse M*N matrix in CRS format
I - row index, 0<=I<M
IRow - output buffer, can be preallocated. In case buffer
size is too small to store I-th row, it is
automatically reallocated.
OUTPUT PARAMETERS:
IRow - array[M], I-th row.
NOTE: this function has O(N) running time, where N is a column count. It
allocates and fills N-element array, even although most of its
elemets are zero.
NOTE: If you have O(non-zeros-per-row) time and memory requirements, use
SparseGetCompressedRow() function. It returns data in compressed
format.
NOTE: when incorrect I (outside of [0,M-1]) or matrix (non CRS/SKS)
is passed, this function throws exception.
-- ALGLIB PROJECT --
Copyright 10.12.2014 by Bochkanov Sergey
*************************************************************************/
void sparsegetrow(const sparsematrix &s, const ae_int_t i, real_1d_array &irow, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function returns I-th row of the sparse matrix IN COMPRESSED FORMAT -
only non-zero elements are returned (with their indexes). Matrix must be
stored in CRS or SKS format.
INPUT PARAMETERS:
S - sparse M*N matrix in CRS format
I - row index, 0<=I<M
ColIdx - output buffer for column indexes, can be preallocated.
In case buffer size is too small to store I-th row, it
is automatically reallocated.
Vals - output buffer for values, can be preallocated. In case
buffer size is too small to store I-th row, it is
automatically reallocated.
OUTPUT PARAMETERS:
ColIdx - column indexes of non-zero elements, sorted by
ascending. Symbolically non-zero elements are counted
(i.e. if you allocated place for element, but it has
zero numerical value - it is counted).
Vals - values. Vals[K] stores value of matrix element with
indexes (I,ColIdx[K]). Symbolically non-zero elements
are counted (i.e. if you allocated place for element,
but it has zero numerical value - it is counted).
NZCnt - number of symbolically non-zero elements per row.
NOTE: when incorrect I (outside of [0,M-1]) or matrix (non CRS/SKS)
is passed, this function throws exception.
NOTE: this function may allocate additional, unnecessary place for ColIdx
and Vals arrays. It is dictated by performance reasons - on SKS
matrices it is faster to allocate space at the beginning with
some "extra"-space, than performing two passes over matrix - first
time to calculate exact space required for data, second time - to
store data itself.
-- ALGLIB PROJECT --
Copyright 10.12.2014 by Bochkanov Sergey
*************************************************************************/
void sparsegetcompressedrow(const sparsematrix &s, const ae_int_t i, integer_1d_array &colidx, real_1d_array &vals, ae_int_t &nzcnt, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs transpose of CRS matrix.
INPUT PARAMETERS
S - sparse matrix in CRS format.
OUTPUT PARAMETERS
S - sparse matrix, transposed.
NOTE: internal temporary copy is allocated for the purposes of
transposition. It is deallocated after transposition.
-- ALGLIB PROJECT --
Copyright 30.01.2018 by Bochkanov Sergey
*************************************************************************/
void sparsetransposecrs(const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs copying with transposition of CRS matrix.
INPUT PARAMETERS
S0 - sparse matrix in CRS format.
OUTPUT PARAMETERS
S1 - sparse matrix, transposed
-- ALGLIB PROJECT --
Copyright 23.07.2018 by Bochkanov Sergey
*************************************************************************/
void sparsecopytransposecrs(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs copying with transposition of CRS matrix (buffered
version which reuses memory already allocated by the target as much as
possible).
INPUT PARAMETERS
S0 - sparse matrix in CRS format.
OUTPUT PARAMETERS
S1 - sparse matrix, transposed; previously allocated memory is
reused if possible.
-- ALGLIB PROJECT --
Copyright 23.07.2018 by Bochkanov Sergey
*************************************************************************/
void sparsecopytransposecrsbuf(const sparsematrix &s0, const sparsematrix &s1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs in-place conversion to desired sparse storage
format.
INPUT PARAMETERS
S0 - sparse matrix in any format.
Fmt - desired storage format of the output, as returned by
SparseGetMatrixType() function:
* 0 for hash-based storage
* 1 for CRS
* 2 for SKS
OUTPUT PARAMETERS
S0 - sparse matrix in requested format.
NOTE: in-place conversion wastes a lot of memory which is used to store
temporaries. If you perform a lot of repeated conversions, we
recommend to use out-of-place buffered conversion functions, like
SparseCopyToBuf(), which can reuse already allocated memory.
-- ALGLIB PROJECT --
Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparseconvertto(const sparsematrix &s0, const ae_int_t fmt, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs out-of-place conversion to desired sparse storage
format. S0 is copied to S1 and converted on-the-fly. Memory allocated in
S1 is reused to maximum extent possible.
INPUT PARAMETERS
S0 - sparse matrix in any format.
Fmt - desired storage format of the output, as returned by
SparseGetMatrixType() function:
* 0 for hash-based storage
* 1 for CRS
* 2 for SKS
OUTPUT PARAMETERS
S1 - sparse matrix in requested format.
-- ALGLIB PROJECT --
Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecopytobuf(const sparsematrix &s0, const ae_int_t fmt, const sparsematrix &s1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs in-place conversion to Hash table storage.
INPUT PARAMETERS
S - sparse matrix in CRS format.
OUTPUT PARAMETERS
S - sparse matrix in Hash table format.
NOTE: this function has no effect when called with matrix which is
already in Hash table mode.
NOTE: in-place conversion involves allocation of temporary arrays. If you
perform a lot of repeated in- place conversions, it may lead to
memory fragmentation. Consider using out-of-place SparseCopyToHashBuf()
function in this case.
-- ALGLIB PROJECT --
Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparseconverttohash(const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs out-of-place conversion to Hash table storage
format. S0 is copied to S1 and converted on-the-fly.
INPUT PARAMETERS
S0 - sparse matrix in any format.
OUTPUT PARAMETERS
S1 - sparse matrix in Hash table format.
NOTE: if S0 is stored as Hash-table, it is just copied without conversion.
NOTE: this function de-allocates memory occupied by S1 before starting
conversion. If you perform a lot of repeated conversions, it may
lead to memory fragmentation. In this case we recommend you to use
SparseCopyToHashBuf() function which re-uses memory in S1 as much as
possible.
-- ALGLIB PROJECT --
Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytohash(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs out-of-place conversion to Hash table storage
format. S0 is copied to S1 and converted on-the-fly. Memory allocated in
S1 is reused to maximum extent possible.
INPUT PARAMETERS
S0 - sparse matrix in any format.
OUTPUT PARAMETERS
S1 - sparse matrix in Hash table format.
NOTE: if S0 is stored as Hash-table, it is just copied without conversion.
-- ALGLIB PROJECT --
Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytohashbuf(const sparsematrix &s0, const sparsematrix &s1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs out-of-place conversion to CRS format. S0 is
copied to S1 and converted on-the-fly.
INPUT PARAMETERS
S0 - sparse matrix in any format.
OUTPUT PARAMETERS
S1 - sparse matrix in CRS format.
NOTE: if S0 is stored as CRS, it is just copied without conversion.
NOTE: this function de-allocates memory occupied by S1 before starting CRS
conversion. If you perform a lot of repeated CRS conversions, it may
lead to memory fragmentation. In this case we recommend you to use
SparseCopyToCRSBuf() function which re-uses memory in S1 as much as
possible.
-- ALGLIB PROJECT --
Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytocrs(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs in-place conversion to SKS format.
INPUT PARAMETERS
S - sparse matrix in any format.
OUTPUT PARAMETERS
S - sparse matrix in SKS format.
NOTE: this function has no effect when called with matrix which is
already in SKS mode.
NOTE: in-place conversion involves allocation of temporary arrays. If you
perform a lot of repeated in- place conversions, it may lead to
memory fragmentation. Consider using out-of-place SparseCopyToSKSBuf()
function in this case.
-- ALGLIB PROJECT --
Copyright 15.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparseconverttosks(const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs out-of-place conversion to SKS storage format.
S0 is copied to S1 and converted on-the-fly.
INPUT PARAMETERS
S0 - sparse matrix in any format.
OUTPUT PARAMETERS
S1 - sparse matrix in SKS format.
NOTE: if S0 is stored as SKS, it is just copied without conversion.
NOTE: this function de-allocates memory occupied by S1 before starting
conversion. If you perform a lot of repeated conversions, it may
lead to memory fragmentation. In this case we recommend you to use
SparseCopyToSKSBuf() function which re-uses memory in S1 as much as
possible.
-- ALGLIB PROJECT --
Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytosks(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs out-of-place conversion to SKS format. S0 is
copied to S1 and converted on-the-fly. Memory allocated in S1 is reused
to maximum extent possible.
INPUT PARAMETERS
S0 - sparse matrix in any format.
OUTPUT PARAMETERS
S1 - sparse matrix in SKS format.
NOTE: if S0 is stored as SKS, it is just copied without conversion.
-- ALGLIB PROJECT --
Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytosksbuf(const sparsematrix &s0, const sparsematrix &s1, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function returns type of the matrix storage format.
INPUT PARAMETERS:
S - sparse matrix.
RESULT:
sparse storage format used by matrix:
0 - Hash-table
1 - CRS (compressed row storage)
2 - SKS (skyline)
NOTE: future versions of ALGLIB may include additional sparse storage
formats.
-- ALGLIB PROJECT --
Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetmatrixtype(const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using Hash table representation.
INPUT PARAMETERS:
S - sparse matrix.
RESULT:
True if matrix type is Hash table
False if matrix type is not Hash table
-- ALGLIB PROJECT --
Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
bool sparseishash(const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using SKS representation.
INPUT PARAMETERS:
S - sparse matrix.
RESULT:
True if matrix type is SKS
False if matrix type is not SKS
-- ALGLIB PROJECT --
Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
bool sparseissks(const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
The function frees all memory occupied by sparse matrix. Sparse matrix
structure becomes unusable after this call.
OUTPUT PARAMETERS
S - sparse matrix to delete
-- ALGLIB PROJECT --
Copyright 24.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsefree(sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
The function returns number of 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
The function returns number of strictly upper triangular non-zero elements
in the matrix. It counts SYMBOLICALLY non-zero elements, i.e. entries
in the sparse matrix data structure. If some element has zero numerical
value, it is still counted.
This function has different cost for different types of matrices:
* for hash-based matrices it involves complete pass over entire hash-table
with O(NNZ) cost, where NNZ is number of non-zero elements
* for CRS and SKS matrix types cost of counting is O(N) (N - matrix size).
RESULT: number of non-zero elements strictly above main diagonal
-- ALGLIB PROJECT --
Copyright 12.02.2014 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetuppercount(const sparsematrix &s, const xparams _xparams = alglib::xdefault);
/*************************************************************************
The function returns number of strictly lower triangular non-zero elements
in the matrix. It counts SYMBOLICALLY non-zero elements, i.e. entries
in the sparse matrix data structure. If some element has zero numerical
value, it is still counted.
This function has different cost for different types of matrices:
* for hash-based matrices it involves complete pass over entire hash-table
with O(NNZ) cost, where NNZ is number of non-zero elements
* for CRS and SKS matrix types cost of counting is O(N) (N - matrix size).
RESULT: number of non-zero elements strictly below main diagonal
-- ALGLIB PROJECT --
Copyright 12.02.2014 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetlowercount(const sparsematrix &s, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_ABLAS) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Cache-oblivous complex "copy-and-transpose"
Input parameters:
M - number of rows
N - number of columns
A - source matrix, MxN submatrix is copied and transposed
IA - submatrix offset (row index)
JA - submatrix offset (column index)
B - destination matrix, must be large enough to store result
IB - submatrix offset (row index)
JB - submatrix offset (column index)
*************************************************************************/
void cmatrixtranspose(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, complex_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Cache-oblivous real "copy-and-transpose"
Input parameters:
M - number of rows
N - number of columns
A - source matrix, MxN submatrix is copied and transposed
IA - submatrix offset (row index)
JA - submatrix offset (column index)
B - destination matrix, must be large enough to store result
IB - submatrix offset (row index)
JB - submatrix offset (column index)
*************************************************************************/
void rmatrixtranspose(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This code enforces symmetricy of the matrix by copying Upper part to lower
one (or vice versa).
INPUT PARAMETERS:
A - matrix
N - number of rows/columns
IsUpper - whether we want to copy upper triangle to lower one (True)
or vice versa (False).
*************************************************************************/
void rmatrixenforcesymmetricity(const real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Copy
Input parameters:
M - number of rows
N - number of columns
A - source matrix, MxN submatrix is copied 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Copy
Input parameters:
N - subvector size
A - source vector, N elements are copied
IA - source offset (first element index)
B - destination vector, must be large enough to store result
IB - destination offset (first element index)
*************************************************************************/
void rvectorcopy(const ae_int_t n, const real_1d_array &a, const ae_int_t ia, const real_1d_array &b, const ae_int_t ib, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Copy
Input parameters:
M - number of rows
N - number of columns
A - source matrix, MxN submatrix is copied 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, const real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Performs generalized copy: B := Beta*B + Alpha*A.
If Beta=0, then previous contents of B is simply ignored. If Alpha=0, then
A is ignored and not referenced. If both Alpha and Beta are zero, B is
filled by zeros.
Input parameters:
M - number of rows
N - number of columns
Alpha- coefficient
A - source matrix, MxN submatrix is copied and transposed
IA - submatrix offset (row index)
JA - submatrix offset (column index)
Beta- coefficient
B - destination matrix, must be large enough to store result
IB - submatrix offset (row index)
JB - submatrix offset (column index)
*************************************************************************/
void rmatrixgencopy(const ae_int_t m, const ae_int_t n, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const double beta, const real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Rank-1 correction: A := A + alpha*u*v'
NOTE: this function expects A to be large enough to store result. No
automatic preallocation happens for smaller arrays. No integrity
checks is performed for sizes of A, u, v.
INPUT PARAMETERS:
M - number of rows
N - number of columns
A - target matrix, MxN submatrix is updated
IA - submatrix offset (row index)
JA - submatrix offset (column index)
Alpha- coefficient
U - vector #1
IU - subvector offset
V - vector #2
IV - subvector offset
-- ALGLIB routine --
16.10.2017
Bochkanov Sergey
*************************************************************************/
void rmatrixger(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const double alpha, const real_1d_array &u, const ae_int_t iu, const real_1d_array &v, const ae_int_t iv, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Rank-1 correction: A := A + u*v'
INPUT PARAMETERS:
M - number of rows
N - number of columns
A - target matrix, MxN submatrix is updated
IA - submatrix offset (row index)
JA - submatrix offset (column index)
U - vector #1
IU - subvector offset
V - vector #2
IV - subvector offset
*************************************************************************/
void cmatrixrank1(const ae_int_t m, const ae_int_t n, complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, complex_1d_array &u, const ae_int_t iu, complex_1d_array &v, const ae_int_t iv, const xparams _xparams = alglib::xdefault);
/*************************************************************************
IMPORTANT: this function is deprecated since ALGLIB 3.13. Use RMatrixGER()
which is more generic version of this function.
Rank-1 correction: A := A + u*v'
INPUT PARAMETERS:
M - number of rows
N - number of columns
A - target matrix, MxN submatrix is updated
IA - submatrix offset (row index)
JA - submatrix offset (column index)
U - vector #1
IU - subvector offset
V - vector #2
IV - subvector offset
*************************************************************************/
void rmatrixrank1(const ae_int_t m, const ae_int_t n, real_2d_array &a, const ae_int_t ia, const ae_int_t ja, real_1d_array &u, const ae_int_t iu, real_1d_array &v, const ae_int_t iv, const xparams _xparams = alglib::xdefault);
/*************************************************************************
*************************************************************************/
void rmatrixgemv(const ae_int_t m, const ae_int_t n, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t opa, const real_1d_array &x, const ae_int_t ix, const double beta, const real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Matrix-vector product: y := op(A)*x
INPUT PARAMETERS:
M - number of rows of op(A)
M>=0
N - number of columns of op(A)
N>=0
A - target matrix
IA - submatrix offset (row index)
JA - submatrix offset (column index)
OpA - operation type:
* OpA=0 => op(A) = A
* OpA=1 => op(A) = A^T
* OpA=2 => op(A) = A^H
X - input vector
IX - subvector offset
IY - subvector offset
Y - preallocated matrix, must be large enough to store result
OUTPUT PARAMETERS:
Y - vector which stores result
if M=0, then subroutine does nothing.
if N=0, Y is filled by zeros.
-- ALGLIB routine --
28.01.2010
Bochkanov Sergey
*************************************************************************/
void cmatrixmv(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t opa, const complex_1d_array &x, const ae_int_t ix, complex_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);
/*************************************************************************
IMPORTANT: this function is deprecated since ALGLIB 3.13. Use RMatrixGEMV()
which is more generic version of this function.
Matrix-vector product: y := op(A)*x
INPUT PARAMETERS:
M - number of rows of op(A)
N - number of columns of op(A)
A - target matrix
IA - submatrix offset (row index)
JA - submatrix offset (column index)
OpA - operation type:
* OpA=0 => op(A) = A
* OpA=1 => op(A) = A^T
X - input vector
IX - subvector offset
IY - subvector offset
Y - preallocated matrix, must be large enough to store result
OUTPUT PARAMETERS:
Y - vector which stores result
if M=0, then subroutine does nothing.
if N=0, Y is filled by zeros.
-- ALGLIB routine --
28.01.2010
Bochkanov Sergey
*************************************************************************/
void rmatrixmv(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t opa, const real_1d_array &x, const ae_int_t ix, const real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);
/*************************************************************************
*************************************************************************/
void rmatrixsymv(const ae_int_t n, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const bool isupper, const real_1d_array &x, const ae_int_t ix, const double beta, const real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);
/*************************************************************************
*************************************************************************/
double rmatrixsyvmv(const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const bool isupper, const real_1d_array &x, const ae_int_t ix, const real_1d_array &tmp, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This subroutine solves linear system op(A)*x=b where:
* A is NxN upper/lower triangular/unitriangular matrix
* X and B are Nx1 vectors
* "op" may be identity transformation, transposition, conjugate transposition
Solution replaces X.
IMPORTANT: * no overflow/underflow/denegeracy tests is performed.
* no integrity checks for operand sizes, out-of-bounds accesses
and so on is performed
INPUT PARAMETERS
N - matrix size, N>=0
A - matrix, actial matrix is stored in A[IA:IA+N-1,JA:JA+N-1]
IA - submatrix offset
JA - submatrix offset
IsUpper - whether matrix is upper triangular
IsUnit - whether matrix is unitriangular
OpType - transformation type:
* 0 - no transformation
* 1 - transposition
X - right part, actual vector is stored in X[IX:IX+N-1]
IX - offset
OUTPUT PARAMETERS
X - solution replaces elements X[IX:IX+N-1]
-- ALGLIB routine / remastering of LAPACK's DTRSV --
(c) 2017 Bochkanov Sergey - converted to ALGLIB
(c) 2016 Reference BLAS level1 routine (LAPACK version 3.7.0)
Reference BLAS is a software package provided by Univ. of Tennessee,
Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.
*************************************************************************/
void rmatrixtrsv(const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const bool isupper, const bool isunit, const ae_int_t optype, const real_1d_array &x, const ae_int_t ix, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This subroutine calculates X*op(A^-1) where:
* X is MxN general matrix
* A is NxN upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition, conjugate transposition
Multiplication result replaces X.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS
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 --
20.01.2018
Bochkanov Sergey
*************************************************************************/
void cmatrixrighttrsm(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, const complex_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This subroutine calculates op(A^-1)*X where:
* X is MxN general matrix
* A is MxM upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition, conjugate transposition
Multiplication result replaces X.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS
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 --
15.12.2009-22.01.2018
Bochkanov Sergey
*************************************************************************/
void cmatrixlefttrsm(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, const complex_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This subroutine calculates X*op(A^-1) where:
* X is MxN general matrix
* A is NxN upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition
Multiplication result replaces X.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS
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 --
15.12.2009-22.01.2018
Bochkanov Sergey
*************************************************************************/
void rmatrixrighttrsm(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, const real_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This subroutine calculates op(A^-1)*X where:
* X is MxN general matrix
* A is MxM upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition
Multiplication result replaces X.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS
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 --
15.12.2009-22.01.2018
Bochkanov Sergey
*************************************************************************/
void rmatrixlefttrsm(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, const real_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This subroutine calculates C=alpha*A*A^H+beta*C or C=alpha*A^H*A+beta*C
where:
* C is NxN Hermitian matrix given by its upper/lower triangle
* A is NxK matrix when A*A^H is calculated, KxN matrix otherwise
Additional info:
* multiplication result replaces C. If Beta=0, C elements are not used in
calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS
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 --
16.12.2009-22.01.2018
Bochkanov Sergey
*************************************************************************/
void cmatrixherk(const ae_int_t n, const ae_int_t k, const double alpha, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const double beta, const complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This subroutine calculates C=alpha*A*A^T+beta*C or C=alpha*A^T*A+beta*C
where:
* C is NxN symmetric matrix given by its upper/lower triangle
* A is NxK matrix when A*A^T is calculated, KxN matrix otherwise
Additional info:
* multiplication result replaces C. If Beta=0, C elements are not used in
calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS
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 --
16.12.2009-22.01.2018
Bochkanov Sergey
*************************************************************************/
void rmatrixsyrk(const ae_int_t n, const ae_int_t k, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const double beta, const real_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This subroutine calculates C = alpha*op1(A)*op2(B) +beta*C where:
* C is MxN general matrix
* op1(A) is MxK matrix
* op2(B) is KxN matrix
* "op" may be identity transformation, transposition, conjugate transposition
Additional info:
* cache-oblivious algorithm is used.
* multiplication result replaces C. If Beta=0, C elements are not used in
calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
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 --
2009-2019
Bochkanov Sergey
*************************************************************************/
void cmatrixgemm(const ae_int_t m, const ae_int_t n, const ae_int_t k, const alglib::complex alpha, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const complex_2d_array &b, const ae_int_t ib, const ae_int_t jb, const ae_int_t optypeb, const alglib::complex beta, const complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This subroutine calculates C = alpha*op1(A)*op2(B) +beta*C where:
* C is MxN general matrix
* op1(A) is MxK matrix
* op2(B) is KxN matrix
* "op" may be identity transformation, transposition
Additional info:
* cache-oblivious algorithm is used.
* multiplication result replaces C. If Beta=0, C elements are not used in
calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
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 --
2009-2019
Bochkanov Sergey
*************************************************************************/
void rmatrixgemm(const ae_int_t m, const ae_int_t n, const ae_int_t k, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const ae_int_t optypeb, const double beta, const real_2d_array &c, const ae_int_t ic, const ae_int_t jc, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This subroutine is an older version of CMatrixHERK(), one with wrong name
(it is HErmitian update, not SYmmetric). It is left here for backward
compatibility.
-- ALGLIB routine --
16.12.2009
Bochkanov Sergey
*************************************************************************/
void cmatrixsyrk(const ae_int_t n, const ae_int_t k, const double alpha, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const double beta, const complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_DLU) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_SPTRF) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_MATGEN) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Generation of a random uniformly distributed (Haar) orthogonal matrix
INPUT PARAMETERS:
N - matrix size, N>=1
OUTPUT PARAMETERS:
A - orthogonal NxN matrix, array[0..N-1,0..N-1]
NOTE: this function uses algorithm described in Stewart, G. W. (1980),
"The Efficient Generation of Random Orthogonal Matrices with an
Application to Condition Estimators".
Speaking short, to generate an (N+1)x(N+1) orthogonal matrix, it:
* takes an NxN one
* takes uniformly distributed unit vector of dimension N+1.
* constructs a Householder reflection from the vector, then applies
it to the smaller matrix (embedded in the larger size with a 1 at
the bottom right corner).
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void rmatrixrndorthogonal(const ae_int_t n, real_2d_array &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Generation of random NxN matrix with given condition number and norm2(A)=1
INPUT PARAMETERS:
N - matrix size
C - condition number (in 2-norm)
OUTPUT PARAMETERS:
A - random matrix with norm2(A)=1 and cond(A)=C
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void rmatrixrndcond(const ae_int_t n, const double c, real_2d_array &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Generation of a random Haar distributed orthogonal complex matrix
INPUT PARAMETERS:
N - matrix size, N>=1
OUTPUT PARAMETERS:
A - orthogonal NxN matrix, array[0..N-1,0..N-1]
NOTE: this function uses algorithm described in Stewart, G. W. (1980),
"The Efficient Generation of Random Orthogonal Matrices with an
Application to Condition Estimators".
Speaking short, to generate an (N+1)x(N+1) orthogonal matrix, it:
* takes an NxN one
* takes uniformly distributed unit vector of dimension N+1.
* constructs a Householder reflection from the vector, then applies
it to the smaller matrix (embedded in the larger size with a 1 at
the bottom right corner).
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void cmatrixrndorthogonal(const ae_int_t n, complex_2d_array &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Generation of random NxN complex matrix with given condition number C and
norm2(A)=1
INPUT PARAMETERS:
N - matrix size
C - condition number (in 2-norm)
OUTPUT PARAMETERS:
A - random matrix with norm2(A)=1 and cond(A)=C
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void cmatrixrndcond(const ae_int_t n, const double c, complex_2d_array &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Generation of random NxN symmetric matrix with given condition number and
norm2(A)=1
INPUT PARAMETERS:
N - matrix size
C - condition number (in 2-norm)
OUTPUT PARAMETERS:
A - random matrix with norm2(A)=1 and cond(A)=C
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void smatrixrndcond(const ae_int_t n, const double c, real_2d_array &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Generation of random NxN symmetric positive definite matrix with given
condition number and norm2(A)=1
INPUT PARAMETERS:
N - matrix size
C - condition number (in 2-norm)
OUTPUT PARAMETERS:
A - random SPD matrix with norm2(A)=1 and cond(A)=C
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void spdmatrixrndcond(const ae_int_t n, const double c, real_2d_array &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Generation of random NxN Hermitian matrix with given condition number and
norm2(A)=1
INPUT PARAMETERS:
N - matrix size
C - condition number (in 2-norm)
OUTPUT PARAMETERS:
A - random matrix with norm2(A)=1 and cond(A)=C
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void hmatrixrndcond(const ae_int_t n, const double c, complex_2d_array &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Generation of random NxN Hermitian positive definite matrix with given
condition number and norm2(A)=1
INPUT PARAMETERS:
N - matrix size
C - condition number (in 2-norm)
OUTPUT PARAMETERS:
A - random HPD matrix with norm2(A)=1 and cond(A)=C
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void hpdmatrixrndcond(const ae_int_t n, const double c, complex_2d_array &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Multiplication of MxN matrix by NxN random Haar distributed orthogonal matrix
INPUT PARAMETERS:
A - matrix, array[0..M-1, 0..N-1]
M, N- matrix size
OUTPUT PARAMETERS:
A - A*Q, where Q is random NxN orthogonal matrix
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void rmatrixrndorthogonalfromtheright(real_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Multiplication of MxN matrix by MxM random Haar distributed orthogonal matrix
INPUT PARAMETERS:
A - matrix, array[0..M-1, 0..N-1]
M, N- matrix size
OUTPUT PARAMETERS:
A - Q*A, where Q is random MxM orthogonal matrix
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void rmatrixrndorthogonalfromtheleft(real_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Multiplication of MxN complex matrix by NxN random Haar distributed
complex orthogonal matrix
INPUT PARAMETERS:
A - matrix, array[0..M-1, 0..N-1]
M, N- matrix size
OUTPUT PARAMETERS:
A - A*Q, where Q is random NxN orthogonal matrix
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void cmatrixrndorthogonalfromtheright(complex_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Multiplication of MxN complex matrix by MxM random Haar distributed
complex orthogonal matrix
INPUT PARAMETERS:
A - matrix, array[0..M-1, 0..N-1]
M, N- matrix size
OUTPUT PARAMETERS:
A - Q*A, where Q is random MxM orthogonal matrix
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void cmatrixrndorthogonalfromtheleft(complex_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Symmetric multiplication of NxN matrix by random Haar distributed
orthogonal matrix
INPUT PARAMETERS:
A - matrix, array[0..N-1, 0..N-1]
N - matrix size
OUTPUT PARAMETERS:
A - Q'*A*Q, where Q is random NxN orthogonal matrix
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void smatrixrndmultiply(real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Hermitian multiplication of NxN matrix by random Haar distributed
complex orthogonal matrix
INPUT PARAMETERS:
A - matrix, array[0..N-1, 0..N-1]
N - matrix size
OUTPUT PARAMETERS:
A - Q^H*A*Q, where Q is random NxN orthogonal matrix
-- ALGLIB routine --
04.12.2009
Bochkanov Sergey
*************************************************************************/
void hmatrixrndmultiply(complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_TRFAC) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
LU decomposition of a general real matrix with row pivoting
A is represented as A = P*L*U, where:
* L is lower unitriangular matrix
* U is upper triangular matrix
* P = P0*P1*...*PK, K=min(M,N)-1,
Pi - permutation matrix for I and Pivots[I]
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS:
A - 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
LU decomposition of a general complex matrix with row pivoting
A is represented as A = P*L*U, where:
* L is lower unitriangular matrix
* U is upper triangular matrix
* P = P0*P1*...*PK, K=min(M,N)-1,
Pi - permutation matrix for I and Pivots[I]
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS:
A - 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Cache-oblivious Cholesky decomposition
The algorithm computes Cholesky decomposition of a Hermitian positive-
definite matrix. The result of an algorithm is a representation of A as
A=U'*U or A=L*L' (here X' denotes conj(X^T)).
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS:
A - 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 --
15.12.2009-22.01.2018
Bochkanov Sergey
*************************************************************************/
bool hpdmatrixcholesky(complex_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Cache-oblivious Cholesky decomposition
The algorithm computes Cholesky decomposition of a symmetric positive-
definite matrix. The result of an algorithm is a representation of A as
A=U^T*U or A=L*L^T
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS:
A - 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Update of Cholesky decomposition: rank-1 update to original A. "Buffered"
version which uses preallocated buffer which is saved between subsequent
function calls.
This function uses internally allocated buffer which is not saved between
subsequent calls. So, if you perform a lot of subsequent updates,
we recommend you to use "buffered" version of this function:
SPDMatrixCholeskyUpdateAdd1Buf().
INPUT PARAMETERS:
A - upper or lower Cholesky factor.
array with elements [0..N-1, 0..N-1].
Exception is thrown if array size is too small.
N - size of matrix A, N>0
IsUpper - if IsUpper=True, then A contains upper Cholesky factor;
otherwise A contains a lower one.
U - array[N], rank-1 update to A: A_mod = A + u*u'
Exception is thrown if array size is too small.
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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Update of Cholesky decomposition: "fixing" some variables.
This function uses internally allocated buffer which is not saved between
subsequent calls. So, if you perform a lot of subsequent updates,
we recommend you to use "buffered" version of this function:
SPDMatrixCholeskyUpdateFixBuf().
"FIXING" EXPLAINED:
Suppose we have N*N positive definite matrix A. "Fixing" some variable
means filling corresponding row/column of A by zeros, and setting
diagonal element to 1.
For example, if we fix 2nd variable in 4*4 matrix A, it becomes Af:
( A00 A01 A02 A03 ) ( Af00 0 Af02 Af03 )
( A10 A11 A12 A13 ) ( 0 1 0 0 )
( A20 A21 A22 A23 ) => ( Af20 0 Af22 Af23 )
( A30 A31 A32 A33 ) ( Af30 0 Af32 Af33 )
If we have Cholesky decomposition of A, it must be recalculated after
variables were fixed. However, it is possible to use efficient
algorithm, which needs O(K*N^2) time to "fix" K variables, given
Cholesky decomposition of original, "unfixed" A.
INPUT PARAMETERS:
A - upper or lower Cholesky factor.
array with elements [0..N-1, 0..N-1].
Exception is thrown if array size is too small.
N - size of matrix A, N>0
IsUpper - if IsUpper=True, then A contains upper Cholesky factor;
otherwise A contains a lower one.
Fix - array[N], I-th element is True if I-th variable must be
fixed. Exception is thrown if array size is too small.
BufR - possibly preallocated buffer; automatically resized if
needed. It is recommended to reuse this buffer if you
perform a lot of subsequent decompositions.
OUTPUT PARAMETERS:
A - updated factorization. If IsUpper=True, then the upper
triangle contains matrix U, and the elements below the main
diagonal are not modified. Similarly, if IsUpper = False.
NOTE: this function always succeeds, so it does not return completion code
NOTE: this function checks sizes of input arrays, but it does NOT checks
for presence of infinities or NAN's.
NOTE: this function is efficient only for moderate amount of updated
variables - say, 0.1*N or 0.3*N. For larger amount of variables it
will still work, but you may get better performance with
straightforward Cholesky.
-- ALGLIB --
03.02.2014
Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdatefix(const real_2d_array &a, const ae_int_t n, const bool isupper, const boolean_1d_array &fix, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Update of Cholesky decomposition: rank-1 update to original A. "Buffered"
version which uses preallocated buffer which is saved between subsequent
function calls.
See comments for SPDMatrixCholeskyUpdateAdd1() for more information.
INPUT PARAMETERS:
A - upper or lower Cholesky factor.
array with elements [0..N-1, 0..N-1].
Exception is thrown if array size is too small.
N - size of matrix A, N>0
IsUpper - if IsUpper=True, then A contains upper Cholesky factor;
otherwise A contains a lower one.
U - array[N], rank-1 update to A: A_mod = A + u*u'
Exception is thrown if array size is too small.
BufR - possibly preallocated buffer; automatically resized if
needed. It is recommended to reuse this buffer if you
perform a lot of subsequent decompositions.
OUTPUT PARAMETERS:
A - updated factorization. If IsUpper=True, then the upper
triangle contains matrix U, and the elements below the main
diagonal are not modified. Similarly, if IsUpper = False.
-- ALGLIB --
03.02.2014
Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdateadd1buf(const real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &u, real_1d_array &bufr, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Update of Cholesky decomposition: "fixing" some variables. "Buffered"
version which uses preallocated buffer which is saved between subsequent
function calls.
See comments for SPDMatrixCholeskyUpdateFix() for more information.
INPUT PARAMETERS:
A - upper or lower Cholesky factor.
array with elements [0..N-1, 0..N-1].
Exception is thrown if array size is too small.
N - size of matrix A, N>0
IsUpper - if IsUpper=True, then A contains upper Cholesky factor;
otherwise A contains a lower one.
Fix - array[N], I-th element is True if I-th variable must be
fixed. Exception is thrown if array size is too small.
BufR - possibly preallocated buffer; automatically resized if
needed. It is recommended to reuse this buffer if you
perform a lot of subsequent decompositions.
OUTPUT PARAMETERS:
A - updated factorization. If IsUpper=True, then the upper
triangle contains matrix U, and the elements below the main
diagonal are not modified. Similarly, if IsUpper = False.
-- ALGLIB --
03.02.2014
Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdatefixbuf(const real_2d_array &a, const ae_int_t n, const bool isupper, const boolean_1d_array &fix, real_1d_array &bufr, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Sparse LU decomposition with column pivoting for sparsity and row pivoting
for stability. Input must be square sparse matrix stored in CRS format.
The algorithm computes LU decomposition of a general square matrix
(rectangular ones are not supported). The result of an algorithm is a
representation of A as A = P*L*U*Q, where:
* L is lower unitriangular matrix
* U is upper triangular matrix
* P = P0*P1*...*PK, K=N-1, Pi - permutation matrix for I and P[I]
* Q = QK*...*Q1*Q0, K=N-1, Qi - permutation matrix for I and Q[I]
This function pivots columns for higher sparsity, and then pivots rows for
stability (larger element at the diagonal).
INPUT PARAMETERS:
A - sparse NxN matrix in CRS format. An exception is generated
if matrix is non-CRS or non-square.
PivotType- pivoting strategy:
* 0 for best pivoting available (2 in current version)
* 1 for row-only pivoting (NOT RECOMMENDED)
* 2 for complete pivoting which produces most sparse outputs
OUTPUT PARAMETERS:
A - the result of factorization, matrices L and U stored in
compact form using CRS sparse storage format:
* lower unitriangular L is stored strictly under main diagonal
* upper triangilar U is stored ON and ABOVE main diagonal
P - row permutation matrix in compact form, array[N]
Q - col permutation matrix in compact form, array[N]
This function always succeeds, i.e. it ALWAYS returns valid factorization,
but for your convenience it also returns boolean value which helps to
detect symbolically degenerate matrices:
* function returns TRUE, if the matrix was factorized AND symbolically
non-degenerate
* function returns FALSE, if the matrix was factorized but U has strictly
zero elements at the diagonal (the factorization is returned anyway).
-- ALGLIB routine --
03.09.2018
Bochkanov Sergey
*************************************************************************/
bool sparselu(const sparsematrix &a, const ae_int_t pivottype, integer_1d_array &p, integer_1d_array &q, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Sparse Cholesky decomposition for skyline matrixm using in-place algorithm
without allocating additional storage.
The algorithm computes Cholesky decomposition of a symmetric positive-
definite sparse matrix. The result of an algorithm is a representation of
A as A=U^T*U or A=L*L^T
This function 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, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_RCOND) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Estimate of a matrix condition number (1-norm)
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
A - matrix. Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double rmatrixrcond1(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Estimate of a matrix condition number (infinity-norm).
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
A - matrix. Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double rmatrixrcondinf(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Condition number estimate of a symmetric positive definite matrix.
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
It should be noted that 1-norm and inf-norm of condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.
Input parameters:
A - symmetric positive definite matrix which is given by its
upper or lower triangle depending on the value of
IsUpper. Array with elements [0..N-1, 0..N-1].
N - size of matrix A.
IsUpper - storage format.
Result:
1/LowerBound(cond(A)), if matrix A is positive definite,
-1, if matrix A is not positive definite, and its condition number
could not be found by this algorithm.
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double spdmatrixrcond(const real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Triangular matrix: estimate of a condition number (1-norm)
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
A - matrix. Array[0..N-1, 0..N-1].
N - size of A.
IsUpper - True, if the matrix is upper triangular.
IsUnit - True, if the matrix has a unit diagonal.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double rmatrixtrrcond1(const real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Triangular matrix: estimate of a matrix condition number (infinity-norm).
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
A - matrix. Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
IsUpper - True, if the matrix is upper triangular.
IsUnit - True, if the matrix has a unit diagonal.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double rmatrixtrrcondinf(const real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Condition number estimate of a Hermitian positive definite matrix.
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
It should be noted that 1-norm and inf-norm of condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.
Input parameters:
A - Hermitian positive definite matrix which is given by its
upper or lower triangle depending on the value of
IsUpper. Array with elements [0..N-1, 0..N-1].
N - size of matrix A.
IsUpper - storage format.
Result:
1/LowerBound(cond(A)), if matrix A is positive definite,
-1, if matrix A is not positive definite, and its condition number
could not be found by this algorithm.
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double hpdmatrixrcond(const complex_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Estimate of a matrix condition number (1-norm)
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
A - matrix. Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double cmatrixrcond1(const complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Estimate of a matrix condition number (infinity-norm).
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
A - matrix. Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double cmatrixrcondinf(const complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition (1-norm)
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
LUA - LU decomposition of a matrix in compact form. Output of
the RMatrixLU subroutine.
N - size of matrix A.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double rmatrixlurcond1(const real_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition
(infinity norm).
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
LUA - LU decomposition of a matrix in compact form. Output of
the RMatrixLU subroutine.
N - size of matrix A.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double rmatrixlurcondinf(const real_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Condition number estimate of a symmetric positive definite matrix given by
Cholesky decomposition.
The algorithm calculates a lower bound of the condition number. In this
case, the algorithm does not return a lower bound of the condition number,
but an inverse number (to avoid an overflow in case of a singular matrix).
It should be noted that 1-norm and inf-norm condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.
Input parameters:
CD - Cholesky decomposition of matrix A,
output of SMatrixCholesky subroutine.
N - size of matrix A.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double spdmatrixcholeskyrcond(const real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Condition number estimate of a Hermitian positive definite matrix given by
Cholesky decomposition.
The algorithm calculates a lower bound of the condition number. In this
case, the algorithm does not return a lower bound of the condition number,
but an inverse number (to avoid an overflow in case of a singular matrix).
It should be noted that 1-norm and inf-norm condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.
Input parameters:
CD - Cholesky decomposition of matrix A,
output of SMatrixCholesky subroutine.
N - size of matrix A.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double hpdmatrixcholeskyrcond(const complex_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition (1-norm)
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
LUA - LU decomposition of a matrix in compact form. Output of
the CMatrixLU subroutine.
N - size of matrix A.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double cmatrixlurcond1(const complex_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition
(infinity norm).
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
LUA - LU decomposition of a matrix in compact form. Output of
the CMatrixLU subroutine.
N - size of matrix A.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double cmatrixlurcondinf(const complex_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Triangular matrix: estimate of a condition number (1-norm)
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
A - matrix. Array[0..N-1, 0..N-1].
N - size of A.
IsUpper - True, if the matrix is upper triangular.
IsUnit - True, if the matrix has a unit diagonal.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double cmatrixtrrcond1(const complex_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Triangular matrix: estimate of a matrix condition number (infinity-norm).
The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).
Input parameters:
A - matrix. Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
IsUpper - True, if the matrix is upper triangular.
IsUnit - True, if the matrix has a unit diagonal.
Result: 1/LowerBound(cond(A))
NOTE:
if k(A) is very large, then matrix is assumed degenerate, k(A)=INF,
0.0 is returned in such cases.
*************************************************************************/
double cmatrixtrrcondinf(const complex_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_MATINV) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Inversion of a matrix given by its LU decomposition.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS:
A - 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, const xparams _xparams = alglib::xdefault);
void rmatrixluinverse(real_2d_array &a, const integer_1d_array &pivots, ae_int_t &info, matinvreport &rep, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Inversion of a general matrix.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix.
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, const xparams _xparams = alglib::xdefault);
void rmatrixinverse(real_2d_array &a, ae_int_t &info, matinvreport &rep, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Inversion of a matrix given by its LU decomposition.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS:
A - 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, const xparams _xparams = alglib::xdefault);
void cmatrixluinverse(complex_2d_array &a, const integer_1d_array &pivots, ae_int_t &info, matinvreport &rep, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Inversion of a general matrix.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix
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, const xparams _xparams = alglib::xdefault);
void cmatrixinverse(complex_2d_array &a, ae_int_t &info, matinvreport &rep, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Inversion of a symmetric positive definite matrix which is given
by Cholesky decomposition.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - 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, const xparams _xparams = alglib::xdefault);
void spdmatrixcholeskyinverse(real_2d_array &a, ae_int_t &info, matinvreport &rep, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Inversion of a symmetric positive definite matrix.
Given an upper or lower triangle of a symmetric positive definite matrix,
the algorithm generates matrix A^-1 and saves the upper or lower triangle
depending on the input.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix to be 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, const xparams _xparams = alglib::xdefault);
void spdmatrixinverse(real_2d_array &a, ae_int_t &info, matinvreport &rep, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Inversion of a Hermitian positive definite matrix which is given
by Cholesky decomposition.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - 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, const xparams _xparams = alglib::xdefault);
void hpdmatrixcholeskyinverse(complex_2d_array &a, ae_int_t &info, matinvreport &rep, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Inversion of a Hermitian positive definite matrix.
Given an upper or lower triangle of a Hermitian positive definite matrix,
the algorithm generates matrix A^-1 and saves the upper or lower triangle
depending on the input.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix to be 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, const xparams _xparams = alglib::xdefault);
void hpdmatrixinverse(complex_2d_array &a, ae_int_t &info, matinvreport &rep, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Triangular matrix inverse (real)
The subroutine inverts the following types of matrices:
* upper triangular
* upper triangular with unit diagonal
* lower triangular
* lower triangular with unit diagonal
In case of an upper (lower) triangular matrix, the inverse matrix will
also be upper (lower) triangular, and after the end of the algorithm, the
inverse matrix replaces the source matrix. The elements below (above) the
main diagonal are not changed by the algorithm.
If the matrix has a unit diagonal, the inverse matrix also has a unit
diagonal, and the diagonal elements are not passed to the algorithm.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix, 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, const xparams _xparams = alglib::xdefault);
void rmatrixtrinverse(real_2d_array &a, const bool isupper, ae_int_t &info, matinvreport &rep, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Triangular matrix inverse (complex)
The subroutine inverts the following types of matrices:
* upper triangular
* upper triangular with unit diagonal
* lower triangular
* lower triangular with unit diagonal
In case of an upper (lower) triangular matrix, the inverse matrix will
also be upper (lower) triangular, and after the end of the algorithm, the
inverse matrix replaces the source matrix. The elements below (above) the
main diagonal are not changed by the algorithm.
If the matrix has a unit diagonal, the inverse matrix also has a unit
diagonal, and the diagonal elements are not passed to the algorithm.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix, 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, const xparams _xparams = alglib::xdefault);
void cmatrixtrinverse(complex_2d_array &a, const bool isupper, ae_int_t &info, matinvreport &rep, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_ORTFAC) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
QR decomposition of a rectangular matrix of size MxN
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix A 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
LQ decomposition of a rectangular matrix of size MxN
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix A 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
QR decomposition of a rectangular complex matrix of size MxN
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix A 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
LQ decomposition of a rectangular complex matrix of size MxN
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix A 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Partial unpacking of matrix Q from the QR decomposition of a matrix A
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Unpacking of matrix R from the QR decomposition of a matrix A
Input parameters:
A - matrices Q and R in compact form.
Output of RMatrixQR subroutine.
M - number of rows in given matrix A. M>=0.
N - number of columns in given matrix A. N>=0.
Output parameters:
R - matrix R, array[0..M-1, 0..N-1].
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void rmatrixqrunpackr(const real_2d_array &a, const ae_int_t m, const ae_int_t n, real_2d_array &r, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Partial unpacking of matrix Q from the LQ decomposition of a matrix A
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Unpacking of matrix L from the LQ decomposition of a matrix A
Input parameters:
A - matrices Q and L in compact form.
Output of RMatrixLQ subroutine.
M - number of rows in given matrix A. M>=0.
N - number of columns in given matrix A. N>=0.
Output parameters:
L - matrix L, array[0..M-1, 0..N-1].
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void rmatrixlqunpackl(const real_2d_array &a, const ae_int_t m, const ae_int_t n, real_2d_array &l, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Partial unpacking of matrix Q from QR decomposition of a complex matrix A.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Unpacking of matrix R from the QR decomposition of a matrix A
Input parameters:
A - matrices Q and R in compact form.
Output of CMatrixQR subroutine.
M - number of rows in given matrix A. M>=0.
N - number of columns in given matrix A. N>=0.
Output parameters:
R - matrix R, array[0..M-1, 0..N-1].
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void cmatrixqrunpackr(const complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_2d_array &r, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Partial unpacking of matrix Q from LQ decomposition of a complex matrix A.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Unpacking of matrix L from the LQ decomposition of a matrix A
Input parameters:
A - matrices Q and L in compact form.
Output of CMatrixLQ subroutine.
M - number of rows in given matrix A. M>=0.
N - number of columns in given matrix A. N>=0.
Output parameters:
L - matrix L, array[0..M-1, 0..N-1].
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void cmatrixlqunpackl(const complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_2d_array &l, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Reduction of a rectangular matrix to bidiagonal form
The algorithm reduces the rectangular matrix A to bidiagonal form by
orthogonal transformations P and Q: A = Q*B*(P^T).
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - source matrix. array[0..M-1, 0..N-1]
M - number of rows in matrix A.
N - number of columns in matrix A.
Output parameters:
A - matrices Q, B, P in compact form (see below).
TauQ - scalar factors which are used to form matrix Q.
TauP - scalar factors which are used to form matrix P.
The main diagonal and one of the secondary diagonals of matrix A are
replaced with bidiagonal matrix B. Other elements contain elementary
reflections which form MxM matrix Q and NxN matrix P, respectively.
If M>=N, B is the upper bidiagonal MxN matrix and is stored in the
corresponding elements of matrix A. Matrix Q is represented as a
product of elementary reflections Q = H(0)*H(1)*...*H(n-1), where
H(i) = 1-tau*v*v'. Here tau is a scalar which is stored in TauQ[i], and
vector v has the following structure: v(0:i-1)=0, v(i)=1, v(i+1:m-1) is
stored in elements A(i+1:m-1,i). Matrix P is as follows: P =
G(0)*G(1)*...*G(n-2), where G(i) = 1 - tau*u*u'. Tau is stored in TauP[i],
u(0:i)=0, u(i+1)=1, u(i+2:n-1) is stored in elements A(i,i+2:n-1).
If M<N, B is the lower bidiagonal MxN matrix and is stored in the
corresponding elements of matrix A. Q = H(0)*H(1)*...*H(m-2), where
H(i) = 1 - tau*v*v', tau is stored in TauQ, v(0:i)=0, v(i+1)=1, v(i+2:m-1)
is stored in elements A(i+2:m-1,i). P = G(0)*G(1)*...*G(m-1),
G(i) = 1-tau*u*u', tau is stored in TauP, u(0:i-1)=0, u(i)=1, u(i+1:n-1)
is stored in A(i,i+1:n-1).
EXAMPLE:
m=6, n=5 (m > n): m=5, n=6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )
Here vi and ui are vectors which form H(i) and G(i), and d and e -
are the diagonal and off-diagonal elements of matrix B.
-- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994.
Sergey Bochkanov, ALGLIB project, translation from FORTRAN to
pseudocode, 2007-2010.
*************************************************************************/
void rmatrixbd(real_2d_array &a, const ae_int_t m, const ae_int_t n, real_1d_array &tauq, real_1d_array &taup, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Unpacking matrix Q which reduces a matrix to bidiagonal form.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
QP - matrices Q and P in compact form.
Output of ToBidiagonal subroutine.
M - number of rows in matrix A.
N - number of columns in matrix A.
TAUQ - scalar factors which are used to form Q.
Output of ToBidiagonal subroutine.
QColumns - required number of columns in matrix Q.
M>=QColumns>=0.
Output parameters:
Q - first QColumns columns of matrix Q.
Array[0..M-1, 0..QColumns-1]
If QColumns=0, the array is not modified.
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixbdunpackq(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &tauq, const ae_int_t qcolumns, real_2d_array &q, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Multiplication by matrix Q which reduces matrix A to bidiagonal form.
The algorithm allows pre- or post-multiply by Q or Q'.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
QP - matrices Q and P in compact form.
Output of ToBidiagonal subroutine.
M - number of rows in matrix A.
N - number of columns in matrix A.
TAUQ - scalar factors which are used to form Q.
Output of ToBidiagonal subroutine.
Z - multiplied matrix.
array[0..ZRows-1,0..ZColumns-1]
ZRows - number of rows in matrix Z. If FromTheRight=False,
ZRows=M, otherwise ZRows can be arbitrary.
ZColumns - number of columns in matrix Z. If FromTheRight=True,
ZColumns=M, otherwise ZColumns can be arbitrary.
FromTheRight - pre- or post-multiply.
DoTranspose - multiply by Q or Q'.
Output parameters:
Z - product of Z and Q.
Array[0..ZRows-1,0..ZColumns-1]
If ZRows=0 or ZColumns=0, the array is not modified.
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixbdmultiplybyq(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &tauq, real_2d_array &z, const ae_int_t zrows, const ae_int_t zcolumns, const bool fromtheright, const bool dotranspose, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Unpacking matrix P which reduces matrix A to bidiagonal form.
The subroutine returns transposed matrix P.
Input parameters:
QP - matrices Q and P in compact form.
Output of ToBidiagonal subroutine.
M - number of rows in matrix A.
N - number of columns in matrix A.
TAUP - scalar factors which are used to form P.
Output of ToBidiagonal subroutine.
PTRows - required number of rows of matrix P^T. N >= PTRows >= 0.
Output parameters:
PT - first PTRows columns of matrix P^T
Array[0..PTRows-1, 0..N-1]
If PTRows=0, the array is not modified.
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixbdunpackpt(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &taup, const ae_int_t ptrows, real_2d_array &pt, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Multiplication by matrix P which reduces matrix A to bidiagonal form.
The algorithm allows pre- or post-multiply by P or P'.
Input parameters:
QP - matrices Q and P in compact form.
Output of RMatrixBD subroutine.
M - number of rows in matrix A.
N - number of columns in matrix A.
TAUP - scalar factors which are used to form P.
Output of RMatrixBD subroutine.
Z - multiplied matrix.
Array whose indexes range within [0..ZRows-1,0..ZColumns-1].
ZRows - number of rows in matrix Z. If FromTheRight=False,
ZRows=N, otherwise ZRows can be arbitrary.
ZColumns - number of columns in matrix Z. If FromTheRight=True,
ZColumns=N, otherwise ZColumns can be arbitrary.
FromTheRight - pre- or post-multiply.
DoTranspose - multiply by P or P'.
Output parameters:
Z - product of Z and P.
Array whose indexes range within [0..ZRows-1,0..ZColumns-1].
If ZRows=0 or ZColumns=0, the array is not modified.
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixbdmultiplybyp(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &taup, real_2d_array &z, const ae_int_t zrows, const ae_int_t zcolumns, const bool fromtheright, const bool dotranspose, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Unpacking of the main and secondary diagonals of bidiagonal decomposition
of matrix A.
Input parameters:
B - output of RMatrixBD subroutine.
M - number of rows in matrix B.
N - number of columns in matrix B.
Output parameters:
IsUpper - True, if the matrix is upper bidiagonal.
otherwise IsUpper is False.
D - the main diagonal.
Array whose index ranges within [0..Min(M,N)-1].
E - the secondary diagonal (upper or lower, depending on
the value of IsUpper).
Array index ranges within [0..Min(M,N)-1], the last
element is not used.
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixbdunpackdiagonals(const real_2d_array &b, const ae_int_t m, const ae_int_t n, bool &isupper, real_1d_array &d, real_1d_array &e, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Reduction of a square matrix to upper Hessenberg form: Q'*A*Q = H,
where Q is an orthogonal matrix, H - Hessenberg matrix.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix A with elements [0..N-1, 0..N-1]
N - size of matrix A.
Output parameters:
A - matrices Q and P in compact form (see below).
Tau - array of scalar factors which are used to form matrix Q.
Array whose index ranges within [0..N-2]
Matrix H is located on the main diagonal, on the lower secondary diagonal
and above the main diagonal of matrix A. The elements which are used to
form matrix Q are situated in array Tau and below the lower secondary
diagonal of matrix A as follows:
Matrix Q is represented as a product of elementary reflections
Q = H(0)*H(2)*...*H(n-2),
where each H(i) is given by
H(i) = 1 - tau * v * (v^T)
where tau is a scalar stored in Tau[I]; v - is a real vector,
so that v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) stored in A(i+2:n-1,i).
-- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1992
*************************************************************************/
void rmatrixhessenberg(real_2d_array &a, const ae_int_t n, real_1d_array &tau, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Unpacking matrix Q which reduces matrix A to upper Hessenberg form
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - output of RMatrixHessenberg subroutine.
N - size of matrix A.
Tau - scalar factors which are used to form Q.
Output of RMatrixHessenberg subroutine.
Output parameters:
Q - matrix Q.
Array whose indexes range within [0..N-1, 0..N-1].
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixhessenbergunpackq(const real_2d_array &a, const ae_int_t n, const real_1d_array &tau, real_2d_array &q, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Unpacking matrix H (the result of matrix A reduction to upper Hessenberg form)
Input parameters:
A - output of RMatrixHessenberg subroutine.
N - size of matrix A.
Output parameters:
H - matrix H. Array whose indexes range within [0..N-1, 0..N-1].
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixhessenbergunpackh(const real_2d_array &a, const ae_int_t n, real_2d_array &h, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Reduction of a symmetric matrix which is given by its higher or lower
triangular part to a tridiagonal matrix using orthogonal similarity
transformation: Q'*A*Q=T.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix to be transformed
array with elements [0..N-1, 0..N-1].
N - size of matrix A.
IsUpper - storage format. If IsUpper = True, then matrix A is given
by its upper triangle, and the lower triangle is not used
and not modified by the algorithm, and vice versa
if IsUpper = False.
Output parameters:
A - matrices T and Q in compact form (see lower)
Tau - array of factors which are forming matrices H(i)
array with elements [0..N-2].
D - main diagonal of symmetric matrix T.
array with elements [0..N-1].
E - secondary diagonal of symmetric matrix T.
array with elements [0..N-2].
If IsUpper=True, the matrix Q is represented as a product of elementary
reflectors
Q = H(n-2) . . . H(2) H(0).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in
A(0:i-1,i+1), and tau in TAU(i).
If IsUpper=False, the matrix Q is represented as a product of elementary
reflectors
Q = H(0) H(2) . . . H(n-2).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i),
and tau in TAU(i).
The contents of A on exit are illustrated by the following examples
with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v1 v2 v3 ) ( d )
( d e v2 v3 ) ( e d )
( d e v3 ) ( v0 e d )
( d e ) ( v0 v1 e d )
( d ) ( v0 v1 v2 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).
-- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1992
*************************************************************************/
void smatrixtd(real_2d_array &a, const ae_int_t n, const bool isupper, real_1d_array &tau, real_1d_array &d, real_1d_array &e, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Unpacking matrix Q which reduces symmetric matrix to a tridiagonal
form.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - the result of a SMatrixTD subroutine
N - size of matrix A.
IsUpper - storage format (a parameter of SMatrixTD subroutine)
Tau - the result of a SMatrixTD subroutine
Output parameters:
Q - transformation matrix.
array with elements [0..N-1, 0..N-1].
-- ALGLIB --
Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void smatrixtdunpackq(const real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &tau, real_2d_array &q, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Reduction of a Hermitian matrix which is given by its higher or lower
triangular part to a real tridiagonal matrix using unitary similarity
transformation: Q'*A*Q = T.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - matrix to be transformed
array with elements [0..N-1, 0..N-1].
N - size of matrix A.
IsUpper - storage format. If IsUpper = True, then matrix A is given
by its upper triangle, and the lower triangle is not used
and not modified by the algorithm, and vice versa
if IsUpper = False.
Output parameters:
A - matrices T and Q in compact form (see lower)
Tau - array of factors which are forming matrices H(i)
array with elements [0..N-2].
D - main diagonal of real symmetric matrix T.
array with elements [0..N-1].
E - secondary diagonal of real symmetric matrix T.
array with elements [0..N-2].
If IsUpper=True, the matrix Q is represented as a product of elementary
reflectors
Q = H(n-2) . . . H(2) H(0).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a complex scalar, and v is a complex vector with
v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in
A(0:i-1,i+1), and tau in TAU(i).
If IsUpper=False, the matrix Q is represented as a product of elementary
reflectors
Q = H(0) H(2) . . . H(n-2).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a complex scalar, and v is a complex vector with
v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i),
and tau in TAU(i).
The contents of A on exit are illustrated by the following examples
with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v1 v2 v3 ) ( d )
( d e v2 v3 ) ( e d )
( d e v3 ) ( v0 e d )
( d e ) ( v0 v1 e d )
( d ) ( v0 v1 v2 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).
-- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1992
*************************************************************************/
void hmatrixtd(complex_2d_array &a, const ae_int_t n, const bool isupper, complex_1d_array &tau, real_1d_array &d, real_1d_array &e, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Unpacking matrix Q which reduces a Hermitian matrix to a real tridiagonal
form.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - the result of a HMatrixTD subroutine
N - size of matrix A.
IsUpper - storage format (a parameter of HMatrixTD subroutine)
Tau - the result of a HMatrixTD subroutine
Output parameters:
Q - transformation matrix.
array with elements [0..N-1, 0..N-1].
-- ALGLIB --
Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void hmatrixtdunpackq(const complex_2d_array &a, const ae_int_t n, const bool isupper, const complex_1d_array &tau, complex_2d_array &q, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_FBLS) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_BDSVD) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Singular value decomposition of a bidiagonal matrix (extended algorithm)
COMMERCIAL EDITION OF ALGLIB:
! Commercial version of ALGLIB includes one important improvement of
! this function, which can be used from C++ and C#:
! * 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, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_SVD) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Singular value decomposition of a rectangular matrix.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
The algorithm calculates the singular value decomposition of a matrix of
size MxN: A = U * S * V^T
The algorithm finds the singular values and, optionally, matrices U and V^T.
The algorithm can find both first min(M,N) columns of matrix U and rows of
matrix V^T (singular vectors), and matrices U and V^T wholly (of sizes MxM
and NxN respectively).
Take into account that the subroutine does not return matrix V but V^T.
Input parameters:
A - matrix to be decomposed.
Array whose indexes range within [0..M-1, 0..N-1].
M - number of rows in matrix A.
N - number of columns in matrix A.
UNeeded - 0, 1 or 2. See the description of the parameter U.
VTNeeded - 0, 1 or 2. See the description of the parameter VT.
AdditionalMemory -
If the parameter:
* equals 0, the algorithm doesn't use additional
memory (lower requirements, lower performance).
* equals 1, the algorithm uses additional
memory of size min(M,N)*min(M,N) of real numbers.
It often speeds up the algorithm.
* equals 2, the algorithm uses additional
memory of size M*min(M,N) of real numbers.
It allows to get a maximum performance.
The recommended value of the parameter is 2.
Output parameters:
W - contains singular values in descending order.
U - if UNeeded=0, U isn't changed, the left singular vectors
are not calculated.
if Uneeded=1, U contains left singular vectors (first
min(M,N) columns of matrix U). Array whose indexes range
within [0..M-1, 0..Min(M,N)-1].
if UNeeded=2, U contains matrix U wholly. Array whose
indexes range within [0..M-1, 0..M-1].
VT - if VTNeeded=0, VT isn't changed, the right singular vectors
are not calculated.
if VTNeeded=1, VT contains right singular vectors (first
min(M,N) rows of matrix V^T). Array whose indexes range
within [0..min(M,N)-1, 0..N-1].
if VTNeeded=2, VT contains matrix V^T wholly. Array whose
indexes range within [0..N-1, 0..N-1].
-- ALGLIB --
Copyright 2005 by Bochkanov Sergey
*************************************************************************/
bool rmatrixsvd(const real_2d_array &a, const ae_int_t m, const ae_int_t n, const ae_int_t uneeded, const ae_int_t vtneeded, const ae_int_t additionalmemory, real_1d_array &w, real_2d_array &u, real_2d_array &vt, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_NORMESTIMATOR) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
This procedure initializes matrix norm estimator.
USAGE:
1. User initializes algorithm state with NormEstimatorCreate() call
2. User calls NormEstimatorEstimateSparse() (or NormEstimatorIteration())
3. User calls NormEstimatorResults() to get solution.
INPUT PARAMETERS:
M - number of rows in the matrix being estimated, M>0
N - number of columns in the matrix being estimated, N>0
NStart - number of random starting vectors
recommended value - at least 5.
NIts - number of iterations to do with best starting vector
recommended value - at least 5.
OUTPUT PARAMETERS:
State - structure which stores algorithm state
NOTE: this algorithm is effectively deterministic, i.e. it always returns
same result when repeatedly called for the same matrix. In fact, algorithm
uses randomized starting vectors, but internal random numbers generator
always generates same sequence of the random values (it is a feature, not
bug).
Algorithm can be made non-deterministic with NormEstimatorSetSeed(0) call.
-- ALGLIB --
Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorcreate(const ae_int_t m, const ae_int_t n, const ae_int_t nstart, const ae_int_t nits, normestimatorstate &state, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function changes seed value used by algorithm. In some cases we need
deterministic processing, i.e. subsequent calls must return equal results,
in other cases we need non-deterministic algorithm which returns different
results for the same matrix on every pass.
Setting zero seed will lead to non-deterministic algorithm, while non-zero
value will make our algorithm deterministic.
INPUT PARAMETERS:
State - norm estimator state, must be initialized with a call
to NormEstimatorCreate()
SeedVal - seed value, >=0. Zero value = non-deterministic algo.
-- ALGLIB --
Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorsetseed(const normestimatorstate &state, const ae_int_t seedval, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function estimates norm of the sparse M*N matrix A.
INPUT PARAMETERS:
State - norm estimator state, must be initialized with a call
to NormEstimatorCreate()
A - sparse M*N matrix, must be converted to CRS format
prior to calling this function.
After this function is over you can call NormEstimatorResults() to get
estimate of the norm(A).
-- ALGLIB --
Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorestimatesparse(const normestimatorstate &state, const sparsematrix &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Matrix norm estimation results
INPUT PARAMETERS:
State - algorithm state
OUTPUT PARAMETERS:
Nrm - estimate of the matrix norm, Nrm>=0
-- ALGLIB --
Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorresults(const normestimatorstate &state, double &nrm, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_HSSCHUR) || !defined(AE_PARTIAL_BUILD)
#endif
#if defined(AE_COMPILE_EVD) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
This function initializes subspace iteration solver. This solver is used
to solve symmetric real eigenproblems where just a few (top K) eigenvalues
and corresponding eigenvectors is required.
This solver can be significantly faster than complete EVD decomposition
in the following case:
* when only just a small fraction of top eigenpairs of dense matrix is
required. When K approaches N, this solver is slower than complete dense
EVD
* when problem matrix is sparse (and/or is not known explicitly, i.e. only
matrix-matrix product can be performed)
USAGE (explicit dense/sparse matrix):
1. User initializes algorithm state with eigsubspacecreate() call
2. [optional] User tunes solver parameters by calling eigsubspacesetcond()
or other functions
3. User calls eigsubspacesolvedense() or eigsubspacesolvesparse() methods,
which take algorithm state and 2D array or alglib.sparsematrix object.
USAGE (out-of-core mode):
1. User initializes algorithm state with eigsubspacecreate() call
2. [optional] User tunes solver parameters by calling eigsubspacesetcond()
or other functions
3. User activates out-of-core mode of the solver and repeatedly calls
communication functions in a loop like below:
> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
> alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
> alglib.eigsubspaceoocgetrequestdata(state, out X)
> [calculate Y=A*X, with X=R^NxM]
> alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)
INPUT PARAMETERS:
N - problem dimensionality, N>0
K - number of top eigenvector to calculate, 0<K<=N.
OUTPUT PARAMETERS:
State - structure which stores algorithm state
NOTE: if you solve many similar EVD problems you may find it useful to
reuse previous subspace as warm-start point for new EVD problem. It
can be done with eigsubspacesetwarmstart() function.
-- ALGLIB --
Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacecreate(const ae_int_t n, const ae_int_t k, eigsubspacestate &state, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Buffered version of constructor which aims to reuse previously allocated
memory as much as possible.
-- ALGLIB --
Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacecreatebuf(const ae_int_t n, const ae_int_t k, const eigsubspacestate &state, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function sets stopping critera for the solver:
* error in eigenvector/value allowed by solver
* maximum number of iterations to perform
INPUT PARAMETERS:
State - solver structure
Eps - eps>=0, with non-zero value used to tell solver that
it can stop after all eigenvalues converged with
error roughly proportional to eps*MAX(LAMBDA_MAX),
where LAMBDA_MAX is a maximum eigenvalue.
Zero value means that no check for precision is
performed.
MaxIts - maxits>=0, with non-zero value used to tell solver
that it can stop after maxits steps (no matter how
precise current estimate is)
NOTE: passing eps=0 and maxits=0 results in automatic selection of
moderate eps as stopping criteria (1.0E-6 in current implementation,
but it may change without notice).
NOTE: very small values of eps are possible (say, 1.0E-12), although the
larger problem you solve (N and/or K), the harder it is to find
precise eigenvectors because rounding errors tend to accumulate.
NOTE: passing non-zero eps results in some performance penalty, roughly
equal to 2N*(2K)^2 FLOPs per iteration. These additional computations
are required in order to estimate current error in eigenvalues via
Rayleigh-Ritz process.
Most of this additional time is spent in construction of ~2Kx2K
symmetric subproblem whose eigenvalues are checked with exact
eigensolver.
This additional time is negligible if you search for eigenvalues of
the large dense matrix, but may become noticeable on highly sparse
EVD problems, where cost of matrix-matrix product is low.
If you set eps to exactly zero, Rayleigh-Ritz phase is completely
turned off.
-- ALGLIB --
Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesetcond(const eigsubspacestate &state, const double eps, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function sets warm-start mode of the solver: next call to the solver
will reuse previous subspace as warm-start point. It can significantly
speed-up convergence when you solve many similar eigenproblems.
INPUT PARAMETERS:
State - solver structure
UseWarmStart- either True or False
-- ALGLIB --
Copyright 12.11.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesetwarmstart(const eigsubspacestate &state, const bool usewarmstart, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function initiates out-of-core mode of subspace eigensolver. It
should be used in conjunction with other out-of-core-related functions of
this subspackage in a loop like below:
> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
> alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
> alglib.eigsubspaceoocgetrequestdata(state, out X)
> [calculate Y=A*X, with X=R^NxM]
> alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)
INPUT PARAMETERS:
State - solver object
MType - matrix type:
* 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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function performs subspace iteration in the out-of-core mode. It
should be used in conjunction with other out-of-core-related functions of
this subspackage in a loop like below:
> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
> alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
> alglib.eigsubspaceoocgetrequestdata(state, out X)
> [calculate Y=A*X, with X=R^NxM]
> alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)
-- ALGLIB --
Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
bool eigsubspaceooccontinue(const eigsubspacestate &state, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function is used to retrieve information about out-of-core request
sent by solver to user code: request type (current version of the solver
sends only requests for matrix-matrix products) and request size (size of
the matrices being multiplied).
This function returns just request metrics; in order to get contents of
the matrices being multiplied, use eigsubspaceoocgetrequestdata().
It should be used in conjunction with other out-of-core-related functions
of this subspackage in a loop like below:
> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
> alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
> alglib.eigsubspaceoocgetrequestdata(state, out X)
> [calculate Y=A*X, with X=R^NxM]
> alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)
INPUT PARAMETERS:
State - solver running in out-of-core mode
OUTPUT PARAMETERS:
RequestType - type of the request to process:
* 0 - for matrix-matrix product A*X, with A being
NxN matrix whose eigenvalues/vectors are needed,
and X being NxREQUESTSIZE one which is returned
by the eigsubspaceoocgetrequestdata().
RequestSize - size of the X matrix (number of columns), usually
it is several times larger than number of vectors
K requested by user.
-- ALGLIB --
Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocgetrequestinfo(const eigsubspacestate &state, ae_int_t &requesttype, ae_int_t &requestsize, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function is used to retrieve information about out-of-core request
sent by solver to user code: matrix X (array[N,RequestSize) which have to
be multiplied by out-of-core matrix A in a product A*X.
This function returns just request data; in order to get size of the data
prior to processing requestm, use eigsubspaceoocgetrequestinfo().
It should be used in conjunction with other out-of-core-related functions
of this subspackage in a loop like below:
> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
> alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
> alglib.eigsubspaceoocgetrequestdata(state, out X)
> [calculate Y=A*X, with X=R^NxM]
> alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)
INPUT PARAMETERS:
State - solver running in out-of-core mode
X - possibly preallocated storage; reallocated if
needed, left unchanged, if large enough to store
request data.
OUTPUT PARAMETERS:
X - array[N,RequestSize] or larger, leading rectangle
is filled with dense matrix X.
-- ALGLIB --
Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocgetrequestdata(const eigsubspacestate &state, real_2d_array &x, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function is used to send user reply to out-of-core request sent by
solver. Usually it is product A*X for returned by solver matrix X.
It should be used in conjunction with other out-of-core-related functions
of this subspackage in a loop like below:
> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
> alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
> alglib.eigsubspaceoocgetrequestdata(state, out X)
> [calculate Y=A*X, with X=R^NxM]
> alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)
INPUT PARAMETERS:
State - solver running in out-of-core mode
AX - array[N,RequestSize] or larger, leading rectangle
is filled with product A*X.
-- ALGLIB --
Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocsendresult(const eigsubspacestate &state, const real_2d_array &ax, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function finalizes out-of-core mode of subspace eigensolver. It
should be used in conjunction with other out-of-core-related functions of
this subspackage in a loop like below:
> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
> alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
> alglib.eigsubspaceoocgetrequestdata(state, out X)
> [calculate Y=A*X, with X=R^NxM]
> alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)
INPUT PARAMETERS:
State - solver state
OUTPUT PARAMETERS:
W - array[K], depending on solver settings:
* top K eigenvalues ordered by descending - if
eigenvectors are returned in Z
* zeros - if invariant subspace is returned in Z
Z - array[N,K], depending on solver settings either:
* matrix of eigenvectors found
* orthogonal basis of K-dimensional invariant subspace
Rep - report with additional parameters
-- ALGLIB --
Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocstop(const eigsubspacestate &state, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function runs eigensolver for dense NxN symmetric matrix A, given by
upper or lower triangle.
This function can not process nonsymmetric matrices.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * multithreading support (C++ and C# versions)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
INPUT PARAMETERS:
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, const xparams _xparams = alglib::xdefault);
/*************************************************************************
This function runs eigensolver for dense NxN symmetric matrix A, given by
upper or lower triangle.
This function can not process nonsymmetric matrices.
INPUT PARAMETERS:
State - solver state
A - NxN symmetric matrix given by one of its triangles
IsUpper - whether upper or lower triangle of A is given (the
other one is not referenced at all).
OUTPUT PARAMETERS:
W - array[K], top K eigenvalues ordered by descending
of their absolute values
Z - array[N,K], matrix of eigenvectors found
Rep - report with additional parameters
-- ALGLIB --
Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesolvesparses(const eigsubspacestate &state, const sparsematrix &a, const bool isupper, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Finding the eigenvalues and eigenvectors of a symmetric matrix
The algorithm finds eigen pairs of a symmetric matrix by reducing it to
tridiagonal form and using the QL/QR algorithm.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - symmetric matrix which is given by its upper or lower
triangular part.
Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
ZNeeded - flag controlling whether the eigenvectors are needed or not.
If ZNeeded is equal to:
* 0, the eigenvectors are not returned;
* 1, the eigenvectors are returned.
IsUpper - storage format.
Output parameters:
D - eigenvalues in ascending order.
Array whose index ranges within [0..N-1].
Z - if ZNeeded is equal to:
* 0, Z hasn't changed;
* 1, Z contains the eigenvectors.
Array whose indexes range within [0..N-1, 0..N-1].
The eigenvectors are stored in the matrix columns.
Result:
True, if the algorithm has converged.
False, if the algorithm hasn't converged (rare case).
-- ALGLIB --
Copyright 2005-2008 by Bochkanov Sergey
*************************************************************************/
bool smatrixevd(const real_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, real_1d_array &d, real_2d_array &z, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Subroutine for finding the eigenvalues (and eigenvectors) of a symmetric
matrix in a given half open interval (A, B] by using a bisection and
inverse iteration
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - symmetric matrix which is given by its upper or lower
triangular part. Array [0..N-1, 0..N-1].
N - size of matrix A.
ZNeeded - flag controlling whether the eigenvectors are needed or not.
If ZNeeded is equal to:
* 0, the eigenvectors are not returned;
* 1, the eigenvectors are returned.
IsUpperA - storage format of matrix A.
B1, B2 - half open interval (B1, B2] to search eigenvalues in.
Output parameters:
M - number of eigenvalues found in a given half-interval (M>=0).
W - array of the eigenvalues found.
Array whose index ranges within [0..M-1].
Z - if ZNeeded is equal to:
* 0, Z hasn't changed;
* 1, Z contains eigenvectors.
Array whose indexes range within [0..N-1, 0..M-1].
The eigenvectors are stored in the matrix columns.
Result:
True, if successful. M contains the number of eigenvalues in the given
half-interval (could be equal to 0), W contains the eigenvalues,
Z contains the eigenvectors (if needed).
False, if the bisection method subroutine wasn't able to find the
eigenvalues in the given interval or if the inverse iteration subroutine
wasn't able to find all the corresponding eigenvectors.
In that case, the eigenvalues and eigenvectors are not returned,
M is equal to 0.
-- ALGLIB --
Copyright 07.01.2006 by Bochkanov Sergey
*************************************************************************/
bool smatrixevdr(const real_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const double b1, const double b2, ae_int_t &m, real_1d_array &w, real_2d_array &z, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Subroutine for finding the eigenvalues and eigenvectors of a symmetric
matrix with given indexes by using bisection and inverse iteration methods.
Input parameters:
A - symmetric matrix which is given by its upper or lower
triangular part. Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
ZNeeded - flag controlling whether the eigenvectors are needed or not.
If ZNeeded is equal to:
* 0, the eigenvectors are not returned;
* 1, the eigenvectors are returned.
IsUpperA - storage format of matrix A.
I1, I2 - index interval for searching (from I1 to I2).
0 <= I1 <= I2 <= N-1.
Output parameters:
W - array of the eigenvalues found.
Array whose index ranges within [0..I2-I1].
Z - if ZNeeded is equal to:
* 0, Z hasn't changed;
* 1, Z contains eigenvectors.
Array whose indexes range within [0..N-1, 0..I2-I1].
In that case, the eigenvectors are stored in the matrix columns.
Result:
True, if successful. W contains the eigenvalues, Z contains the
eigenvectors (if needed).
False, if the bisection method subroutine wasn't able to find the
eigenvalues in the given interval or if the inverse iteration subroutine
wasn't able to find all the corresponding eigenvectors.
In that case, the eigenvalues and eigenvectors are not returned.
-- ALGLIB --
Copyright 07.01.2006 by Bochkanov Sergey
*************************************************************************/
bool smatrixevdi(const real_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const ae_int_t i1, const ae_int_t i2, real_1d_array &w, real_2d_array &z, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Finding the eigenvalues and eigenvectors of a Hermitian matrix
The algorithm finds eigen pairs of a Hermitian matrix by reducing it to
real tridiagonal form and using the QL/QR algorithm.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
A - Hermitian matrix which is given by its upper or lower
triangular part.
Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
IsUpper - storage format.
ZNeeded - flag controlling whether the eigenvectors are needed or
not. If ZNeeded is equal to:
* 0, the eigenvectors are not returned;
* 1, the eigenvectors are returned.
Output parameters:
D - eigenvalues in ascending order.
Array whose index ranges within [0..N-1].
Z - if ZNeeded is equal to:
* 0, Z hasn't changed;
* 1, Z contains the eigenvectors.
Array whose indexes range within [0..N-1, 0..N-1].
The eigenvectors are stored in the matrix columns.
Result:
True, if the algorithm has converged.
False, if the algorithm hasn't converged (rare case).
Note:
eigenvectors of Hermitian matrix are defined up to multiplication by
a complex number L, such that |L|=1.
-- ALGLIB --
Copyright 2005, 23 March 2007 by Bochkanov Sergey
*************************************************************************/
bool hmatrixevd(const complex_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, real_1d_array &d, complex_2d_array &z, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Subroutine for finding the eigenvalues (and eigenvectors) of a Hermitian
matrix in a given half-interval (A, B] by using a bisection and inverse
iteration
Input parameters:
A - Hermitian matrix which is given by its upper or lower
triangular part. Array whose indexes range within
[0..N-1, 0..N-1].
N - size of matrix A.
ZNeeded - flag controlling whether the eigenvectors are needed or
not. If ZNeeded is equal to:
* 0, the eigenvectors are not returned;
* 1, the eigenvectors are returned.
IsUpperA - storage format of matrix A.
B1, B2 - half-interval (B1, B2] to search eigenvalues in.
Output parameters:
M - number of eigenvalues found in a given half-interval, M>=0
W - array of the eigenvalues found.
Array whose index ranges within [0..M-1].
Z - if ZNeeded is equal to:
* 0, Z hasn't changed;
* 1, Z contains eigenvectors.
Array whose indexes range within [0..N-1, 0..M-1].
The eigenvectors are stored in the matrix columns.
Result:
True, if successful. M contains the number of eigenvalues in the given
half-interval (could be equal to 0), W contains the eigenvalues,
Z contains the eigenvectors (if needed).
False, if the bisection method subroutine wasn't able to find the
eigenvalues in the given interval or if the inverse iteration
subroutine wasn't able to find all the corresponding eigenvectors.
In that case, the eigenvalues and eigenvectors are not returned, M is
equal to 0.
Note:
eigen vectors of Hermitian matrix are defined up to multiplication by
a complex number L, such as |L|=1.
-- ALGLIB --
Copyright 07.01.2006, 24.03.2007 by Bochkanov Sergey.
*************************************************************************/
bool hmatrixevdr(const complex_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const double b1, const double b2, ae_int_t &m, real_1d_array &w, complex_2d_array &z, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Subroutine for finding the eigenvalues and eigenvectors of a Hermitian
matrix with given indexes by using bisection and inverse iteration methods
Input parameters:
A - Hermitian matrix which is given by its upper or lower
triangular part.
Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
ZNeeded - flag controlling whether the eigenvectors are needed or
not. If ZNeeded is equal to:
* 0, the eigenvectors are not returned;
* 1, the eigenvectors are returned.
IsUpperA - storage format of matrix A.
I1, I2 - index interval for searching (from I1 to I2).
0 <= I1 <= I2 <= N-1.
Output parameters:
W - array of the eigenvalues found.
Array whose index ranges within [0..I2-I1].
Z - if ZNeeded is equal to:
* 0, Z hasn't changed;
* 1, Z contains eigenvectors.
Array whose indexes range within [0..N-1, 0..I2-I1].
In that case, the eigenvectors are stored in the matrix
columns.
Result:
True, if successful. W contains the eigenvalues, Z contains the
eigenvectors (if needed).
False, if the bisection method subroutine wasn't able to find the
eigenvalues in the given interval or if the inverse iteration
subroutine wasn't able to find all the corresponding eigenvectors.
In that case, the eigenvalues and eigenvectors are not returned.
Note:
eigen vectors of Hermitian matrix are defined up to multiplication by
a complex number L, such as |L|=1.
-- ALGLIB --
Copyright 07.01.2006, 24.03.2007 by Bochkanov Sergey.
*************************************************************************/
bool hmatrixevdi(const complex_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const ae_int_t i1, const ae_int_t i2, real_1d_array &w, complex_2d_array &z, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Finding the eigenvalues and eigenvectors of a tridiagonal symmetric matrix
The algorithm finds the eigen pairs of a tridiagonal symmetric matrix by
using an QL/QR algorithm with implicit shifts.
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
Input parameters:
D - the main diagonal of a tridiagonal matrix.
Array whose index ranges within [0..N-1].
E - the secondary diagonal of a tridiagonal matrix.
Array whose index ranges within [0..N-2].
N - size of matrix A.
ZNeeded - flag controlling whether the eigenvectors are needed or not.
If ZNeeded is equal to:
* 0, the eigenvectors are not needed;
* 1, the eigenvectors of a tridiagonal matrix
are multiplied by the square matrix Z. It is used if the
tridiagonal matrix is obtained by the similarity
transformation of a symmetric matrix;
* 2, the eigenvectors of a tridiagonal matrix replace the
square matrix Z;
* 3, matrix Z contains the first row of the eigenvectors
matrix.
Z - if ZNeeded=1, Z contains the square matrix by which the
eigenvectors are multiplied.
Array whose indexes range within [0..N-1, 0..N-1].
Output parameters:
D - eigenvalues in ascending order.
Array whose index ranges within [0..N-1].
Z - if ZNeeded is equal to:
* 0, Z hasn't changed;
* 1, Z contains the product of a given matrix (from the left)
and the eigenvectors matrix (from the right);
* 2, Z contains the eigenvectors.
* 3, Z contains the first row of the eigenvectors matrix.
If ZNeeded<3, Z is the array whose indexes range within [0..N-1, 0..N-1].
In that case, the eigenvectors are stored in the matrix columns.
If ZNeeded=3, Z is the array whose indexes range within [0..0, 0..N-1].
Result:
True, if the algorithm has converged.
False, if the algorithm hasn't converged.
-- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
*************************************************************************/
bool smatrixtdevd(real_1d_array &d, const real_1d_array &e, const ae_int_t n, const ae_int_t zneeded, real_2d_array &z, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Subroutine for finding the tridiagonal matrix eigenvalues/vectors in a
given half-interval (A, B] by using bisection and inverse iteration.
Input parameters:
D - the main diagonal of a tridiagonal matrix.
Array whose index ranges within [0..N-1].
E - the secondary diagonal of a tridiagonal matrix.
Array whose index ranges within [0..N-2].
N - size of matrix, N>=0.
ZNeeded - flag controlling whether the eigenvectors are needed or not.
If ZNeeded is equal to:
* 0, the eigenvectors are not needed;
* 1, the eigenvectors of a tridiagonal matrix are multiplied
by the square matrix Z. It is used if the tridiagonal
matrix is obtained by the similarity transformation
of a symmetric matrix.
* 2, the eigenvectors of a tridiagonal matrix replace matrix Z.
A, B - half-interval (A, B] to search eigenvalues in.
Z - if ZNeeded is equal to:
* 0, Z isn't used and remains unchanged;
* 1, Z contains the square matrix (array whose indexes range
within [0..N-1, 0..N-1]) which reduces the given symmetric
matrix to tridiagonal form;
* 2, Z isn't used (but changed on the exit).
Output parameters:
D - array of the eigenvalues found.
Array whose index ranges within [0..M-1].
M - number of eigenvalues found in the given half-interval (M>=0).
Z - if ZNeeded is equal to:
* 0, doesn't contain any information;
* 1, contains the product of a given NxN matrix Z (from the
left) and NxM matrix of the eigenvectors found (from the
right). Array whose indexes range within [0..N-1, 0..M-1].
* 2, contains the matrix of the eigenvectors found.
Array whose indexes range within [0..N-1, 0..M-1].
Result:
True, if successful. In that case, M contains the number of eigenvalues
in the given half-interval (could be equal to 0), D contains the eigenvalues,
Z contains the eigenvectors (if needed).
It should be noted that the subroutine changes the size of arrays D and Z.
False, if the bisection method subroutine wasn't able to find the
eigenvalues in the given interval or if the inverse iteration subroutine
wasn't able to find all the corresponding eigenvectors. In that case,
the eigenvalues and eigenvectors are not returned, M is equal to 0.
-- ALGLIB --
Copyright 31.03.2008 by Bochkanov Sergey
*************************************************************************/
bool smatrixtdevdr(real_1d_array &d, const real_1d_array &e, const ae_int_t n, const ae_int_t zneeded, const double a, const double b, ae_int_t &m, real_2d_array &z, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Subroutine for finding tridiagonal matrix eigenvalues/vectors with given
indexes (in ascending order) by using the bisection and inverse iteraion.
Input parameters:
D - the main diagonal of a tridiagonal matrix.
Array whose index ranges within [0..N-1].
E - the secondary diagonal of a tridiagonal matrix.
Array whose index ranges within [0..N-2].
N - size of matrix. N>=0.
ZNeeded - flag controlling whether the eigenvectors are needed or not.
If ZNeeded is equal to:
* 0, the eigenvectors are not needed;
* 1, the eigenvectors of a tridiagonal matrix are multiplied
by the square matrix Z. It is used if the
tridiagonal matrix is obtained by the similarity transformation
of a symmetric matrix.
* 2, the eigenvectors of a tridiagonal matrix replace
matrix Z.
I1, I2 - index interval for searching (from I1 to I2).
0 <= I1 <= I2 <= N-1.
Z - if ZNeeded is equal to:
* 0, Z isn't used and remains unchanged;
* 1, Z contains the square matrix (array whose indexes range within [0..N-1, 0..N-1])
which reduces the given symmetric matrix to tridiagonal form;
* 2, Z isn't used (but changed on the exit).
Output parameters:
D - array of the eigenvalues found.
Array whose index ranges within [0..I2-I1].
Z - if ZNeeded is equal to:
* 0, doesn't contain any information;
* 1, contains the product of a given NxN matrix Z (from the left) and
Nx(I2-I1) matrix of the eigenvectors found (from the right).
Array whose indexes range within [0..N-1, 0..I2-I1].
* 2, contains the matrix of the eigenvalues found.
Array whose indexes range within [0..N-1, 0..I2-I1].
Result:
True, if successful. In that case, D contains the eigenvalues,
Z contains the eigenvectors (if needed).
It should be noted that the subroutine changes the size of arrays D and Z.
False, if the bisection method subroutine wasn't able to find the eigenvalues
in the given interval or if the inverse iteration subroutine wasn't able
to find all the corresponding eigenvectors. In that case, the eigenvalues
and eigenvectors are not returned.
-- ALGLIB --
Copyright 25.12.2005 by Bochkanov Sergey
*************************************************************************/
bool smatrixtdevdi(real_1d_array &d, const real_1d_array &e, const ae_int_t n, const ae_int_t zneeded, const ae_int_t i1, const ae_int_t i2, real_2d_array &z, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Finding eigenvalues and eigenvectors of a general (unsymmetric) matrix
! COMMERCIAL EDITION OF ALGLIB:
!
! Commercial Edition of ALGLIB includes following important improvements
! of this function:
! * high-performance native backend with same C# interface (C# version)
! * hardware vendor (Intel) implementations of linear algebra primitives
! (C++ and C# versions, x86/x64 platform)
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.
The algorithm finds eigenvalues and eigenvectors of a general matrix by
using the QR algorithm with multiple shifts. The algorithm can find
eigenvalues and both left and right eigenvectors.
The right eigenvector is a vector x such that A*x = w*x, and the left
eigenvector is a vector y such that y'*A = w*y' (here y' implies a complex
conjugate transposition of vector y).
Input parameters:
A - matrix. Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
VNeeded - flag controlling whether eigenvectors are needed or not.
If VNeeded is equal to:
* 0, eigenvectors are not returned;
* 1, right eigenvectors are returned;
* 2, left eigenvectors are returned;
* 3, both left and right eigenvectors are returned.
Output parameters:
WR - real parts of eigenvalues.
Array whose index ranges within [0..N-1].
WR - imaginary parts of eigenvalues.
Array whose index ranges within [0..N-1].
VL, VR - arrays of left and right eigenvectors (if they are needed).
If WI[i]=0, the respective eigenvalue is a real number,
and it corresponds to the column number I of matrices VL/VR.
If WI[i]>0, we have a pair of complex conjugate numbers with
positive and negative imaginary parts:
the first eigenvalue WR[i] + sqrt(-1)*WI[i];
the second eigenvalue WR[i+1] + sqrt(-1)*WI[i+1];
WI[i]>0
WI[i+1] = -WI[i] < 0
In that case, the eigenvector corresponding to the first
eigenvalue is located in i and i+1 columns of matrices
VL/VR (the column number i contains the real part, and the
column number i+1 contains the imaginary part), and the vector
corresponding to the second eigenvalue is a complex conjugate to
the first vector.
Arrays whose indexes range within [0..N-1, 0..N-1].
Result:
True, if the algorithm has converged.
False, if the algorithm has not converged.
Note 1:
Some users may ask the following question: what if WI[N-1]>0?
WI[N] must contain an eigenvalue which is complex conjugate to the
N-th eigenvalue, but the array has only size N?
The answer is as follows: such a situation cannot occur because the
algorithm finds a pairs of eigenvalues, therefore, if WI[i]>0, I is
strictly less than N-1.
Note 2:
The algorithm performance depends on the value of the internal parameter
NS of the InternalSchurDecomposition subroutine which defines the number
of shifts in the QR algorithm (similarly to the block width in block-matrix
algorithms of linear algebra). If you require maximum performance
on your machine, it is recommended to adjust this parameter manually.
See also the InternalTREVC subroutine.
The algorithm is based on the LAPACK 3.0 library.
*************************************************************************/
bool rmatrixevd(const real_2d_array &a, const ae_int_t n, const ae_int_t vneeded, real_1d_array &wr, real_1d_array &wi, real_2d_array &vl, real_2d_array &vr, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_SCHUR) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Subroutine performing the Schur decomposition of a general matrix by using
the QR algorithm with multiple shifts.
COMMERCIAL EDITION OF ALGLIB:
! Commercial version of ALGLIB includes one important improvement of
! this function, which can be used from C++ and C#:
! * 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, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_SPDGEVD) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Algorithm for solving the following generalized symmetric positive-definite
eigenproblem:
A*x = lambda*B*x (1) or
A*B*x = lambda*x (2) or
B*A*x = lambda*x (3).
where A is a symmetric matrix, B - symmetric positive-definite matrix.
The problem is solved by reducing it to an ordinary symmetric eigenvalue
problem.
Input parameters:
A - symmetric matrix which is given by its upper or lower
triangular part.
Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrices A and B.
IsUpperA - storage format of matrix A.
B - symmetric positive-definite matrix which is given by
its upper or lower triangular part.
Array whose indexes range within [0..N-1, 0..N-1].
IsUpperB - storage format of matrix B.
ZNeeded - if ZNeeded is equal to:
* 0, the eigenvectors are not returned;
* 1, the eigenvectors are returned.
ProblemType - if ProblemType is equal to:
* 1, the following problem is solved: A*x = lambda*B*x;
* 2, the following problem is solved: A*B*x = lambda*x;
* 3, the following problem is solved: B*A*x = lambda*x.
Output parameters:
D - eigenvalues in ascending order.
Array whose index ranges within [0..N-1].
Z - if ZNeeded is equal to:
* 0, Z hasn't changed;
* 1, Z contains eigenvectors.
Array whose indexes range within [0..N-1, 0..N-1].
The eigenvectors are stored in matrix columns. It should
be noted that the eigenvectors in such problems do not
form an orthogonal system.
Result:
True, if the problem was solved successfully.
False, if the error occurred during the Cholesky decomposition of matrix
B (the matrix isn't positive-definite) or during the work of the iterative
algorithm for solving the symmetric eigenproblem.
See also the GeneralizedSymmetricDefiniteEVDReduce subroutine.
-- ALGLIB --
Copyright 1.28.2006 by Bochkanov Sergey
*************************************************************************/
bool smatrixgevd(const real_2d_array &a, const ae_int_t n, const bool isuppera, const real_2d_array &b, const bool isupperb, const ae_int_t zneeded, const ae_int_t problemtype, real_1d_array &d, real_2d_array &z, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Algorithm for reduction of the following generalized symmetric positive-
definite eigenvalue problem:
A*x = lambda*B*x (1) or
A*B*x = lambda*x (2) or
B*A*x = lambda*x (3)
to the symmetric eigenvalues problem C*y = lambda*y (eigenvalues of this and
the given problems are the same, and the eigenvectors of the given problem
could be obtained by multiplying the obtained eigenvectors by the
transformation matrix x = R*y).
Here A is a symmetric matrix, B - symmetric positive-definite matrix.
Input parameters:
A - symmetric matrix which is given by its upper or lower
triangular part.
Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrices A and B.
IsUpperA - storage format of matrix A.
B - symmetric positive-definite matrix which is given by
its upper or lower triangular part.
Array whose indexes range within [0..N-1, 0..N-1].
IsUpperB - storage format of matrix B.
ProblemType - if ProblemType is equal to:
* 1, the following problem is solved: A*x = lambda*B*x;
* 2, the following problem is solved: A*B*x = lambda*x;
* 3, the following problem is solved: B*A*x = lambda*x.
Output parameters:
A - symmetric matrix which is given by its upper or lower
triangle depending on IsUpperA. Contains matrix C.
Array whose indexes range within [0..N-1, 0..N-1].
R - upper triangular or low triangular transformation matrix
which is used to obtain the eigenvectors of a given problem
as the product of eigenvectors of C (from the right) and
matrix R (from the left). If the matrix is upper
triangular, the elements below the main diagonal
are equal to 0 (and vice versa). Thus, we can perform
the multiplication without taking into account the
internal structure (which is an easier though less
effective way).
Array whose indexes range within [0..N-1, 0..N-1].
IsUpperR - type of matrix R (upper or lower triangular).
Result:
True, if the problem was reduced successfully.
False, if the error occurred during the Cholesky decomposition of
matrix B (the matrix is not positive-definite).
-- ALGLIB --
Copyright 1.28.2006 by Bochkanov Sergey
*************************************************************************/
bool smatrixgevdreduce(real_2d_array &a, const ae_int_t n, const bool isuppera, const real_2d_array &b, const bool isupperb, const ae_int_t problemtype, real_2d_array &r, bool &isupperr, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_INVERSEUPDATE) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula
The algorithm updates matrix A^-1 when adding a number to an element
of matrix A.
Input parameters:
InvA - inverse of matrix A.
Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
UpdRow - row where the element to be updated is stored.
UpdColumn - column where the element to be updated is stored.
UpdVal - a number to be added to the element.
Output parameters:
InvA - inverse of modified matrix A.
-- ALGLIB --
Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void rmatrixinvupdatesimple(real_2d_array &inva, const ae_int_t n, const ae_int_t updrow, const ae_int_t updcolumn, const double updval, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula
The algorithm updates matrix A^-1 when adding a vector to a row
of matrix A.
Input parameters:
InvA - inverse of matrix A.
Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
UpdRow - the row of A whose vector V was added.
0 <= Row <= N-1
V - the vector to be added to a row.
Array whose index ranges within [0..N-1].
Output parameters:
InvA - inverse of modified matrix A.
-- ALGLIB --
Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void rmatrixinvupdaterow(real_2d_array &inva, const ae_int_t n, const ae_int_t updrow, const real_1d_array &v, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula
The algorithm updates matrix A^-1 when adding a vector to a column
of matrix A.
Input parameters:
InvA - inverse of matrix A.
Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
UpdColumn - the column of A whose vector U was added.
0 <= UpdColumn <= N-1
U - the vector to be added to a column.
Array whose index ranges within [0..N-1].
Output parameters:
InvA - inverse of modified matrix A.
-- ALGLIB --
Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void rmatrixinvupdatecolumn(real_2d_array &inva, const ae_int_t n, const ae_int_t updcolumn, const real_1d_array &u, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula
The algorithm computes the inverse of matrix A+u*v' by using the given matrix
A^-1 and the vectors u and v.
Input parameters:
InvA - inverse of matrix A.
Array whose indexes range within [0..N-1, 0..N-1].
N - size of matrix A.
U - the vector modifying the matrix.
Array whose index ranges within [0..N-1].
V - the vector modifying the matrix.
Array whose index ranges within [0..N-1].
Output parameters:
InvA - inverse of matrix A + u*v'.
-- ALGLIB --
Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void rmatrixinvupdateuv(real_2d_array &inva, const ae_int_t n, const real_1d_array &u, const real_1d_array &v, const xparams _xparams = alglib::xdefault);
#endif
#if defined(AE_COMPILE_MATDET) || !defined(AE_PARTIAL_BUILD)
/*************************************************************************
Determinant calculation of the matrix given by its LU decomposition.
Input parameters:
A - LU decomposition of the matrix (output of
RMatrixLU subroutine).
Pivots - table of permutations which were made during
the LU decomposition.
Output of RMatrixLU subroutine.
N - (optional) size of matrix A:
* if given, only principal NxN submatrix is processed and
overwritten. other elements are unchanged.
* if not given, automatically determined from matrix size
(A must be square matrix)
Result: matrix determinant.
-- ALGLIB --
Copyright 2005 by Bochkanov Sergey
*************************************************************************/
double rmatrixludet(const real_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double rmatrixludet(const real_2d_array &a, const integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Calculation of the determinant of a general matrix
Input parameters:
A - matrix, array[0..N-1, 0..N-1]
N - (optional) size of matrix A:
* if given, only principal NxN submatrix is processed and
overwritten. other elements are unchanged.
* if not given, automatically determined from matrix size
(A must be square matrix)
Result: determinant of matrix A.
-- ALGLIB --
Copyright 2005 by Bochkanov Sergey
*************************************************************************/
double rmatrixdet(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double rmatrixdet(const real_2d_array &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Determinant calculation of the matrix given by its LU decomposition.
Input parameters:
A - LU decomposition of the matrix (output of
RMatrixLU subroutine).
Pivots - table of permutations which were made during
the LU decomposition.
Output of RMatrixLU subroutine.
N - (optional) size of matrix A:
* if given, only principal NxN submatrix is processed and
overwritten. other elements are unchanged.
* if not given, automatically determined from matrix size
(A must be square matrix)
Result: matrix determinant.
-- ALGLIB --
Copyright 2005 by Bochkanov Sergey
*************************************************************************/
alglib::complex cmatrixludet(const complex_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, const xparams _xparams = alglib::xdefault);
alglib::complex cmatrixludet(const complex_2d_array &a, const integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Calculation of the determinant of a general matrix
Input parameters:
A - matrix, array[0..N-1, 0..N-1]
N - (optional) size of matrix A:
* if given, only principal NxN submatrix is processed and
overwritten. other elements are unchanged.
* if not given, automatically determined from matrix size
(A must be square matrix)
Result: determinant of matrix A.
-- ALGLIB --
Copyright 2005 by Bochkanov Sergey
*************************************************************************/
alglib::complex cmatrixdet(const complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
alglib::complex cmatrixdet(const complex_2d_array &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Determinant calculation of the matrix given by the Cholesky decomposition.
Input parameters:
A - Cholesky decomposition,
output of SMatrixCholesky subroutine.
N - (optional) size of matrix A:
* if given, only principal NxN submatrix is processed and
overwritten. other elements are unchanged.
* if not given, automatically determined from matrix size
(A must be square matrix)
As the determinant is equal to the product of squares of diagonal elements,
it's not necessary to specify which triangle - lower or upper - the matrix
is stored in.
Result:
matrix determinant.
-- ALGLIB --
Copyright 2005-2008 by Bochkanov Sergey
*************************************************************************/
double spdmatrixcholeskydet(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double spdmatrixcholeskydet(const real_2d_array &a, const xparams _xparams = alglib::xdefault);
/*************************************************************************
Determinant calculation of the symmetric positive definite matrix.
Input parameters:
A - matrix. Array 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, const xparams _xparams = alglib::xdefault);
double spdmatrixdet(const real_2d_array &a, const xparams _xparams = alglib::xdefault);
#endif
}
/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (FUNCTIONS)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
#if defined(AE_COMPILE_SPARSE) || !defined(AE_PARTIAL_BUILD)
void sparsecreate(ae_int_t m,
ae_int_t n,
ae_int_t k,
sparsematrix* s,
ae_state *_state);
void sparsecreatebuf(ae_int_t m,
ae_int_t n,
ae_int_t k,
sparsematrix* s,
ae_state *_state);
void sparsecreatecrs(ae_int_t m,
ae_int_t n,
/* Integer */ 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 sparsecreatesksband(ae_int_t m,
ae_int_t n,
ae_int_t bw,
sparsematrix* s,
ae_state *_state);
void sparsecreatesksbandbuf(ae_int_t m,
ae_int_t n,
ae_int_t bw,
sparsematrix* s,
ae_state *_state);
void sparsecopy(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 sparsegemv(sparsematrix* s,
double alpha,
ae_int_t ops,
/* Real */ ae_vector* x,
ae_int_t ix,
double beta,
/* Real */ ae_vector* y,
ae_int_t iy,
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);
void sparseinitduidx(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 sparsetransposecrs(sparsematrix* s, ae_state *_state);
void sparsecopytransposecrs(sparsematrix* s0,
sparsematrix* s1,
ae_state *_state);
void sparsecopytransposecrsbuf(sparsematrix* s0,
sparsematrix* s1,
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, ae_bool make_automatic);
void _sparsematrix_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _sparsematrix_clear(void* _p);
void _sparsematrix_destroy(void* _p);
void _sparsebuffers_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _sparsebuffers_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _sparsebuffers_clear(void* _p);
void _sparsebuffers_destroy(void* _p);
#endif
#if defined(AE_COMPILE_ABLAS) || !defined(AE_PARTIAL_BUILD)
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 gemmparallelsize(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 generatereflection(/* Real */ ae_vector* x,
ae_int_t n,
double* tau,
ae_state *_state);
void applyreflectionfromtheleft(/* Real */ ae_matrix* c,
double tau,
/* Real */ ae_vector* v,
ae_int_t m1,
ae_int_t m2,
ae_int_t n1,
ae_int_t n2,
/* Real */ ae_vector* work,
ae_state *_state);
void applyreflectionfromtheright(/* Real */ ae_matrix* c,
double tau,
/* Real */ ae_vector* v,
ae_int_t m1,
ae_int_t m2,
ae_int_t n1,
ae_int_t n2,
/* Real */ ae_vector* work,
ae_state *_state);
void cmatrixtranspose(ae_int_t m,
ae_int_t n,
/* Complex */ 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 rvectorcopy(ae_int_t n,
/* Real */ ae_vector* a,
ae_int_t ia,
/* Real */ ae_vector* b,
ae_int_t ib,
ae_state *_state);
void rmatrixcopy(ae_int_t m,
ae_int_t n,
/* Real */ ae_matrix* a,
ae_int_t ia,
ae_int_t ja,
/* Real */ ae_matrix* b,
ae_int_t ib,
ae_int_t jb,
ae_state *_state);
void rmatrixgencopy(ae_int_t m,
ae_int_t n,
double alpha,
/* Real */ ae_matrix* a,
ae_int_t ia,
ae_int_t ja,
double beta,
/* Real */ ae_matrix* b,
ae_int_t ib,
ae_int_t jb,
ae_state *_state);
void rmatrixger(ae_int_t m,
ae_int_t n,
/* Real */ ae_matrix* a,
ae_int_t ia,
ae_int_t ja,
double alpha,
/* Real */ ae_vector* u,
ae_int_t iu,
/* Real */ ae_vector* v,
ae_int_t iv,
ae_state *_state);
void cmatrixrank1(ae_int_t m,
ae_int_t n,
/* Complex */ ae_matrix* a,
ae_int_t ia,
ae_int_t ja,
/* Complex */ 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 rmatrixgemv(ae_int_t m,
ae_int_t n,
double alpha,
/* Real */ ae_matrix* a,
ae_int_t ia,
ae_int_t ja,
ae_int_t opa,
/* Real */ ae_vector* x,
ae_int_t ix,
double beta,
/* Real */ ae_vector* y,
ae_int_t iy,
ae_state *_state);
void cmatrixmv(ae_int_t m,
ae_int_t n,
/* Complex */ 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 rmatrixsymv(ae_int_t n,
double alpha,
/* Real */ ae_matrix* a,
ae_int_t ia,
ae_int_t ja,
ae_bool isupper,
/* Real */ ae_vector* x,
ae_int_t ix,
double beta,
/* Real */ ae_vector* y,
ae_int_t iy,
ae_state *_state);
double rmatrixsyvmv(ae_int_t n,
/* Real */ ae_matrix* a,
ae_int_t ia,
ae_int_t ja,
ae_bool isupper,
/* Real */ ae_vector* x,
ae_int_t ix,
/* Real */ ae_vector* tmp,
ae_state *_state);
void rmatrixtrsv(ae_int_t n,
/* Real */ ae_matrix* a,
ae_int_t ia,
ae_int_t ja,
ae_bool isupper,
ae_bool isunit,
ae_int_t optype,
/* Real */ ae_vector* x,
ae_int_t ix,
ae_state *_state);
void cmatrixrighttrsm(ae_int_t m,
ae_int_t n,
/* Complex */ ae_matrix* a,
ae_int_t i1,
ae_int_t j1,
ae_bool isupper,
ae_bool isunit,
ae_int_t optype,
/* Complex */ ae_matrix* x,
ae_int_t i2,
ae_int_t j2,
ae_state *_state);
ae_bool _trypexec_cmatrixrighttrsm(ae_int_t m,
ae_int_t n,
/* Complex */ 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);
ae_bool _trypexec_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);
ae_bool _trypexec_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);
ae_bool _trypexec_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);
ae_bool _trypexec_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);
ae_bool _trypexec_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);
ae_bool _trypexec_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);
ae_bool _trypexec_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);
#endif
#if defined(AE_COMPILE_DLU) || !defined(AE_PARTIAL_BUILD)
void cmatrixluprec(/* Complex */ ae_matrix* a,
ae_int_t offs,
ae_int_t m,
ae_int_t n,
/* Integer */ ae_vector* pivots,
/* Complex */ ae_vector* tmp,
ae_state *_state);
void rmatrixluprec(/* Real */ ae_matrix* a,
ae_int_t offs,
ae_int_t m,
ae_int_t n,
/* Integer */ ae_vector* pivots,
/* Real */ ae_vector* tmp,
ae_state *_state);
void cmatrixplurec(/* Complex */ ae_matrix* a,
ae_int_t offs,
ae_int_t m,
ae_int_t n,
/* Integer */ ae_vector* pivots,
/* Complex */ ae_vector* tmp,
ae_state *_state);
void rmatrixplurec(/* Real */ ae_matrix* a,
ae_int_t offs,
ae_int_t m,
ae_int_t n,
/* Integer */ ae_vector* pivots,
/* Real */ ae_vector* tmp,
ae_state *_state);
#endif
#if defined(AE_COMPILE_SPTRF) || !defined(AE_PARTIAL_BUILD)
ae_bool sptrflu(sparsematrix* a,
ae_int_t pivottype,
/* Integer */ ae_vector* pr,
/* Integer */ ae_vector* pc,
sluv2buffer* buf,
ae_state *_state);
void _sluv2list1matrix_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _sluv2list1matrix_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _sluv2list1matrix_clear(void* _p);
void _sluv2list1matrix_destroy(void* _p);
void _sluv2sparsetrail_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _sluv2sparsetrail_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _sluv2sparsetrail_clear(void* _p);
void _sluv2sparsetrail_destroy(void* _p);
void _sluv2densetrail_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _sluv2densetrail_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _sluv2densetrail_clear(void* _p);
void _sluv2densetrail_destroy(void* _p);
void _sluv2buffer_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _sluv2buffer_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _sluv2buffer_clear(void* _p);
void _sluv2buffer_destroy(void* _p);
#endif
#if defined(AE_COMPILE_MATGEN) || !defined(AE_PARTIAL_BUILD)
void rmatrixrndorthogonal(ae_int_t n,
/* Real */ ae_matrix* a,
ae_state *_state);
void rmatrixrndcond(ae_int_t n,
double c,
/* Real */ ae_matrix* a,
ae_state *_state);
void cmatrixrndorthogonal(ae_int_t n,
/* Complex */ ae_matrix* a,
ae_state *_state);
void cmatrixrndcond(ae_int_t n,
double c,
/* Complex */ ae_matrix* a,
ae_state *_state);
void smatrixrndcond(ae_int_t n,
double c,
/* Real */ ae_matrix* a,
ae_state *_state);
void spdmatrixrndcond(ae_int_t n,
double c,
/* Real */ ae_matrix* a,
ae_state *_state);
void hmatrixrndcond(ae_int_t n,
double c,
/* Complex */ ae_matrix* a,
ae_state *_state);
void hpdmatrixrndcond(ae_int_t n,
double c,
/* Complex */ ae_matrix* a,
ae_state *_state);
void rmatrixrndorthogonalfromtheright(/* Real */ ae_matrix* a,
ae_int_t m,
ae_int_t n,
ae_state *_state);
void rmatrixrndorthogonalfromtheleft(/* Real */ ae_matrix* a,
ae_int_t m,
ae_int_t n,
ae_state *_state);
void cmatrixrndorthogonalfromtheright(/* Complex */ ae_matrix* a,
ae_int_t m,
ae_int_t n,
ae_state *_state);
void cmatrixrndorthogonalfromtheleft(/* Complex */ ae_matrix* a,
ae_int_t m,
ae_int_t n,
ae_state *_state);
void smatrixrndmultiply(/* Real */ ae_matrix* a,
ae_int_t n,
ae_state *_state);
void hmatrixrndmultiply(/* Complex */ ae_matrix* a,
ae_int_t n,
ae_state *_state);
#endif
#if defined(AE_COMPILE_TRFAC) || !defined(AE_PARTIAL_BUILD)
void rmatrixlu(/* Real */ ae_matrix* a,
ae_int_t m,
ae_int_t n,
/* Integer */ ae_vector* pivots,
ae_state *_state);
void cmatrixlu(/* Complex */ ae_matrix* a,
ae_int_t m,
ae_int_t n,
/* Integer */ ae_vector* pivots,
ae_state *_state);
ae_bool hpdmatrixcholesky(/* Complex */ ae_matrix* a,
ae_int_t n,
ae_bool isupper,
ae_state *_state);
ae_bool spdmatrixcholesky(/* Real */ ae_matrix* a,
ae_int_t n,
ae_bool isupper,
ae_state *_state);
void spdmatrixcholeskyupdateadd1(/* Real */ ae_matrix* a,
ae_int_t n,
ae_bool isupper,
/* Real */ 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 sparselu(sparsematrix* a,
ae_int_t pivottype,
/* Integer */ ae_vector* p,
/* Integer */ ae_vector* q,
ae_state *_state);
ae_bool sparsecholeskyskyline(sparsematrix* a,
ae_int_t n,
ae_bool isupper,
ae_state *_state);
ae_bool 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);
#endif
#if defined(AE_COMPILE_RCOND) || !defined(AE_PARTIAL_BUILD)
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);
#endif
#if defined(AE_COMPILE_MATINV) || !defined(AE_PARTIAL_BUILD)
void 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 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 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 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 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 spdmatrixcholeskyinverserec(/* Real */ ae_matrix* a,
ae_int_t offs,
ae_int_t n,
ae_bool isupper,
/* Real */ ae_vector* tmp,
ae_state *_state);
ae_bool _trypexec_spdmatrixcholeskyinverserec(/* Real */ ae_matrix* a,
ae_int_t offs,
ae_int_t n,
ae_bool isupper,
/* Real */ ae_vector* tmp, ae_state *_state);
void _matinvreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _matinvreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _matinvreport_clear(void* _p);
void _matinvreport_destroy(void* _p);
#endif
#if defined(AE_COMPILE_ORTFAC) || !defined(AE_PARTIAL_BUILD)
void rmatrixqr(/* Real */ ae_matrix* a,
ae_int_t m,
ae_int_t n,
/* Real */ ae_vector* tau,
ae_state *_state);
void rmatrixlq(/* Real */ ae_matrix* a,
ae_int_t m,
ae_int_t n,
/* Real */ ae_vector* tau,
ae_state *_state);
void cmatrixqr(/* Complex */ ae_matrix* a,
ae_int_t m,
ae_int_t n,
/* Complex */ ae_vector* tau,
ae_state *_state);
void cmatrixlq(/* Complex */ ae_matrix* a,
ae_int_t m,
ae_int_t n,
/* Complex */ ae_vector* tau,
ae_state *_state);
void rmatrixqrunpackq(/* Real */ 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 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 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 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);
#endif
#if defined(AE_COMPILE_FBLS) || !defined(AE_PARTIAL_BUILD)
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, ae_bool make_automatic);
void _fblslincgstate_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _fblslincgstate_clear(void* _p);
void _fblslincgstate_destroy(void* _p);
#endif
#if defined(AE_COMPILE_BDSVD) || !defined(AE_PARTIAL_BUILD)
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);
#endif
#if defined(AE_COMPILE_SVD) || !defined(AE_PARTIAL_BUILD)
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);
#endif
#if defined(AE_COMPILE_NORMESTIMATOR) || !defined(AE_PARTIAL_BUILD)
void normestimatorcreate(ae_int_t m,
ae_int_t n,
ae_int_t nstart,
ae_int_t nits,
normestimatorstate* state,
ae_state *_state);
void normestimatorsetseed(normestimatorstate* state,
ae_int_t seedval,
ae_state *_state);
ae_bool normestimatoriteration(normestimatorstate* state,
ae_state *_state);
void normestimatorestimatesparse(normestimatorstate* state,
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, ae_bool make_automatic);
void _normestimatorstate_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _normestimatorstate_clear(void* _p);
void _normestimatorstate_destroy(void* _p);
#endif
#if defined(AE_COMPILE_HSSCHUR) || !defined(AE_PARTIAL_BUILD)
void rmatrixinternalschurdecomposition(/* Real */ ae_matrix* h,
ae_int_t n,
ae_int_t tneeded,
ae_int_t zneeded,
/* Real */ ae_vector* wr,
/* Real */ ae_vector* wi,
/* Real */ ae_matrix* z,
ae_int_t* info,
ae_state *_state);
ae_bool upperhessenbergschurdecomposition(/* Real */ ae_matrix* h,
ae_int_t n,
/* Real */ ae_matrix* s,
ae_state *_state);
void internalschurdecomposition(/* Real */ ae_matrix* h,
ae_int_t n,
ae_int_t tneeded,
ae_int_t zneeded,
/* Real */ ae_vector* wr,
/* Real */ ae_vector* wi,
/* Real */ ae_matrix* z,
ae_int_t* info,
ae_state *_state);
#endif
#if defined(AE_COMPILE_EVD) || !defined(AE_PARTIAL_BUILD)
void eigsubspacecreate(ae_int_t n,
ae_int_t k,
eigsubspacestate* state,
ae_state *_state);
void eigsubspacecreatebuf(ae_int_t n,
ae_int_t k,
eigsubspacestate* state,
ae_state *_state);
void eigsubspacesetcond(eigsubspacestate* state,
double eps,
ae_int_t maxits,
ae_state *_state);
void eigsubspacesetwarmstart(eigsubspacestate* state,
ae_bool usewarmstart,
ae_state *_state);
void eigsubspaceoocstart(eigsubspacestate* state,
ae_int_t mtype,
ae_state *_state);
ae_bool eigsubspaceooccontinue(eigsubspacestate* state, ae_state *_state);
void eigsubspaceoocgetrequestinfo(eigsubspacestate* state,
ae_int_t* requesttype,
ae_int_t* requestsize,
ae_state *_state);
void eigsubspaceoocgetrequestdata(eigsubspacestate* state,
/* Real */ ae_matrix* x,
ae_state *_state);
void eigsubspaceoocsendresult(eigsubspacestate* state,
/* Real */ 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 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, ae_bool make_automatic);
void _eigsubspacestate_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _eigsubspacestate_clear(void* _p);
void _eigsubspacestate_destroy(void* _p);
void _eigsubspacereport_init(void* _p, ae_state *_state, ae_bool make_automatic);
void _eigsubspacereport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _eigsubspacereport_clear(void* _p);
void _eigsubspacereport_destroy(void* _p);
#endif
#if defined(AE_COMPILE_SCHUR) || !defined(AE_PARTIAL_BUILD)
ae_bool rmatrixschur(/* Real */ ae_matrix* a,
ae_int_t n,
/* Real */ ae_matrix* s,
ae_state *_state);
#endif
#if defined(AE_COMPILE_SPDGEVD) || !defined(AE_PARTIAL_BUILD)
ae_bool smatrixgevd(/* Real */ 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);
#endif
#if defined(AE_COMPILE_INVERSEUPDATE) || !defined(AE_PARTIAL_BUILD)
void rmatrixinvupdatesimple(/* Real */ ae_matrix* inva,
ae_int_t n,
ae_int_t updrow,
ae_int_t updcolumn,
double updval,
ae_state *_state);
void rmatrixinvupdaterow(/* Real */ ae_matrix* inva,
ae_int_t n,
ae_int_t updrow,
/* Real */ 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);
#endif
#if defined(AE_COMPILE_MATDET) || !defined(AE_PARTIAL_BUILD)
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
}
#endif