/usr/include/fflas-ffpack/ffpack/ffpack.h is in fflas-ffpack-common 1.6.0-1.
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1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 | /* -*- mode: C++; tab-width: 8; indent-tabs-mode: t; c-basic-offset: 8 -*- */
// vim:sts=8:sw=8:ts=8:noet:sr:cino=>s,f0,{0,g0,(0,\:0,t0,+0,=s
/* ffpack.h
* Copyright (C) 2005 Clement Pernet
*
* Written by Clement Pernet <Clement.Pernet@imag.fr>
*
*
* ========LICENCE========
* This file is part of the library FFLAS-FFPACK.
*
* FFLAS-FFPACK is free software: you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public
* License as published by the Free Software Foundation; either
* version 2.1 of the License, or (at your option) any later version.
*
* This library 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
* Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with this library; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
* ========LICENCE========
*.
*/
/** @file ffpack.h
* \brief Set of elimination based routines for dense linear algebra.
* Matrices are supposed over finite prime field of characteristic less than 2^26.
*/
#ifndef __FFLASFFPACK_ffpack_H
#define __FFLASFFPACK_ffpack_H
#include "fflas-ffpack/fflas/fflas.h"
#include <list>
#include <vector>
#include <iostream> // std::cout
// The use of the small size LQUP is currently disabled:
// need for a better handling of element base (double, float, generic) combined
// with different thresholds.
// TransPosed version has to be implemented too.
#ifndef __FFPACK_LUDIVINE_CUTOFF
#define __FFPACK_LUDIVINE_CUTOFF 0
#endif
#ifndef __FFPACK_CHARPOLY_THRESHOLD
#define __FFPACK_CHARPOLY_THRESHOLD 30
#endif
/** @brief <b>F</b>inite <b>F</b>ield <b>PACK</b>
* Set of elimination based routines for dense linear algebra.
*
* This namespace enlarges the set of BLAS routines of the class FFLAS, with higher
* level routines based on elimination.
\ingroup ffpack
*/
namespace FFPACK {
// public:
enum FFPACK_LUDIVINE_TAG
{
FfpackLQUP=1,
FfpackSingular=2
};
enum FFPACK_CHARPOLY_TAG
{
FfpackLUK=1,
FfpackKG=2,
FfpackHybrid=3,
FfpackKGFast=4,
FfpackDanilevski=5,
FfpackArithProg=6,
FfpackKGFastG=7
};
class CharpolyFailed{};
enum FFPACK_MINPOLY_TAG
{
FfpackDense=1,
FfpackKGF=2
};
/** Apply a permutation submatrix of P (between ibeg and iend) to a matrix
* to (iend-ibeg) vectors of size M stored in A (as column for NoTrans
* and rows for Trans).
* Side==FFLAS::FflasLeft for row permutation Side==FFLAS::FflasRight for a column
* permutation
* Trans==FFLAS::FflasTrans for the inverse permutation of P
* @param F
* @param Side
* @param Trans
* @param M
* @param ibeg
* @param iend
* @param A
* @param lda
* @param P
* @warning not sure the submatrix is still a permutation and the one we expect in all cases... examples for iend=2, ibeg=1 and P=[2,2,2]
*/
template<class Field>
void
applyP( const Field& F,
const FFLAS::FFLAS_SIDE Side,
const FFLAS::FFLAS_TRANSPOSE Trans,
const size_t M, const int ibeg, const int iend,
typename Field::Element * A, const size_t lda, const size_t * P )
{
if ( Side == FFLAS::FflasRight ) {
typename Field::Element tmp;
if ( Trans == FFLAS::FflasTrans )
for (size_t j = 0 ; j < M ; ++j){
for ( size_t i=(size_t)ibeg; i<(size_t) iend; ++i)
if ( P[i]> i ) {
F.assign(tmp,A[j*lda+P[i]]);
F.assign(A[j*lda+P[i]],A[j*lda+i]);
F.assign(A[j*lda+i],tmp);
// std::swap(A[j*lda+P[i]],A[j*lda+i]);
}
//FFLAS::fswap( F, M, A + P[i]*1, lda, A + i*1, lda );
}
else // Trans == FFLAS::FflasNoTrans
for (size_t j = 0 ; j < M ; ++j){
for (int i=iend; i-->ibeg; )
if ( P[i]>(size_t)i ) {
F.assign(tmp,A[j*lda+P[i]]);
F.assign(A[j*lda+P[i]],A[j*lda+(size_t)i]);
F.assign(A[j*lda+(size_t)i],tmp);
// std::swap(A[j*lda+P[i]],A[j*lda+(size_t)i]);
}
//FFLAS::fswap( F, M, A + P[i]*1, lda, A + i*1, lda );
}
}
else { // Side == FFLAS::FflasLeft
if ( Trans == FFLAS::FflasNoTrans )
for (size_t i=(size_t)ibeg; i<(size_t)iend; ++i){
if ( P[i]> (size_t) i )
FFLAS::fswap( F, M,
A + P[i]*lda, 1,
A + i*lda, 1 );
}
else // Trans == FFLAS::FflasTrans
for (int i=iend; i-->ibeg; ){
if ( P[i]> (size_t) i ){
FFLAS::fswap( F, M,
A + P[i]*lda, 1,
A + (size_t)i*lda, 1 );
}
}
}
}
/** Computes the rank of the given matrix using a LQUP factorization.
* The input matrix is modified.
* @param F field
* @param M row dimension of the matrix
* @param N column dimension of the matrix
* @param A input matrix
* @param lda leading dimension of A
*/
template <class Field>
size_t
Rank( const Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda)
{
size_t *P = new size_t[N];
size_t *Q = new size_t[M];
size_t R = LUdivine (F, FFLAS::FflasNonUnit, FFLAS::FflasNoTrans, M, N,
A, lda, P, Q, FfpackLQUP);
delete[] Q;
delete[] P;
return R;
}
/** Returns true if the given matrix is singular.
* The method is a block elimination with early termination
*
* using LQUP factorization with early termination.
* @warning The input matrix is modified.
* @param F field
* @param M row dimension of the matrix
* @param N column dimension of the matrix
* @param A input matrix
* @param lda leading dimension of A
*/
template <class Field>
bool
IsSingular( const Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda)
{
size_t *P = new size_t[N];
size_t *Q = new size_t[M];
bool singular = !LUdivine (F, FFLAS::FflasNonUnit, FFLAS::FflasNoTrans, M, N,
A, lda, P, Q, FfpackSingular);
delete[] P;
delete[] Q;
return singular;
}
/** Returns the determinant of the given matrix.
* The method is a block elimination with early termination
* @warning The input matrix is modified.
* @param F field
* @param M row dimension of the matrix
* @param N column dimension of the matrix
* @param A input matrix
* @param lda leading dimension of A
*/
/// using LQUP factorization with early termination.
template <class Field>
typename Field::Element
Det( const Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda)
{
typename Field::Element det; F.init(det);
bool singular;
size_t *P = new size_t[N];
size_t *Q = new size_t[M];
singular = !LUdivine (F, FFLAS::FflasNonUnit, FFLAS::FflasNoTrans, M, N,
A, lda, P, Q, FfpackSingular);
if (singular){
F.assign(det,F.zero);
delete[] P;
delete[] Q;
return det;
}
else{
F.assign(det,F.one);
typename Field::Element *Ai=A;
for (; Ai < A+ M*lda+N; Ai+=lda+1 )
F.mulin( det, *Ai );
int count=0;
for (size_t i=0;i<N;++i)
if (P[i] != i) ++count;
if ((count&1) == 1)
F.negin(det);
}
delete[] P;
delete[] Q;
return det;
}
// forward declaration
template<class Field>
void
solveLB2( const Field& F, const FFLAS::FFLAS_SIDE Side,
const size_t M, const size_t N, const size_t R,
typename Field::Element * L, const size_t ldl,
const size_t * Q,
typename Field::Element * B, const size_t ldb ) ;
/** Solve the system \f$A X = B\f$ or \f$X A = B\f$.
* Solving using the \c LQUP decomposition of \p A
* already computed inplace with \c LUdivine(FFLAS::FflasNoTrans, FFLAS::FflasNonUnit).
* Version for A square.
* If A is rank deficient, a solution is returned if the system is consistent,
* Otherwise an info is 1
*
* @param F field
* @param Side Determine wheter the resolution is left or right looking.
* @param M row dimension of \p B
* @param N col dimension of \p B
* @param R rank of \p A
* @param A input matrix
* @param lda leading dimension of \p A
* @param P column permutation of the \c LQUP decomposition of \p A
* @param Q column permutation of the \c LQUP decomposition of \p A
* @param B Right/Left hand side matrix. Initially stores \p B, finally stores the solution \p X.
* @param ldb leading dimension of \p B
* @param info Success of the computation: 0 if successfull, >0 if system is inconsistent
*/
template <class Field>
void
fgetrs (const Field& F,
const FFLAS::FFLAS_SIDE Side,
const size_t M, const size_t N, const size_t R,
typename Field::Element *A, const size_t lda,
const size_t *P, const size_t *Q,
typename Field::Element *B, const size_t ldb,
int * info)
{
*info =0;
if (Side == FFLAS::FflasLeft) { // Left looking solve A X = B
solveLB2 (F, FFLAS::FflasLeft, M, N, R, A, lda, Q, B, ldb);
applyP (F, FFLAS::FflasLeft, FFLAS::FflasNoTrans,
N, 0,(int) R, B, ldb, Q);
bool consistent = true;
for (size_t i = R; i < M; ++i)
for (size_t j = 0; j < N; ++j)
if (!F.isZero (*(B + i*ldb + j)))
consistent = false;
if (!consistent) {
std::cerr<<"System is inconsistent"<<std::endl;
*info = 1;
}
// The last rows of B are now supposed to be 0
#if 0
for (size_t i = R; i < M; ++i)
for (size_t j = 0; j < N; ++j)
*(B + i*ldb + j) = F.zero;
#endif
ftrsm (F, FFLAS::FflasLeft, FFLAS::FflasUpper, FFLAS::FflasNoTrans, FFLAS::FflasNonUnit,
R, N, F.one, A, lda , B, ldb);
applyP (F, FFLAS::FflasLeft, FFLAS::FflasTrans,
N, 0,(int) R, B, ldb, P);
}
else { // Right Looking X A = B
applyP (F, FFLAS::FflasRight, FFLAS::FflasTrans,
M, 0,(int) R, B, ldb, P);
ftrsm (F, FFLAS::FflasRight, FFLAS::FflasUpper, FFLAS::FflasNoTrans, FFLAS::FflasNonUnit,
M, R, F.one, A, lda , B, ldb);
fgemm (F, FFLAS::FflasNoTrans, FFLAS::FflasNoTrans, M, N-R, R, F.one,
B, ldb, A+R, lda, F.mOne, B+R, ldb);
bool consistent = true;
for (size_t i = 0; i < M; ++i)
for (size_t j = R; j < N; ++j)
if (!F.isZero (*(B + i*ldb + j)))
consistent = false;
if (!consistent) {
std::cerr<<"System is inconsistent"<<std::endl;
*info = 1;
}
// The last cols of B are now supposed to be 0
applyP (F, FFLAS::FflasRight, FFLAS::FflasNoTrans,
M, 0,(int) R, B, ldb, Q);
solveLB2 (F, FFLAS::FflasRight, M, N, R, A, lda, Q, B, ldb);
}
}
/** Solve the system A X = B or X A = B.
* Solving using the LQUP decomposition of A
* already computed inplace with LUdivine(FFLAS::FflasNoTrans, FFLAS::FflasNonUnit).
* Version for A rectangular.
* If A is rank deficient, a solution is returned if the system is consistent,
* Otherwise an info is 1
*
* @param F field
* @param Side Determine wheter the resolution is left or right looking.
* @param M row dimension of A
* @param N col dimension of A
* @param NRHS number of columns (if Side = FFLAS::FflasLeft) or row (if Side = FFLAS::FflasRight) of the matrices X and B
* @param R rank of A
* @param A input matrix
* @param lda leading dimension of A
* @param P column permutation of the LQUP decomposition of A
* @param Q column permutation of the LQUP decomposition of A
* @param X solution matrix
* @param ldx leading dimension of X
* @param B Right/Left hand side matrix.
* @param ldb leading dimension of B
* @param info Succes of the computation: 0 if successfull, >0 if system is inconsistent
*/
template <class Field>
typename Field::Element *
fgetrs (const Field& F,
const FFLAS::FFLAS_SIDE Side,
const size_t M, const size_t N, const size_t NRHS, const size_t R,
typename Field::Element *A, const size_t lda,
const size_t *P, const size_t *Q,
typename Field::Element *X, const size_t ldx,
const typename Field::Element *B, const size_t ldb,
int * info)
{
*info =0;
typename Field::Element* W;
size_t ldw;
if (Side == FFLAS::FflasLeft) { // Left looking solve A X = B
// Initializing X to 0 (to be optimized)
for (size_t i = 0; i <N; ++i)
for (size_t j=0; j< NRHS; ++j)
F.assign (*(X+i*ldx+j), F.zero);
if (M > N){ // Cannot copy B into X
W = new typename Field::Element [M*NRHS];
ldw = NRHS;
for (size_t i=0; i < M; ++i)
FFLAS::fcopy (F, NRHS, W + i*ldw, 1, B + i*ldb, 1);
solveLB2 (F, FFLAS::FflasLeft, M, NRHS, R, A, lda, Q, W, ldw);
applyP (F, FFLAS::FflasLeft, FFLAS::FflasNoTrans,
NRHS, 0,(int) R, W, ldw, Q);
bool consistent = true;
for (size_t i = R; i < M; ++i)
for (size_t j = 0; j < NRHS; ++j)
if (!F.isZero (*(W + i*ldw + j)))
consistent = false;
if (!consistent) {
std::cerr<<"System is inconsistent"<<std::endl;
*info = 1;
delete[] W;
return X;
}
// Here the last rows of W are supposed to be 0
ftrsm (F, FFLAS::FflasLeft, FFLAS::FflasUpper, FFLAS::FflasNoTrans, FFLAS::FflasNonUnit,
R, NRHS, F.one, A, lda , W, ldw);
for (size_t i=0; i < R; ++i)
FFLAS::fcopy (F, NRHS, X + i*ldx, 1, W + i*ldw, 1);
delete[] W;
applyP (F, FFLAS::FflasLeft, FFLAS::FflasTrans,
NRHS, 0,(int) R, X, ldx, P);
}
else { // Copy B to X directly
for (size_t i=0; i < M; ++i)
FFLAS::fcopy (F, NRHS, X + i*ldx, 1, B + i*ldb, 1);
solveLB2 (F, FFLAS::FflasLeft, M, NRHS, R, A, lda, Q, X, ldx);
applyP (F, FFLAS::FflasLeft, FFLAS::FflasNoTrans,
NRHS, 0,(int) R, X, ldx, Q);
bool consistent = true;
for (size_t i = R; i < M; ++i)
for (size_t j = 0; j < NRHS; ++j)
if (!F.isZero (*(X + i*ldx + j)))
consistent = false;
if (!consistent) {
std::cerr<<"System is inconsistent"<<std::endl;
*info = 1;
return X;
}
// Here the last rows of W are supposed to be 0
ftrsm (F, FFLAS::FflasLeft, FFLAS::FflasUpper, FFLAS::FflasNoTrans, FFLAS::FflasNonUnit,
R, NRHS, F.one, A, lda , X, ldx);
applyP (F, FFLAS::FflasLeft, FFLAS::FflasTrans,
NRHS, 0,(int) R, X, ldx, P);
}
return X;
}
else { // Right Looking X A = B
for (size_t i = 0; i <NRHS; ++i)
for (size_t j=0; j< M; ++j)
F.assign (*(X+i*ldx+j), F.zero);
if (M < N) {
W = new typename Field::Element [NRHS*N];
ldw = N;
for (size_t i=0; i < NRHS; ++i)
FFLAS::fcopy (F, N, W + i*ldw, 1, B + i*ldb, 1);
applyP (F, FFLAS::FflasRight, FFLAS::FflasTrans,
NRHS, 0,(int) R, W, ldw, P);
ftrsm (F, FFLAS::FflasRight, FFLAS::FflasUpper, FFLAS::FflasNoTrans, FFLAS::FflasNonUnit,
NRHS, R, F.one, A, lda , W, ldw);
fgemm (F, FFLAS::FflasNoTrans, FFLAS::FflasNoTrans, NRHS, N-R, R, F.one,
W, ldw, A+R, lda, F.mOne, W+R, ldw);
bool consistent = true;
for (size_t i = 0; i < NRHS; ++i)
for (size_t j = R; j < N; ++j)
if (!F.isZero (*(W + i*ldw + j)))
consistent = false;
if (!consistent) {
std::cerr<<"System is inconsistent"<<std::endl;
*info = 1;
delete[] W;
return X;
}
// The last N-R cols of W are now supposed to be 0
for (size_t i=0; i < NRHS; ++i)
FFLAS::fcopy (F, R, X + i*ldx, 1, W + i*ldb, 1);
delete[] W;
applyP (F, FFLAS::FflasRight, FFLAS::FflasNoTrans,
NRHS, 0,(int) R, X, ldx, Q);
solveLB2 (F, FFLAS::FflasRight, NRHS, M, R, A, lda, Q, X, ldx);
}
else {
for (size_t i=0; i < NRHS; ++i)
FFLAS::fcopy (F, N, X + i*ldx, 1, B + i*ldb, 1);
applyP (F, FFLAS::FflasRight, FFLAS::FflasTrans,
NRHS, 0,(int) R, X, ldx, P);
ftrsm (F, FFLAS::FflasRight, FFLAS::FflasUpper, FFLAS::FflasNoTrans, FFLAS::FflasNonUnit,
NRHS, R, F.one, A, lda , X, ldx);
fgemm (F, FFLAS::FflasNoTrans, FFLAS::FflasNoTrans, NRHS, N-R, R, F.one,
X, ldx, A+R, lda, F.mOne, X+R, ldx);
bool consistent = true;
for (size_t i = 0; i < NRHS; ++i)
for (size_t j = R; j < N; ++j)
if (!F.isZero (*(X + i*ldx + j)))
consistent = false;
if (!consistent) {
std::cerr<<"System is inconsistent"<<std::endl;
*info = 1;
return X;
}
// The last N-R cols of W are now supposed to be 0
applyP (F, FFLAS::FflasRight, FFLAS::FflasNoTrans,
NRHS, 0,(int) R, X, ldx, Q);
solveLB2 (F, FFLAS::FflasRight, NRHS, M, R, A, lda, Q, X, ldx);
}
return X;
}
}
/** @brief Square system solver
* @param F The computation domain
* @param Side Determine wheter the resolution is left or right looking
* @param M row dimension of B
* @param N col dimension of B
* @param A input matrix
* @param lda leading dimension of A
* @param B Right/Left hand side matrix. Initially contains B, finally contains the solution X.
* @param ldb leading dimension of B
* @param info Success of the computation: 0 if successfull, >0 if system is inconsistent
* @return the rank of the system
*
* Solve the system A X = B or X A = B.
* Version for A square.
* If A is rank deficient, a solution is returned if the system is consistent,
* Otherwise an info is 1
*/
template <class Field>
size_t
fgesv (const Field& F,
const FFLAS::FFLAS_SIDE Side,
const size_t M, const size_t N,
typename Field::Element *A, const size_t lda,
typename Field::Element *B, const size_t ldb,
int * info)
{
size_t Na;
if (Side == FFLAS::FflasLeft)
Na = M;
else
Na = N;
size_t* P = new size_t[Na];
size_t* Q = new size_t[Na];
size_t R = LUdivine (F, FFLAS::FflasNonUnit, FFLAS::FflasNoTrans, Na, Na, A, lda, P, Q, FfpackLQUP);
fgetrs (F, Side, M, N, R, A, lda, P, Q, B, ldb, info);
delete[] P;
delete[] Q;
return R;
}
/** @brief Rectangular system solver
* @param F The computation domain
* @param Side Determine wheter the resolution is left or right looking
* @param M row dimension of A
* @param N col dimension of A
* @param NRHS number of columns (if Side = FFLAS::FflasLeft) or row (if Side = FFLAS::FflasRight) of the matrices X and B
* @param A input matrix
* @param lda leading dimension of A
* @param B Right/Left hand side matrix. Initially contains B, finally contains the solution X.
* @param ldb leading dimension of B
* @param X
* @param ldx
* @param info Success of the computation: 0 if successfull, >0 if system is inconsistent
* @return the rank of the system
*
* Solve the system A X = B or X A = B.
* Version for A square.
* If A is rank deficient, a solution is returned if the system is consistent,
* Otherwise an info is 1
*/
template <class Field>
size_t
fgesv (const Field& F,
const FFLAS::FFLAS_SIDE Side,
const size_t M, const size_t N, const size_t NRHS,
typename Field::Element *A, const size_t lda,
typename Field::Element *X, const size_t ldx,
const typename Field::Element *B, const size_t ldb,
int * info)
{
size_t* P = new size_t[N];
size_t* Q = new size_t[M];
size_t R = LUdivine (F, FFLAS::FflasNonUnit, FFLAS::FflasNoTrans, M, N, A, lda, P, Q, FfpackLQUP);
fgetrs (F, Side, M, N, NRHS, R, A, lda, P, Q, X, ldx, B, ldb, info);
delete[] P;
delete[] Q;
return R;
}
/** Solve the system Ax=b.
* Solving using LQUP factorization and
* two triangular system resolutions.
* The input matrix is modified.
* @param F The computation domain
* @param M row dimension of the matrix
* @param A input matrix
* @param lda leading dimension of A
* @param x solution vector
* @param incx increment of x
* @param b right hand side vector
* @param incb increment of b
*/
/// Solve linear system using LQUP factorization.
template <class Field>
typename Field::Element*
Solve( const Field& F, const size_t M,
typename Field::Element * A, const size_t lda,
typename Field::Element * x, const int incx,
const typename Field::Element * b, const int incb )
{
size_t *P = new size_t[M];
size_t *rowP = new size_t[M];
if (LUdivine( F, FFLAS::FflasNonUnit, FFLAS::FflasNoTrans, M, M, A, lda, P, rowP, FfpackLQUP) < M){
std::cerr<<"SINGULAR MATRIX"<<std::endl;
delete[] P;
delete[] rowP;
return x;
}
else{
FFLAS::fcopy( F, M, x, incx, b, incb );
ftrsv(F, FFLAS::FflasLower, FFLAS::FflasNoTrans, FFLAS::FflasUnit, M,
A, lda , x, incx);
ftrsv(F, FFLAS::FflasUpper, FFLAS::FflasNoTrans, FFLAS::FflasNonUnit, M,
A, lda , x, incx);
applyP( F, FFLAS::FflasRight, FFLAS::FflasTrans,
M, 0,(int) M, x, incx, P );
delete[] rowP;
delete[] P;
return x;
}
}
/** Computes a basis of the Left/Right nullspace of the matrix A.
* return the dimension of the nullspace.
*
* @param F The computation domain
* @param Side
* @param M
* @param N
* @param A input matrix of dimension M x N, A is modified
* @param lda
* @param NS output matrix of dimension N x NSdim (allocated here)
* @param ldn
* @param NSdim the dimension of the Nullspace (N-rank(A))
*
*/
template <class Field>
size_t NullSpaceBasis (const Field& F, const FFLAS::FFLAS_SIDE Side,
const size_t M, const size_t N,
typename Field::Element* A, const size_t lda,
typename Field::Element*& NS, size_t& ldn,
size_t& NSdim)
{
if (Side == FFLAS::FflasRight) { // Right NullSpace
size_t* P = new size_t[N];
size_t* Qt = new size_t[M];
size_t R = LUdivine (F, FFLAS::FflasNonUnit, FFLAS::FflasNoTrans, M, N, A, lda, P, Qt);
ldn = N-R;
NSdim = ldn;
NS = new typename Field::Element [N*ldn];
for (size_t i=0; i<R; ++i)
FFLAS::fcopy (F, ldn, NS + i*ldn, 1, A + R + i*lda, 1);
ftrsm (F, FFLAS::FflasLeft, FFLAS::FflasUpper, FFLAS::FflasNoTrans, FFLAS::FflasNonUnit, R, ldn,
F.mOne, A, lda, NS, ldn);
for (size_t i=R; i<N; ++i){
for (size_t j=0; j < ldn; ++j)
F.assign (*(NS+i*ldn+j), F.zero);
F.assign (*(NS + i*ldn + i-R), F.one);
}
applyP (F, FFLAS::FflasLeft, FFLAS::FflasTrans,
NSdim, 0,(int) R, NS, ldn, P);
delete [] P;
delete [] Qt;
return N-R;
}
else { // Left NullSpace
size_t* P = new size_t[M];
size_t* Qt = new size_t[N];
size_t R = LUdivine (F, FFLAS::FflasNonUnit, FFLAS::FflasTrans, M, N, A, lda, P, Qt);
ldn = M;
NSdim = M-R;
NS = new typename Field::Element [NSdim*ldn];
for (size_t i=0; i<NSdim; ++i)
FFLAS::fcopy (F, R, NS + i*ldn, 1, A + (R + i)*lda, 1);
ftrsm (F, FFLAS::FflasRight, FFLAS::FflasLower, FFLAS::FflasNoTrans, FFLAS::FflasNonUnit, NSdim, R,
F.mOne, A, lda, NS, ldn);
for (size_t i=0; i<NSdim; ++i){
for (size_t j=R; j < M; ++j)
F.assign (*(NS+i*ldn+j), F.zero);
F.assign (*(NS + i*ldn + i+R), F.one);
}
applyP (F, FFLAS::FflasRight, FFLAS::FflasNoTrans,
NSdim, 0,(int) R, NS, ldn, P);
delete [] P;
delete [] Qt;
return N-R;
}
}
/** Computes the row rank profile of A.
*
* @param F
* @param M
* @param N
* @param A input matrix of dimension M x N
* @param lda
* @param rkprofile return the rank profile as an array of row indexes, of dimension r=rank(A)
*
* rkprofile is allocated during the computation.
* @returns R
*/
template <class Field>
size_t RowRankProfile (const Field& F, const size_t M, const size_t N,
typename Field::Element* A, const size_t lda,
size_t* &rkprofile)
{
size_t *P = new size_t[N];
size_t *Q = new size_t[M];
size_t R;
R = LUdivine (F, FFLAS::FflasNonUnit, FFLAS::FflasNoTrans, M, N, A, lda, P, Q);
rkprofile = new size_t[R];
for (size_t i=0; i<R; ++i)
rkprofile[i] = Q[i];
delete[] P;
delete[] Q;
return R;
}
/** Computes the column rank profile of A.
*
* @param F
* @param M
* @param N
* @param A input matrix of dimension
* @param lda
* @param rkprofile return the rank profile as an array of row indexes, of dimension r=rank(A)
*
* A is modified
* rkprofile is allocated during the computation.
* @returns R
*/
template <class Field>
size_t ColumnRankProfile (const Field& F, const size_t M, const size_t N,
typename Field::Element* A, const size_t lda,
size_t* &rkprofile)
{
size_t *P = new size_t[N];
size_t *Q = new size_t[M];
size_t R;
R = LUdivine (F, FFLAS::FflasNonUnit, FFLAS::FflasTrans, M, N, A, lda, P, Q);
rkprofile = new size_t[R];
for (size_t i=0; i<R; ++i)
rkprofile[i] = Q[i];
delete[] P;
delete[] Q;
return R;
}
/** RowRankProfileSubmatrixIndices.
* Computes the indices of the submatrix r*r X of A whose rows correspond to
* the row rank profile of A.
*
* @param F
* @param M
* @param N
* @param A input matrix of dimension
* @param rowindices array of the row indices of X in A
* @param colindices array of the col indices of X in A
* @param lda
* @param[out] R
*
* rowindices and colindices are allocated during the computation.
* A is modified
* @returns R
*/
template <class Field>
size_t RowRankProfileSubmatrixIndices (const Field& F,
const size_t M, const size_t N,
typename Field::Element* A,
const size_t lda,
size_t*& rowindices,
size_t*& colindices,
size_t& R)
{
size_t *P = new size_t[N];
size_t *Q = new size_t[M];
R = LUdivine (F, FFLAS::FflasNonUnit, FFLAS::FflasNoTrans, M, N, A, lda, P, Q);
rowindices = new size_t[M];
colindices = new size_t[N];
for (size_t i=0; i<R; ++i){
rowindices [i] = Q [i];
}
for (size_t i=0; i<N; ++i)
colindices [i] = i;
size_t tmp;
for (size_t i=0; i<R; ++i){
if (i != P[i]){
tmp = colindices[i];
colindices[i] = colindices[P[i]];
colindices[P[i]] = tmp;
}
}
delete[] P;
delete[] Q;
return R;
}
/** Computes the indices of the submatrix r*r X of A whose columns correspond to
* the column rank profile of A.
*
* @param F
* @param M
* @param N
* @param A input matrix of dimension
* @param rowindices array of the row indices of X in A
* @param colindices array of the col indices of X in A
* @param lda
* @param[out] R
*
* rowindices and colindices are allocated during the computation.
* @warning A is modified
* \return R
*/
template <class Field>
size_t ColRankProfileSubmatrixIndices (const Field& F,
const size_t M, const size_t N,
typename Field::Element* A,
const size_t lda,
size_t*& rowindices,
size_t*& colindices,
size_t& R)
{
size_t *P = new size_t[M];
size_t *Q = new size_t[N];
R = LUdivine (F, FFLAS::FflasNonUnit, FFLAS::FflasTrans, M, N, A, lda, P, Q);
rowindices = new size_t[M];
colindices = new size_t[N];
for (size_t i=0; i<R; ++i)
colindices [i] = Q [i];
for (size_t i=0; i<N; ++i)
rowindices [i] = i;
size_t tmp;
for (size_t i=0; i<R; ++i){
if (i != P[i]){
tmp = rowindices[i];
rowindices[i] = rowindices[P[i]];
rowindices[P[i]] = tmp;
}
}
delete[] P;
delete[] Q;
return R;
}
/** Compute the r*r submatrix X of A, by picking the row rank profile rows of A.
*
* @param F
* @param M
* @param N
* @param A input matrix of dimension M x N
* @param lda
* @param X the output matrix
* @param[out] R
*
* A is not modified
* X is allocated during the computation.
* @return R
*/
template <class Field>
size_t RowRankProfileSubmatrix (const Field& F,
const size_t M, const size_t N,
typename Field::Element* A,
const size_t lda,
typename Field::Element*& X, size_t& R)
{
size_t * rowindices, * colindices;
typename Field::Element * A2 = FFLAS::MatCopy (F, M, N, A, lda);
RowRankProfileSubmatrixIndices (F, M, N, A2, N, rowindices, colindices, R);
X = new typename Field::Element[R*R];
for (size_t i=0; i<R; ++i)
for (size_t j=0; j<R; ++j)
F.assign (*(X + i*R + j), *(A + rowindices[i]*lda + colindices[j]));
delete[] A2;
delete[] rowindices;
delete[] colindices;
return R;
}
/** Compute the \f$ r\times r\f$ submatrix X of A, by picking the row rank profile rows of A.
*
*
* @param F
* @param M
* @param N
* @param A input matrix of dimension M x N
* @param lda
* @param X the output matrix
* @param[out] R
*
* A is not modified
* X is allocated during the computation.
* \returns R
*/
template <class Field>
size_t ColRankProfileSubmatrix (const Field& F, const size_t M, const size_t N,
typename Field::Element* A, const size_t lda,
typename Field::Element*& X, size_t& R)
{
size_t * rowindices, * colindices;
typename Field::Element * A2 = FFLAS::MatCopy (F, M, N, A, lda);
ColRankProfileSubmatrixIndices (F, M, N, A2, N, rowindices, colindices, R);
X = new typename Field::Element[R*R];
for (size_t i=0; i<R; ++i)
for (size_t j=0; j<R; ++j)
F.assign (*(X + i*R + j), *(A + rowindices[i]*lda + colindices[j]));
delete[] A2;
delete[] colindices;
delete[] rowindices;
return R;
}
/** LQUPtoInverseOfFullRankMinor.
* Suppose A has been factorized as L.Q.U.P, with rank r.
* Then Qt.A.Pt has an invertible leading principal r x r submatrix
* This procedure efficiently computes the inverse of this minor and puts it into X.
* @note It changes the lower entries of A_factors in the process (NB: unless A was nonsingular and square)
*
* @param F
* @param rank rank of the matrix.
* @param A_factors matrix containing the L and U entries of the factorization
* @param lda
* @param QtPointer theLQUP->getQ()->getPointer() (note: getQ returns Qt!)
* @param X desired location for output
* @param ldx
*/
template <class Field>
typename Field::Element*
LQUPtoInverseOfFullRankMinor( const Field& F, const size_t rank,
typename Field::Element * A_factors, const size_t lda,
const size_t* QtPointer,
typename Field::Element * X, const size_t ldx)
{
// upper entries are okay, just need to move up bottom ones
const size_t* srcRow = QtPointer;
for (size_t row=0; row<rank; row++, srcRow++)
if (*srcRow != row) {
typename Field::Element* oldRow = A_factors + (*srcRow) * lda;
typename Field::Element* newRow = A_factors + row * lda;
for (size_t col=0; col<row; col++, oldRow++, newRow++)
F.assign(*newRow, *oldRow);
}
// X <- (Qt.L.Q)^(-1)
//invL( F, rank, A_factors, lda, X, ldx);
ftrtri (F, FFLAS::FflasLower, FFLAS::FflasUnit, rank, A_factors, lda);
for (size_t i=0; i<rank; ++i)
FFLAS::fcopy (F, rank, A_factors+i*lda, 1, X+i*ldx,1);
// X = U^-1.X
ftrsm( F, FFLAS::FflasLeft, FFLAS::FflasUpper, FFLAS::FflasNoTrans,
FFLAS::FflasNonUnit, rank, rank, F.one, A_factors, lda, X, ldx);
return X;
}
//---------------------------------------------------------------------
// TURBO: rank computation algorithm
//---------------------------------------------------------------------
template <class Field>
size_t
TURBO (const Field& F, const size_t M, const size_t N,
typename Field::Element* A, const size_t lda, size_t * P, size_t * Q, const size_t cutoff);
/** @brief Compute the LQUP factorization of the given matrix.
* Using
* a block algorithm and return its rank.
* The permutations P and Q are represented
* using LAPACK's convention.
* @param F field
* @param Diag precise whether U should have a unit diagonal or not
* @param trans UNKOWN TAG, probably the \c LU of \f$A^t\f$
* @param M matrix row dimension
* @param N matrix column dimension
* @param A input matrix
* @param lda leading dimension of \p A
* @param P the column permutation
* @param Qt the transpose of the row permutation \p Q
* @param LuTag flag for setting the earling termination if the matrix
* is singular
* @param cutoff UNKOWN TAG, probably a switch to a faster algo below \c cutoff
*
* @return the rank of \p A
* @bib
* - Jeannerod CP, <i>\c LSP Matrix Decomposition Revisited</i>, 2006
* - Pernet C, Brassel M <i>\c LUdivine, une divine factorisation \c LU</i>, 2002
* .
*/
template <class Field>
size_t
LUdivine (const Field& F, const FFLAS::FFLAS_DIAG Diag, const FFLAS::FFLAS_TRANSPOSE trans,
const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
size_t* P, size_t* Qt
, const FFPACK_LUDIVINE_TAG LuTag=FfpackLQUP
, const size_t cutoff=__FFPACK_LUDIVINE_CUTOFF
);
//! LUpdate
template <class Field>
size_t LUpdate (const Field& F,
const FFLAS::FFLAS_DIAG Diag, const FFLAS::FFLAS_TRANSPOSE trans,
const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
const size_t R,
const size_t K,
typename Field::Element * B, const size_t ldb,
size_t*P, size_t *Q
, const FFPACK::FFPACK_LUDIVINE_TAG LuTag =FFPACK::FfpackLQUP
, const size_t cutoff =__FFPACK_LUDIVINE_CUTOFF
);
template<class Element>
class callLUdivine_small;
//! LUdivine small case
template <class Field>
size_t
LUdivine_small (const Field& F, const FFLAS::FFLAS_DIAG Diag, const FFLAS::FFLAS_TRANSPOSE trans,
const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
size_t* P, size_t* Q,
const FFPACK_LUDIVINE_TAG LuTag=FfpackLQUP);
//! LUdivine gauss
template <class Field>
size_t
LUdivine_gauss (const Field& F, const FFLAS::FFLAS_DIAG Diag,
const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
size_t* P, size_t* Q,
const FFPACK_LUDIVINE_TAG LuTag=FfpackLQUP);
/** Compute the inverse of a triangular matrix.
* @param F
* @param Uplo whether the matrix is upper of lower triangular
* @param Diag whether the matrix if unit diagonal
* @param N
* @param A
* @param lda
*
*/
template<class Field>
void
ftrtri (const Field& F, const FFLAS::FFLAS_UPLO Uplo, const FFLAS::FFLAS_DIAG Diag,
const size_t N, typename Field::Element * A, const size_t lda)
{
if (N == 1){
if (Diag == FFLAS::FflasNonUnit)
F.invin (*A);
}
else {
size_t N1 = N/2;
size_t N2 = N - N1;
ftrtri (F, Uplo, Diag, N1, A, lda);
ftrtri (F, Uplo, Diag, N2, A + N1*(lda+1), lda);
if (Uplo == FFLAS::FflasUpper){
ftrmm (F, FFLAS::FflasLeft, Uplo, FFLAS::FflasNoTrans, Diag, N1, N2,
F.one, A, lda, A + N1, lda);
ftrmm (F, FFLAS::FflasRight, Uplo, FFLAS::FflasNoTrans, Diag, N1, N2,
F.mOne, A + N1*(lda+1), lda, A + N1, lda);
}
else {
ftrmm (F, FFLAS::FflasLeft, Uplo, FFLAS::FflasNoTrans, Diag, N2, N1,
F.one, A + N1*(lda+1), lda, A + N1*lda, lda);
ftrmm (F, FFLAS::FflasRight, Uplo, FFLAS::FflasNoTrans, Diag, N2, N1,
F.mOne, A, lda, A + N1*lda, lda);
}
}
}
/** Compute the product UL.
* Product UL of the upper, resp lower triangular matrices U and L
* stored one above the other in the square matrix A.
* Diag == Unit if the matrix U is unit diagonal
* @param F
* @param diag
* @param N
* @param A
* @param lda
*
*/
template<class Field>
void
ftrtrm (const Field& F, const FFLAS::FFLAS_DIAG diag, const size_t N,
typename Field::Element * A, const size_t lda)
{
if (N == 1)
return;
size_t N1 = N/2;
size_t N2 = N-N1;
ftrtrm (F, diag, N1, A, lda);
fgemm (F, FFLAS::FflasNoTrans, FFLAS::FflasNoTrans, N1, N1, N2, F.one,
A+N1, lda, A+N1*lda, lda, F.one, A, lda);
ftrmm (F, FFLAS::FflasRight, FFLAS::FflasLower, FFLAS::FflasNoTrans,
(diag == FFLAS::FflasUnit) ? FFLAS::FflasNonUnit : FFLAS::FflasUnit,
N1, N2, F.one, A + N1*(lda+1), lda, A + N1, lda);
ftrmm (F, FFLAS::FflasLeft, FFLAS::FflasUpper, FFLAS::FflasNoTrans, diag, N2, N1,
F.one, A + N1*(lda+1), lda, A + N1*lda, lda);
ftrtrm (F, diag, N2, A + N1*(lda+1), lda);
}
/*****************/
/* ECHELON FORMS */
/*****************/
/** Compute the Column Echelon form of the input matrix in-place.
*
* After the computation A = [ M \ V ] such that AU = C is a column echelon
* decomposition of A, with U = P^T [ V ] and C = M + Q [ Ir ]
* [ 0 In-r ] [ 0 ]
* Qt = Q^T
* If transform=false, the matrix U is not computed.
* See also test-colechelon for an example of use
* @param F
* @param M
* @param N
* @param A
* @param lda
* @param P
* @param Qt
* @param transform
*/
template <class Field>
size_t
ColumnEchelonForm (const Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
size_t* P, size_t* Qt, bool transform = true);
/** Compute the Row Echelon form of the input matrix in-place.
*
* After the computation A = [ L \ M ] such that L A = R is a row echelon
* decomposition of A, with L = [ L 0 ] P and R = M + [Ir 0] Q^T
* [ In-r]
* Qt = Q^T
* If transform=false, the matrix L is not computed.
* See also test-rowechelon for an example of use
* @param F
* @param M
* @param N
* @param A
* @param lda
* @param P
* @param Qt
* @param transform
*/
template <class Field>
size_t
RowEchelonForm (const Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
size_t* P, size_t* Qt, const bool transform = false);
/** Compute the Reduced Column Echelon form of the input matrix in-place.
*
* After the computation A = [ V ] such that AU = R is a reduced col echelon
* [ M 0 ]
* decomposition of A, where U = P^T [ V ] and R = Q [ Ir ]
* [ 0 In-r ] [ M 0 ]
* Qt = Q^T
* If transform=false, the matrix U is not computed and the matrix A = R
*
* @param F
* @param M
* @param N
* @param A
* @param lda
* @param P
* @param Qt
* @param transform
*/
template <class Field>
size_t
ReducedColumnEchelonForm (const Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
size_t* P, size_t* Qt, const bool transform = true);
/** Compute the Reduced Row Echelon form of the input matrix in-place.
*
* After the computation A = [ V1 M ] such that L A = R is a reduced row echelon
* [ V2 0 ]
* decomposition of A, where L = [ V1 0 ] P and R = [ Ir M ] Q^T
* [ V2 In-r ] [ 0 ]
* Qt = Q^T
* If transform=false, the matrix U is not computed and the matrix A = R
* @param F
* @param M
* @param N
* @param A
* @param lda
* @param P
* @param Qt
* @param transform
*/
template <class Field>
size_t
ReducedRowEchelonForm (const Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
size_t* P, size_t* Qt, const bool transform = true);
/** Variant by the block recursive algorithm.
* (See A. Storjohann Thesis 2000)
* !!!!!! Warning !!!!!!
* This code is NOT WORKING properly for some echelon structures.
* This is due to a limitation of the way we represent permutation matrices
* (LAPACK's standard):
* - a composition of transpositions Tij of the form
* P = T_{1,j1} o T_{2,j2] o...oT_{r,jr}, with jk>k for all 0 < k <= r <= n
* - The permutation on the columns, performed by this block recursive algorithm
* cannot be represented with such a composition.
* Consequently this function should only be used for benchmarks
*/
template <class Field>
size_t
ReducedRowEchelonForm2 (const Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
size_t* P, size_t* Qt, const bool transform = true){
for (size_t i=0; i<N; ++i)
Qt[i] = i;
return REF (F, M, N, A, lda, 0, 0, N, P, Qt);
}
//! REF
template <class Field>
size_t
REF (const Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
const size_t colbeg, const size_t rowbeg, const size_t colsize,
size_t* Qt, size_t* P);
/*****************/
/* INVERSION */
/*****************/
/** Invert the given matrix in place
* or computes its nullity if it is singular.
* An inplace 2n^3 algorithm is used.
* @param F The computation domain
* @param M order of the matrix
* @param [in,out] A input matrix (\f$M \times M\f$)
* @param lda leading dimension of A
* @param nullity dimension of the kernel of A
* @return pointer to \f$A \gets A^{-1}\f$
*/
template <class Field>
typename Field::Element*
Invert (const Field& F, const size_t M,
typename Field::Element * A, const size_t lda,
int& nullity)
{
size_t * P = new size_t[M];
size_t * Q = new size_t[M];
size_t R = ReducedColumnEchelonForm (F, M, M, A, lda, P, Q);
nullity = (int)(M - R);
applyP (F, FFLAS::FflasLeft, FFLAS::FflasTrans,
M, 0, (int)R, A, lda, P);
delete [] P;
delete [] Q;
return A;
}
/** Invert the given matrix in place
* or computes its nullity if it is singular.
*
* X is preallocated.
*
* @param F The computation domain
* @param M order of the matrix
* @param [in] A input matrix (\f$M \times M\f$)
* @param lda leading dimension of \p A
* @param [out] X output matrix
* @param ldx leading dimension of \p X
* @param nullity dimension of the kernel of \p A
* @return pointer to \f$X = A^{-1}\f$
*/
template <class Field>
typename Field::Element*
Invert (const Field& F, const size_t M,
const typename Field::Element * A, const size_t lda,
typename Field::Element * X, const size_t ldx,
int& nullity)
{
FFLAS::fcopy(F,M,M,X,ldx,A,lda);
Invert (F, M, X, lda, nullity);
return X;
}
/** Invert the given matrix or computes its nullity if it is singular.
* An 2n^3 algorithm is used.
* This routine can be \% faster than Invert but is not totally inplace.
* X is preallocated.
* @warning A is overwritten here !
* @warning not tested.
* @param F
* @param M order of the matrix
* @param [in,out] A input matrix (\f$M \times M\f$)
* @param lda leading dimension of A
* @param [out] X output matrix
* @param ldx leading dimension of X
* @param nullity dimension of the kernel of A
* @return pointer to \f$X = A^{-1}\f$
*/
template <class Field>
typename Field::Element*
Invert2( const Field& F, const size_t M,
typename Field::Element * A, const size_t lda,
typename Field::Element * X, const size_t ldx,
int& nullity)
{
size_t *P = new size_t[M];
size_t *rowP = new size_t[M];
#if 0 /* timer remnants */
Timer t1;
t1.clear();
t1.start();
#endif
nullity = int(M - LUdivine( F, FFLAS::FflasNonUnit, FFLAS::FflasNoTrans, M, M, A, lda, P, rowP, FfpackLQUP));
#if 0/* timer remnants */
t1.stop();
cerr<<"LU --> "<<t1.usertime()<<endl;
#endif
if (nullity > 0){
delete[] P;
delete[] rowP;
return NULL;
}
else {
// Initializing X to 0
#if 0/* timer remnants */
t1.clear();
t1.start();
#endif
//! @todo this init is not necessary (done after ftrtri)
for (size_t i=0; i<M; ++i)
for (size_t j=0; j<M;++j)
F.assign(*(X+i*ldx+j), F.zero);
// X = L^-1 in n^3/3
ftrtri (F, FFLAS::FflasLower, FFLAS::FflasUnit, M, A, lda);
for (size_t i=0; i<M; ++i){
for (size_t j=i; j<M; ++j)
F.assign(*(X +i*ldx+j), F.zero);
F.assign (*(X+i*(ldx+1)), F.one);
}
for (size_t i=1; i<M; ++i)
FFLAS::fcopy (F, i, (X+i*ldx), 1, (A+i*lda), 1);
#if 0/* timer remnants */
t1.stop();
cerr<<"U^-1 --> "<<t1.usertime()<<endl;
invL( F, M, A, lda, X, ldx );
// X = Q^-1.X is not necessary since Q = Id
// X = U^-1.X
t1.clear();
t1.start();
#endif
ftrsm( F, FFLAS::FflasLeft, FFLAS::FflasUpper, FFLAS::FflasNoTrans, FFLAS::FflasNonUnit,
M, M, F.one, A, lda , X, ldx);
#if 0/* timer remnants */
t1.stop();
cerr<<"ftrsm --> "<<t1.usertime()<<endl;
#endif
// X = P^-1.X
applyP( F, FFLAS::FflasLeft, FFLAS::FflasTrans,
M, 0,(int) M, X, ldx, P );
delete[] P;
delete[] rowP;
return X;
}
}
/*****************************/
/* CHARACTERISTIC POLYNOMIAL */
/*****************************/
/**
* Compute the characteristic polynomial of A using Krylov
* Method, and LUP factorization of the Krylov matrix
*/
template <class Field, class Polynomial>
std::list<Polynomial>&
CharPoly( const Field& F, std::list<Polynomial>& charp, const size_t N,
typename Field::Element * A, const size_t lda,
const FFPACK_CHARPOLY_TAG CharpTag= FfpackArithProg);
template<class Polynomial, class Field>
Polynomial & mulpoly(const Field& F, Polynomial &res, const Polynomial & P1, const Polynomial & P2)
{
size_t i,j;
// Warning: assumes that res is allocated to the size of the product
res.resize(P1.size()+P2.size()-1);
for (i=0;i<res.size();i++)
F.assign(res[i], F.zero);
for ( i=0;i<P1.size();i++)
for ( j=0;j<P2.size();j++)
F.axpyin(res[i+j],P1[i],P2[j]);
return res;
}
template <class Field, class Polynomial>
std::list<Polynomial>&
CharPoly( const Field& F, Polynomial& charp, const size_t N,
typename Field::Element * A, const size_t lda,
const FFPACK_CHARPOLY_TAG CharpTag= FfpackArithProg)
{
std::list<Polynomial> factor_list;
CharPoly (F, factor_list, N, A, lda, CharpTag);
typename std::list<std::vector<typename Field::Element> >::const_iterator it;
it = factor_list.begin();
// std::vector<Element>* tmp = new std::vector<Element> (n+1);
charp.resize(N+1);
Polynomial P = *(it++);
while( it!=factor_list.end() ){
mulpoly (F,charp, P, *it);
P = charp;
// delete &(*it);
++it;
}
return charp;
}
/**********************/
/* MINIMAL POLYNOMIAL */
/**********************/
/** Compute the minimal polynomial.
* Minpoly of (A,v) using an LUP
* factorization of the Krylov Base (v, Av, .., A^kv)
* U,X must be (n+1)*n
* U contains the Krylov matrix and X, its LSP factorization
*/
template <class Field, class Polynomial>
Polynomial&
MinPoly( const Field& F, Polynomial& minP, const size_t N,
const typename Field::Element *A, const size_t lda,
typename Field::Element* X, const size_t ldx, size_t* P,
const FFPACK::FFPACK_MINPOLY_TAG MinTag= FFPACK::FfpackDense,
const size_t kg_mc=0, const size_t kg_mb=0, const size_t kg_j=0 );
//! Solve L X = B or X L = B in place.
//! L is M*M if Side == FFLAS::FflasLeft and N*N if Side == FFLAS::FflasRight, B is M*N.
//! Only the R non trivial column of L are stored in the M*R matrix L
//! Requirement : so that L could be expanded in-place
template<class Field>
void
solveLB( const Field& F, const FFLAS::FFLAS_SIDE Side,
const size_t M, const size_t N, const size_t R,
typename Field::Element * L, const size_t ldl,
const size_t * Q,
typename Field::Element * B, const size_t ldb )
{
size_t LM = (Side == FFLAS::FflasRight)?N:M;
int i = (int)R ;
for (; i--; ){ // much faster for
if ( Q[i] > (size_t) i){
//for (size_t j=0; j<=Q[i]; ++j)
//F.init( *(L+Q[i]+j*ldl), 0 );
//std::cerr<<"1 deplacement "<<i<<"<-->"<<Q[i]<<endl;
FFLAS::fcopy( F, LM-Q[i]-1, L+Q[i]*(ldl+1)+ldl,ldl, L+(Q[i]+1)*ldl+i, ldl );
for ( size_t j=Q[i]*ldl; j<LM*ldl; j+=ldl)
F.assign( *(L+i+j), F.zero );
}
}
ftrsm( F, Side, FFLAS::FflasLower, FFLAS::FflasNoTrans, FFLAS::FflasUnit, M, N, F.one, L, ldl , B, ldb);
//write_field(F,std::cerr<<"dans solveLB "<<endl,L,N,N,ldl);
// Undo the permutation of L
for (size_t ii=0; ii<R; ++ii){
if ( Q[ii] > (size_t) ii){
//for (size_t j=0; j<=Q[ii]; ++j)
//F.init( *(L+Q[ii]+j*ldl), 0 );
FFLAS::fcopy( F, LM-Q[ii]-1, L+(Q[ii]+1)*ldl+ii, ldl, L+Q[ii]*(ldl+1)+ldl,ldl );
for ( size_t j=Q[ii]*ldl; j<LM*ldl; j+=ldl)
F.assign( *(L+Q[ii]+j), F.zero );
}
}
}
//! Solve L X = B in place.
//! L is M*M or N*N, B is M*N.
//! Only the R non trivial column of L are stored in the M*R matrix L
template<class Field>
void
solveLB2( const Field& F, const FFLAS::FFLAS_SIDE Side,
const size_t M, const size_t N, const size_t R,
typename Field::Element * L, const size_t ldl,
const size_t * Q,
typename Field::Element * B, const size_t ldb )
{
typename Field::Element * Lcurr,* Rcurr,* Bcurr;
size_t ib, Ldim;
int k;
if ( Side == FFLAS::FflasLeft ){
size_t j = 0;
while ( j<R ) {
ib = Q[j];
k = (int)ib ;
while ((j<R) && ( (int) Q[j] == k) ) {k++;j++;}
Ldim = (size_t)k-ib;
Lcurr = L + j-Ldim + ib*ldl;
Bcurr = B + ib*ldb;
Rcurr = Lcurr + Ldim*ldl;
ftrsm( F, Side, FFLAS::FflasLower, FFLAS::FflasNoTrans, FFLAS::FflasUnit, Ldim, N, F.one,
Lcurr, ldl , Bcurr, ldb );
fgemm( F, FFLAS::FflasNoTrans, FFLAS::FflasNoTrans, M-k, N, Ldim, F.mOne,
Rcurr , ldl, Bcurr, ldb, F.one, Bcurr+Ldim*ldb, ldb);
}
}
else{ // Side == FFLAS::FflasRight
int j=(int)R-1;
while ( j >= 0 ) {
ib = Q[j];
k = (int) ib;
while ( (j >= 0) && ( (int)Q[j] == k) ) {--k;--j;}
Ldim = ib-(size_t)k;
Lcurr = L + j+1 + (k+1)*ldl;
Bcurr = B + ib+1;
Rcurr = Lcurr + Ldim*ldl;
fgemm (F, FFLAS::FflasNoTrans, FFLAS::FflasNoTrans, M, Ldim, N-ib-1, F.mOne,
Bcurr, ldb, Rcurr, ldl, F.one, Bcurr-Ldim, ldb);
ftrsm (F, Side, FFLAS::FflasLower, FFLAS::FflasNoTrans, FFLAS::FflasUnit, M, Ldim, F.one,
Lcurr, ldl , Bcurr-Ldim, ldb );
}
}
}
template<class Field>
void trinv_left( const Field& F, const size_t N, const typename Field::Element * L, const size_t ldl,
typename Field::Element * X, const size_t ldx )
{
for (size_t i=0; i<N; ++i)
FFLAS::fcopy (F, N, X+i*ldx, 1, L+i*ldl, 1);
ftrtri (F, FFLAS::FflasLower, FFLAS::FflasUnit, N, X, ldx);
//invL(F,N,L,ldl,X,ldx);
}
template <class Field>
size_t KrylovElim( const Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda, size_t*P,
size_t *Q, const size_t deg, size_t *iterates, size_t * inviterates, const size_t maxit,size_t virt);
template <class Field>
size_t SpecRankProfile (const Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda, const size_t deg, size_t *rankProfile);
template <class Field, class Polynomial>
std::list<Polynomial>&
CharpolyArithProg (const Field& F, std::list<Polynomial>& frobeniusForm,
const size_t N, typename Field::Element * A, const size_t lda, const size_t c);
template <class Field>
void CompressRows (Field& F, const size_t M,
typename Field::Element * A, const size_t lda,
typename Field::Element * tmp, const size_t ldtmp,
const size_t * d, const size_t nb_blocs);
template <class Field>
void CompressRowsQK (Field& F, const size_t M,
typename Field::Element * A, const size_t lda,
typename Field::Element * tmp, const size_t ldtmp,
const size_t * d,const size_t deg, const size_t nb_blocs);
template <class Field>
void DeCompressRows (Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
typename Field::Element * tmp, const size_t ldtmp,
const size_t * d, const size_t nb_blocs);
template <class Field>
void DeCompressRowsQK (Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
typename Field::Element * tmp, const size_t ldtmp,
const size_t * d, const size_t deg, const size_t nb_blocs);
template <class Field>
void CompressRowsQA (Field& F, const size_t M,
typename Field::Element * A, const size_t lda,
typename Field::Element * tmp, const size_t ldtmp,
const size_t * d, const size_t nb_blocs);
template <class Field>
void DeCompressRowsQA (Field& F, const size_t M, const size_t N,
typename Field::Element * A, const size_t lda,
typename Field::Element * tmp, const size_t ldtmp,
const size_t * d, const size_t nb_blocs);
namespace Protected {
// Subroutine for Keller-Gehrig charpoly algorithm
// Compute the new d after a LSP ( d[i] can be zero )
template<class Field>
size_t
newD( const Field& F, size_t * d, bool& KeepOn,
const size_t l, const size_t N,
typename Field::Element * X,
const size_t* Q,
std::vector<std::vector<typename Field::Element> >& minpt);
template<class Field>
size_t
updateD(const Field& F, size_t * d, size_t k,
std::vector<std::vector<typename Field::Element> >& minpt );
//---------------------------------------------------------------------
// RectangleCopyTURBO: Copy A to T, with respect to the row permutation
// defined by the lsp factorization of located in
// A-dist2pivot
//---------------------------------------------------------------------
template <class Field>
void
RectangleCopyTURBO( const Field& F, const size_t M, const size_t N,
const size_t dist2pivot, const size_t rank,
typename Field::Element * T, const size_t ldt,
const typename Field::Element * A, const size_t lda )
{
const typename Field::Element * Ai = A;
typename Field::Element * T1i = T, T2i = T + rank*ldt;
size_t x = dist2pivot;
for (; Ai<A+M*lda; Ai+=lda){
while ( F.isZero(*(Ai-x)) ) { // test if the pivot is 0
FFLAS::fcopy( F, N, T2i, 1, Ai, 1);
Ai += lda;
T2i += ldt;
}
FFLAS::fcopy( F, N, T1i, 1, Ai, 1);
T1i += ldt;
x--;
}
}
//---------------------------------------------------------------------
// LUdivine_construct: (Specialisation of LUdivine)
// LUP factorisation of X, the Krylov base matrix of A^t and v, in A.
// X contains the nRowX first vectors v, vA, .., vA^{nRowX-1}
// A contains the LUP factorisation of the nUsedRowX first row of X.
// When all rows of X have been factorized in A, and rank is full,
// then X is updated by the following scheme: X <= ( X; X.B ), where
// B = A^2^i.
// This enables to make use of Matrix multiplication, and stop computing
// Krylov vector, when the rank is not longer full.
// P is the permutation matrix stored in an array of indexes
//---------------------------------------------------------------------
template <class Field>
size_t
LUdivine_construct( const Field& F, const FFLAS::FFLAS_DIAG Diag,
const size_t M, const size_t N,
const typename Field::Element * A, const size_t lda,
typename Field::Element * X, const size_t ldx,
typename Field::Element * u, size_t* P,
bool computeX, const FFPACK_MINPOLY_TAG MinTag= FFPACK::FfpackDense
, const size_t kg_mc =0
, const size_t kg_mb =0
, const size_t kg_j =0
);
template <class Field, class Polynomial>
std::list<Polynomial>&
KellerGehrig( const Field& F, std::list<Polynomial>& charp, const size_t N,
const typename Field::Element * A, const size_t lda );
template <class Field, class Polynomial>
int
KGFast ( const Field& F, std::list<Polynomial>& charp, const size_t N,
typename Field::Element * A, const size_t lda,
size_t * kg_mc, size_t* kg_mb, size_t* kg_j );
template <class Field, class Polynomial>
std::list<Polynomial>&
KGFast_generalized (const Field& F, std::list<Polynomial>& charp,
const size_t N,
typename Field::Element * A, const size_t lda);
template<class Field>
void
fgemv_kgf( const Field& F, const size_t N,
const typename Field::Element * A, const size_t lda,
const typename Field::Element * X, const size_t incX,
typename Field::Element * Y, const size_t incY,
const size_t kg_mc, const size_t kg_mb, const size_t kg_j );
template <class Field, class Polynomial>
std::list<Polynomial>&
LUKrylov( const Field& F, std::list<Polynomial>& charp, const size_t N,
typename Field::Element * A, const size_t lda,
typename Field::Element * U, const size_t ldu);
template <class Field, class Polynomial>
std::list<Polynomial>&
Danilevski (const Field& F, std::list<Polynomial>& charp,
const size_t N, typename Field::Element * A, const size_t lda);
template <class Field, class Polynomial>
std::list<Polynomial>&
LUKrylov_KGFast( const Field& F, std::list<Polynomial>& charp, const size_t N,
typename Field::Element * A, const size_t lda,
typename Field::Element * X, const size_t ldx);
} // Protected
} // FFPACK
#include "ffpack_ludivine.inl"
#include "ffpack_minpoly.inl"
#include "ffpack_charpoly_kglu.inl"
#include "ffpack_charpoly_kgfast.inl"
#include "ffpack_charpoly_kgfastgeneralized.inl"
#include "ffpack_charpoly_danilevski.inl"
#include "ffpack_charpoly.inl"
#include "ffpack_krylovelim.inl"
#include "ffpack_frobenius.inl"
#include "ffpack_echelonforms.inl"
#endif // __FFLASFFPACK_ffpack_H
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