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// vim:sts=4:sw=4:ts=4:noet:sr:cino=>s,f0,{0,g0,(0,\:0,t0,+0,=s
/* ffpack.h
* Copyright (C) 2005 Clement Pernet
* 2014 FFLAS-FFPACK group
*
* Written by Clement Pernet <Clement.Pernet@imag.fr>
* Brice Boyer (briceboyer) <boyer.brice@gmail.com>
*
*
* ========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-ffpack-config.h>
#ifdef __FFLASFFPACK_USE_OPENMP
#include <omp.h>
#endif
#include "fflas-ffpack/fflas/fflas.h"
//#include "parallel.h"
#include <list>
#include <vector>
#include <iostream> // std::cout
#include <algorithm>
// 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 { /* tags */
enum FFPACK_LU_TAG
{
FfpackSlabRecursive = 1,
FfpackTileRecursive = 2,
FfpackSingular = 3
};
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
};
}
namespace FFPACK { /* Permutations */
/*****************/
/* PERMUTATIONS */
/*****************/
void LAPACKPerm2MathPerm (size_t * MathP, const size_t * LapackP,
const size_t N);
void MathPerm2LAPACKPerm (size_t * LapackP, const size_t * MathP,
const size_t N);
template <class Field>
void MatrixApplyS (const Field& F, typename Field::Element_ptr A, const size_t lda, const size_t width,
const size_t M2,
const size_t R1, const size_t R2,
const size_t R3, const size_t R4);
template <class Element>
void PermApplyS (Element* A, const size_t lda, const size_t width,
const size_t M2,
const size_t R1, const size_t R2,
const size_t R3, const size_t R4);
template <class Field>
void MatrixApplyT (const Field& F, typename Field::Element_ptr A, const size_t lda, const size_t width,
const size_t N2,
const size_t R1, const size_t R2,
const size_t R3, const size_t R4);
template <class Element>
void PermApplyT (Element* A, const size_t lda, const size_t width,
const size_t N2,
const size_t R1, const size_t R2,
const size_t R3, const size_t R4);
void composePermutationsP (size_t * MathP,
const size_t * P1,
const size_t * P2,
const size_t R, const size_t N);
void composePermutationsQ (size_t * MathP,
const size_t * Q1,
const size_t * Q2,
const size_t R, const size_t N);
void cyclic_shift_mathPerm (size_t * P, const size_t s);
template<typename Base_t>
void cyclic_shift_row_col(Base_t * A, size_t m, size_t n, size_t lda);
template<class Field>
void cyclic_shift_row(const Field& F, typename Field::Element_ptr A, size_t m, size_t n, size_t lda);
template<class Field>
void cyclic_shift_col(const Field& F, typename Field::Element_ptr A, size_t m, size_t n, size_t lda);
/** Apply a permutation P, stored in the LAPACK format (a sequence of transpositions)
* between indices ibeg and iend of P 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 size_t ibeg, const size_t iend,
typename Field::Element_ptr A, const size_t lda, const size_t * P );
/** Apply a R-monotonically increasing permutation P, to the matrix A.
* The permutation represented by P is defined as follows:
* - the first R values of P is a LAPACK reprensentation (a sequence of transpositions)
* - the remaining iend-ibeg-R values of the permutation are in a monotonically increasing progression
* 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
* @param R
*/
template<class Field>
void
MonotonicApplyP (const Field& F,
const FFLAS::FFLAS_SIDE Side,
const FFLAS::FFLAS_TRANSPOSE Trans,
const size_t M, const size_t ibeg, const size_t iend,
typename Field::Element_ptr A, const size_t lda, const size_t * P, const size_t R);
template<class Field>
void
MonotonicCompress (const Field& F,
const FFLAS::FFLAS_SIDE Side,
const size_t M,
typename Field::Element_ptr A, const size_t lda, const size_t incA, const size_t * P,
const size_t R, const size_t maxpiv, const size_t rowstomove,
const std::vector<bool> &ispiv);
template<class Field>
void
MonotonicCompressMorePivots (const Field& F, const FFLAS::FFLAS_SIDE Side, const size_t M,
typename Field::Element_ptr A, const size_t lda, const size_t incA,
const size_t * MathP, const size_t R, const size_t rowstomove, const size_t lenP);
template<class Field>
void
MonotonicCompressCycles (const Field& F, const FFLAS::FFLAS_SIDE Side, const size_t M,
typename Field::Element_ptr A, const size_t lda, const size_t incA,
const size_t * MathP, const size_t lenP);
template<class Field>
void
MonotonicExpand (const Field& F, const FFLAS::FFLAS_SIDE Side, const size_t M,
typename Field::Element_ptr A, const size_t lda, const size_t incA,
const size_t * MathP, const size_t R, const size_t maxpiv,
const size_t rowstomove, const std::vector<bool> &ispiv);
//! Parallel applyP with OPENMP tasks
template<class Field>
void
papplyP( const Field& F,
const FFLAS::FFLAS_SIDE Side,
const FFLAS::FFLAS_TRANSPOSE Trans,
const size_t m, const size_t ibeg, const size_t iend,
typename Field::Element_ptr A, const size_t lda, const size_t * P );
//! Parallel applyT with OPENMP tasks
template <class Field>
void pMatrixApplyT (const Field& F, typename Field::Element_ptr A, const size_t lda,
const size_t width, const size_t N2,
const size_t R1, const size_t R2,
const size_t R3, const size_t R4) ;
//! Parallel applyS tasks with OPENMP tasks
template <class Field>
void pMatrixApplyS (const Field& F, typename Field::Element_ptr A, const size_t lda,
const size_t width, const size_t M2,
const size_t R1, const size_t R2,
const size_t R3, const size_t R4) ;
template<class Field>
size_t
pPLUQ(const Field& Fi, const FFLAS::FFLAS_DIAG Diag,
const size_t M, const size_t N,
typename Field::Element_ptr A, const size_t lda,
size_t* P, size_t* Q, int nt);
//#endif
} // FFPACK permutations
// #include "ffpack_permutation.inl"
namespace FFPACK { /* fgetrs, fgesv */
/** 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_ptr A, const size_t lda,
const size_t *P, const size_t *Q,
typename Field::Element_ptr B, const size_t ldb,
int * info);
/** 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_ptr
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_ptr A, const size_t lda,
const size_t *P, const size_t *Q,
typename Field::Element_ptr X, const size_t ldx,
typename Field::ConstElement_ptr B, const size_t ldb,
int * info);
/** @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_ptr A, const size_t lda,
typename Field::Element_ptr B, const size_t ldb,
int * info);
/** @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_ptr A, const size_t lda,
typename Field::Element_ptr X, const size_t ldx,
typename Field::ConstElement_ptr B, const size_t ldb,
int * info);
/** 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
*/
} // FFPACK fgesv, fgetrs
// #include "ffpack_fgesv.inl"
// #include "ffpack_fgetrs.inl"
namespace FFPACK { /* ftrtr */
/** 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_ptr A, const size_t lda);
template<class Field>
void trinv_left( const Field& F, const size_t N, typename Field::ConstElement_ptr L, const size_t ldl,
typename Field::Element_ptr X, const size_t ldx );
/** 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_ptr A, const size_t lda);
} // FFPACK ftrtr
// #include "ffpack_ftrtr.inl"
namespace FFPACK { /* PLUQ */
/** @brief Compute the PLUQ 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 whether U should have a unit diagonal or not
* @param trans, \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 row permutation
* @param Q the column permutation
* @return the rank of \p A
* @bib
* - Dumas J-G., Pernet C., and Sultan Z. <i>\c Simultaneous computation of the row and column rank profiles </i>, ISSAC'13, 2013
* .
*/
template<class Field>
size_t
PLUQ (const Field& F, const FFLAS::FFLAS_DIAG Diag,
const size_t M, const size_t N,
typename Field::Element_ptr A, const size_t lda,
size_t*P, size_t *Q);
} // FFPACK PLUQ
// #include "ffpack_pluq.inl"
namespace FFPACK { /* ludivine */
/** @brief Compute the CUP 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 whether U should have a unit diagonal or not
* @param trans \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 C-P, Pernet, C. and Storjohann, A. <i>\c Rank-profile revealing Gaussian elimination and the CUP matrix decomposition </i>, J. of Symbolic Comp., 2013
* - 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_ptr A, const size_t lda,
size_t* P, size_t* Qt,
const FFPACK_LU_TAG LuTag = FfpackSlabRecursive,
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_ptr A, const size_t lda,
size_t* P, size_t* Q,
const FFPACK_LU_TAG LuTag=FfpackSlabRecursive);
//! 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_ptr A, const size_t lda,
size_t* P, size_t* Q,
const FFPACK_LU_TAG LuTag=FfpackSlabRecursive);
namespace Protected {
//---------------------------------------------------------------------
// 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,
typename Field::ConstElement_ptr A, const size_t lda,
typename Field::Element_ptr X, const size_t ldx,
typename Field::Element_ptr u, size_t* P,
bool computeX, const FFPACK_MINPOLY_TAG MinTag= FfpackDense
, const size_t kg_mc =0
, const size_t kg_mb =0
, const size_t kg_j =0
);
} // Protected
} //FFPACK ludivine, turbo
// #include "ffpack_ludivine.inl"
namespace FFPACK { /* echelon */
/*****************/
/* ECHELON FORMS */
/*****************/
/** Compute the Column Echelon form of the input matrix in-place.
*
* If LuTag == FfpackTileRecursive, then 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 ]
* If LuTag == FfpackTileRecursive then A = [ N \ V ] such that the same holds with M = Q N
*
* Qt = Q^T
* If transform=false, the matrix V is not computed.
* See also test-colechelon for an example of use
* @param F
* @param M
* @param N
* @param A
* @param lda
* @param P the column permutation
* @param Qt the row position of the pivots in the echelon form
* @param transform
*/
template <class Field>
size_t
ColumnEchelonForm (const Field& F, const size_t M, const size_t N,
typename Field::Element_ptr A, const size_t lda,
size_t* P, size_t* Qt, bool transform = false,
const FFPACK_LU_TAG LuTag=FfpackSlabRecursive);
/** Compute the Row Echelon form of the input matrix in-place.
*
* If LuTag == FfpackTileRecursive, then after the computation A = [ L \ M ]
* such that X A = R is a row echelon decomposition of A,
* with X = [ L 0 ] P and R = M + [Ir 0] Q^T
* [ In-r]
* If LuTag == FfpackTileRecursive then A = [ L \ N ] such that the same holds with M = N Q^T
* 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 the row permutation
* @param Qt the column position of the pivots in the echelon form
* @param transform
*/
template <class Field>
size_t
RowEchelonForm (const Field& F, const size_t M, const size_t N,
typename Field::Element_ptr A, const size_t lda,
size_t* P, size_t* Qt, const bool transform = false,
const FFPACK_LU_TAG LuTag=FfpackSlabRecursive);
/** Compute the Reduced Column Echelon form of the input matrix in-place.
*
* After the computation A = [ V ] such that AX = R is a reduced col echelon
* [ M 0 ]
* decomposition of A, where X = P^T [ V ] and R = Q [ Ir ]
* [ 0 In-r ] [ M 0 ]
* Qt = Q^T
* If transform=false, the matrix X 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_ptr A, const size_t lda,
size_t* P, size_t* Qt, const bool transform = false,
const FFPACK_LU_TAG LuTag=FfpackSlabRecursive);
/** Compute the Reduced Row Echelon form of the input matrix in-place.
*
* After the computation A = [ V1 M ] such that X A = R is a reduced row echelon
* [ V2 0 ]
* decomposition of A, where X = [ V1 0 ] P and R = [ Ir M ] Q^T
* [ V2 In-r ] [ 0 ]
* Qt = Q^T
* If transform=false, the matrix X 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_ptr A, const size_t lda,
size_t* P, size_t* Qt, const bool transform = false,
const FFPACK_LU_TAG LuTag=FfpackSlabRecursive);
/** 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_ptr A, const size_t lda,
size_t* P, size_t* Qt, const bool transform = true);
//! REF
template <class Field>
size_t
REF (const Field& F, const size_t M, const size_t N,
typename Field::Element_ptr A, const size_t lda,
const size_t colbeg, const size_t rowbeg, const size_t colsize,
size_t* Qt, size_t* P);
} // FFPACK
// #include "ffpack_echelonforms.inl"
namespace FFPACK { /* invert */
/*****************/
/* INVERSION */
/*****************/
/** @brief Invert the given matrix in place
* or computes its nullity if it is singular.
*
* An inplace \f$2n^3\f$ 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$A\f$ and \f$A \gets A^{-1}\f$
*/
template <class Field>
typename Field::Element_ptr
Invert (const Field& F, const size_t M,
typename Field::Element_ptr A, const size_t lda,
int& nullity);
/** @brief Invert the given matrix in place
* or computes its nullity if it is singular.
*
* @pre \p X is preallocated and should be large enough to store the
* \f$ m \times m\f$ matrix \p A.
*
* @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 this is the inverse of \p A if \p A is invertible
* (non \c NULL and \f$ \mathtt{nullity} = 0\f$). It is untouched
* otherwise.
* @param ldx leading dimension of \p X
* @param nullity dimension of the kernel of \p A
* @return pointer to \f$X = A^{-1}\f$
*/
template <class Field>
typename Field::Element_ptr
Invert (const Field& F, const size_t M,
typename Field::ConstElement_ptr A, const size_t lda,
typename Field::Element_ptr X, const size_t ldx,
int& nullity);
/** @brief Invert the given matrix or computes its nullity if it is singular.
*
* An \f$2n^3f\f$ algorithm is used.
* This routine can be \% faster than FFPACK::Invert but is not totally inplace.
*
* @pre \p X is preallocated and should be large enough to store the
* \f$ m \times m\f$ matrix \p A.
*
* @warning A is overwritten here !
* @bug not tested.
* @param F
* @param M order of the matrix
* @param [in,out] A input matrix (\f$M \times M\f$). On output, \p A
* is modified and represents a "psycological" factorisation \c LU.
* @param lda leading dimension of A
* @param [out] X this is the inverse of \p A if \p A is invertible
* (non \c NULL and \f$ \mathtt{nullity} = 0\f$). It is untouched
* otherwise.
* @param ldx leading dimension of \p X
* @param nullity dimension of the kernel of \p A
* @return pointer to \f$X = A^{-1}\f$
*/
template <class Field>
typename Field::Element_ptr
Invert2( const Field& F, const size_t M,
typename Field::Element_ptr A, const size_t lda,
typename Field::Element_ptr X, const size_t ldx,
int& nullity);
} // FFPACK invert
// #include "ffpack_invert.inl"
namespace FFPACK { /* charpoly */
/*****************************/
/* 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_ptr 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);
template <class Field, class Polynomial>
Polynomial&
CharPoly( const Field& F, Polynomial& charp, const size_t N,
typename Field::Element_ptr A, const size_t lda,
const FFPACK_CHARPOLY_TAG CharpTag= FfpackArithProg);
namespace Protected {
template <class Field, class Polynomial>
std::list<Polynomial>&
KellerGehrig( const Field& F, std::list<Polynomial>& charp, const size_t N,
typename Field::ConstElement_ptr 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_ptr 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_ptr A, const size_t lda);
template<class Field>
void
fgemv_kgf( const Field& F, const size_t N,
typename Field::ConstElement_ptr A, const size_t lda,
typename Field::ConstElement_ptr X, const size_t incX,
typename Field::Element_ptr 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_ptr A, const size_t lda,
typename Field::Element_ptr 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_ptr 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_ptr A, const size_t lda,
typename Field::Element_ptr X, const size_t ldx);
} // Protected
} // FFPACK charpoly
// #include "ffpack_charpoly_kglu.inl"
// #include "ffpack_charpoly_kgfast.inl"
// #include "ffpack_charpoly_kgfastgeneralized.inl"
// #include "ffpack_charpoly_danilevski.inl"
// #include "ffpack_charpoly.inl"
namespace FFPACK { /* frobenius, charpoly */
template <class Field, class Polynomial>
std::list<Polynomial>&
CharpolyArithProg (const Field& F, std::list<Polynomial>& frobeniusForm,
const size_t N, typename Field::Element_ptr A, const size_t lda, const size_t c);
} // FFPACK frobenius
// #include "ffpack_frobenius.inl"
namespace FFPACK { /* minpoly */
/**********************/
/* 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,
typename Field::ConstElement_ptr A, const size_t lda,
typename Field::Element_ptr X, const size_t ldx, size_t* P,
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 );
} // FFPACK minpoly
// #include "ffpack_minpoly.inl"
namespace FFPACK { /* Krylov Elim */
template <class Field>
size_t KrylovElim( const Field& F, const size_t M, const size_t N,
typename Field::Element_ptr 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_ptr A, const size_t lda, const size_t deg, size_t *rankProfile);
} // FFPACK KrylovElim
// #include "ffpack_krylovelim.inl"
namespace FFPACK { /* Solutions */
/********/
/* RANK */
/********/
/** 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_ptr A, const size_t lda) ;
/********/
/* DET */
/********/
/** Returns true if the given matrix is singular.
* The method is a block elimination with early termination
*
* using LQUP factorization with early termination.
* If <code>M != N</code>,
* then the matrix is virtually padded with zeros to make it square and
* it's determinant is zero.
* @warning The input matrix is modified.
* @param F field
* @param M row dimension of the matrix
* @param N column dimension of the matrix.
* @param [in,out] 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_ptr A, const size_t lda);
/** @brief Returns the determinant of the given matrix.
* @details The method is a block elimination with early termination
* using LQUP factorization with early termination.
* If <code>M != N</code>,
* then the matrix is virtually padded with zeros to make it square and
* it's determinant is zero.
* @warning The input matrix is modified.
* @param F field
* @param M row dimension of the matrix
* @param N column dimension of the matrix.
* @param [in,out] A input matrix
* @param lda leading dimension of A
*/
template <class Field>
typename Field::Element
Det( const Field& F, const size_t M, const size_t N,
typename Field::Element_ptr A, const size_t lda);
/*********/
/* SOLVE */
/*********/
/// Solve linear system using LQUP factorization.
template <class Field>
typename Field::Element_ptr
Solve( const Field& F, const size_t M,
typename Field::Element_ptr A, const size_t lda,
typename Field::Element_ptr x, const int incx,
typename Field::ConstElement_ptr b, const int incb );
//! 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_ptr L, const size_t ldl,
const size_t * Q,
typename Field::Element_ptr B, const size_t ldb );
//! 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_ptr L, const size_t ldl,
const size_t * Q,
typename Field::Element_ptr B, const size_t ldb );
/*************/
/* NULLSPACE */
/*************/
/** Computes a vector of the Left/Right nullspace of the matrix A.
*
* @param F The computation domain
* @param Side
* @param M
* @param N
* @param[in,out] A input matrix of dimension M x N, A is modified to its LU version
* @param lda
* @param[out] X output vector
* @param incX
*
*/
template <class Field>
void RandomNullSpaceVector (const Field& F, const FFLAS::FFLAS_SIDE Side,
const size_t M, const size_t N,
typename Field::Element_ptr A, const size_t lda,
typename Field::Element_ptr X, const size_t incX);
/** 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[in,out] A input matrix of dimension M x N, A is modified
* @param lda
* @param[out] NS output matrix of dimension N x NSdim (allocated here)
* @param[out] ldn
* @param[out] 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_ptr A, const size_t lda,
typename Field::Element_ptr& NS, size_t& ldn,
size_t& NSdim);
/*****************/
/* RANK PROFILES */
/*****************/
/** @brief 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)
* @param LuTag: chooses the elimination algorithm. SlabRecursive for LUdivine, TileRecursive for PLUQ
*
* A is modified
* 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_ptr A, const size_t lda,
size_t* &rkprofile,
const FFPACK_LU_TAG LuTag=FfpackSlabRecursive);
/** @brief 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)
* @param LuTag: chooses the elimination algorithm. SlabRecursive for LUdivine, TileRecursive for PLUQ
*
* 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_ptr A, const size_t lda,
size_t* &rkprofile,
const FFPACK_LU_TAG LuTag=FfpackSlabRecursive);
/** @brief Recovers the column/row rank profile from the permutation of an LU decomposition.
*
* Works with both the CUP/PLE decompositions (obtained by LUdivine) or the PLUQ decomposition
* Assumes that the output vector containing the rank profile is already allocated.
* @param P the permutation carrying the rank profile information
* @param N the row/col dimension for a row/column rank profile
* @param R the rank of the matrix (
* @param rkprofile return the rank profile as an array of indices
* @param LuTag: chooses the elimination algorithm. SlabRecursive for LUdivine, TileRecursive for PLUQ
*
* A is modified
*
*/
void RankProfileFromLU (const size_t* P, const size_t N, const size_t R,
size_t* rkprofile, const FFPACK_LU_TAG LuTag);
/** @brief Recovers the row and column rank profiles of any leading submatrix from the PLUQ decomposition.
*
* Only works with the PLUQ decomposition
* Assumes that the output vectors containing the rank profiles are already allocated.
*
* @param P the permutation carrying the rank profile information
* @param M the row dimension of the initial matrix
* @param N the column dimension of the initial matrix
* @param R the rank of the initial matrix
* @param LSm the row dimension of the leading submatrix considered
* @param LSn the column dimension of the leading submatrix considered
* @param P the row permutation of the PLUQ decomposition
* @param Q the column permutation of the PLUQ decomposition
* @param RRP return the row rank profile of the leading
* @param LuTag: chooses the elimination algorithm. SlabRecursive for LUdivine, TileRecursive for PLUQ
* @return the rank of the LSm x LSn leading submatrix
*
* A is modified
* @bib
* - Dumas J-G., Pernet C., and Sultan Z. <i>\c Simultaneous computation of the row and column rank profiles </i>, ISSAC'13.
*/
size_t LeadingSubmatrixRankProfiles (const size_t M, const size_t N, const size_t R,
const size_t LSm, const size_t LSn,
const size_t* P, const size_t* Q,
size_t* RRP, size_t* CRP);
/** 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_ptr A,
const size_t lda,
size_t*& rowindices,
size_t*& colindices,
size_t& 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_ptr A,
const size_t lda,
size_t*& rowindices,
size_t*& colindices,
size_t& R);
/** Computes 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_ptr A,
const size_t lda,
typename Field::Element_ptr& X, size_t& 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_ptr A, const size_t lda,
typename Field::Element_ptr& X, size_t& R);
/*********************************************/
/* Accessors to Triangular and Echelon forms */
/*********************************************/
/** Extracts a triangular matrix from a compact storage A=L\U of rank R.
* if OnlyNonZeroVectors is false, then T and A have the same dimensions
* Otherwise, T is R x N if UpLo = FflasUpper, else T is M x R
* @param F: base field
* @param UpLo: selects if the upper or lower triangular matrix is returned
* @param diag: selects if the triangular matrix unit-diagonal
* @param M: row dimension of T
* @param N: column dimension of T
* @param R: rank of the triangular matrix (how many rows/columns need to be copied)
* @param A: input matrix
* @param lda: leading dimension of A
* @param T: output matrix
* @param ldt: leading dimension of T
* @param OnlyNonZeroVectors: decides whether the last zero rows/columns should be ignored
*/
template <class Field>
void
getTriangular (const Field& F, const FFLAS::FFLAS_UPLO Uplo,
const FFLAS::FFLAS_DIAG diag,
const size_t M, const size_t N, const size_t R,
typename Field::ConstElement_ptr A, const size_t lda,
typename Field::Element_ptr T, const size_t ldt,
const bool OnlyNonZeroVectors = false);
/** Cleans up a compact storage A=L\U to reveal a triangular matrix of rank R.
* @param F: base field
* @param UpLo: selects if the upper or lower triangular matrix is revealed
* @param diag: selects if the triangular matrix unit-diagonal
* @param M: row dimension of A
* @param N: column dimension of A
* @param R: rank of the triangular matrix
* @param A: input/output matrix
* @param lda: leading dimension of A
*/
template <class Field>
void
getTriangular (const Field& F, const FFLAS::FFLAS_UPLO Uplo,
const FFLAS::FFLAS_DIAG diag,
const size_t M, const size_t N, const size_t R,
typename Field::Element_ptr A, const size_t lda);
/** Extracts a matrix in echelon form from a compact storage A=L\U of rank R obtained by
* RowEchelonForm or ColumnEchelonForm.
* Either L or U is in Echelon form (depending on Uplo)
* The echelon structure is defined by the first R values of the array P.
* row and column dimension of T are greater or equal to that of A
* @param F: base field
* @param UpLo: selects if the upper or lower triangular matrix is returned
* @param diag: selects if the echelon matrix has unit pivots
* @param M: row dimension of T
* @param N: column dimension of T
* @param R: rank of the triangular matrix (how many rows/columns need to be copied)
* @param P: positions of the R pivots
* @param A: input matrix
* @param lda: leading dimension of A
* @param T: output matrix
* @param ldt: leading dimension of T
* @param OnlyNonZeroVectors: decides whether the last zero rows/columns should be ignored
* @param LuTag: which factorized form (CUP/PLE if FfpackSlabRecursive, PLUQ if FfpackTileRecursive)
*/
template <class Field>
void
getEchelonForm (const Field& F, const FFLAS::FFLAS_UPLO Uplo,
const FFLAS::FFLAS_DIAG diag,
const size_t M, const size_t N, const size_t R, const size_t* P,
typename Field::ConstElement_ptr A, const size_t lda,
typename Field::Element_ptr T, const size_t ldt,
const bool OnlyNonZeroVectors = false,
const FFPACK_LU_TAG LuTag = FfpackSlabRecursive);
/** Cleans up a compact storage A=L\U obtained by RowEchelonForm or ColumnEchelonForm
* to reveal an echelon form of rank R.
* Either L or U is in Echelon form (depending on Uplo)
* The echelon structure is defined by the first R values of the array P.
* @param F: base field
* @param UpLo: selects if the upper or lower triangular matrix is returned
* @param diag: selects if the echelon matrix has unit pivots
* @param M: row dimension of A
* @param N: column dimension of A
* @param R: rank of the triangular matrix (how many rows/columns need to be copied)
* @param P: positions of the R pivots
* @param A: input/output matrix
* @param lda: leading dimension of A
* @param LuTag: which factorized form (CUP/PLE if FfpackSlabRecursive, PLUQ if FfpackTileRecursive)
*/
template <class Field>
void
getEchelonForm (const Field& F, const FFLAS::FFLAS_UPLO Uplo,
const FFLAS::FFLAS_DIAG diag,
const size_t M, const size_t N, const size_t R, const size_t* P,
typename Field::Element_ptr A, const size_t lda,
const FFPACK_LU_TAG LuTag = FfpackSlabRecursive);
/** Extracts a transformation matrix to echelon form from a compact storage A=L\U
* of rank R obtained by RowEchelonForm or ColumnEchelonForm.
* If Uplo == FflasLower:
* T is N x N (already allocated) such that A T = C is a transformation of A in
* Column echelon form
* Else
* T is M x M (already allocated) such that T A = E is a transformation of A in
* Row Echelon form
* @param F: base field
* @param UpLo: Lower means Transformation to Column Echelon Form, Upper, to Row Echelon Form
* @param diag: selects if the echelon matrix has unit pivots
* @param M: row dimension of A
* @param N: column dimension of A
* @param R: rank of the triangular matrix
* @param P: permutation matrix
* @param A: input matrix
* @param lda: leading dimension of A
* @param T: output matrix
* @param ldt: leading dimension of T
* @param LuTag: which factorized form (CUP/PLE if FfpackSlabRecursive, PLUQ if FfpackTileRecursive)
*/
template <class Field>
void
getEchelonTransform (const Field& F, const FFLAS::FFLAS_UPLO Uplo,
const FFLAS::FFLAS_DIAG diag,
const size_t M, const size_t N, const size_t R, const size_t* P, const size_t* Q,
typename Field::ConstElement_ptr A, const size_t lda,
typename Field::Element_ptr T, const size_t ldt,
const FFPACK_LU_TAG LuTag = FfpackSlabRecursive);
/** Extracts a matrix in echelon form from a compact storage A=L\U of rank R obtained by
* ReducedRowEchelonForm or ReducedColumnEchelonForm with transform = true.
* Either L or U is in Echelon form (depending on Uplo)
* The echelon structure is defined by the first R values of the array P.
* row and column dimension of T are greater or equal to that of A
* @param F: base field
* @param UpLo: selects if the upper or lower triangular matrix is returned
* @param diag: selects if the echelon matrix has unit pivots
* @param M: row dimension of T
* @param N: column dimension of T
* @param R: rank of the triangular matrix (how many rows/columns need to be copied)
* @param P: positions of the R pivots
* @param A: input matrix
* @param lda: leading dimension of A
* @param ldt: leading dimension of T
* @param LuTag: which factorized form (CUP/PLE if FfpackSlabRecursive, PLUQ if FfpackTileRecursive)
* @param OnlyNonZeroVectors: decides whether the last zero rows/columns should be ignored
*/
template <class Field>
void
getReducedEchelonForm (const Field& F, const FFLAS::FFLAS_UPLO Uplo,
const size_t M, const size_t N, const size_t R, const size_t* P,
typename Field::ConstElement_ptr A, const size_t lda,
typename Field::Element_ptr T, const size_t ldt,
const bool OnlyNonZeroVectors = false,
const FFPACK_LU_TAG LuTag = FfpackSlabRecursive);
/** Cleans up a compact storage A=L\U of rank R obtained by ReducedRowEchelonForm or
* ReducedColumnEchelonForm with transform = true.
* Either L or U is in Echelon form (depending on Uplo)
* The echelon structure is defined by the first R values of the array P.
* @param F: base field
* @param UpLo: selects if the upper or lower triangular matrix is returned
* @param diag: selects if the echelon matrix has unit pivots
* @param M: row dimension of A
* @param N: column dimension of A
* @param R: rank of the triangular matrix (how many rows/columns need to be copied)
* @param P: positions of the R pivots
* @param A: input/output matrix
* @param lda: leading dimension of A
* @param LuTag: which factorized form (CUP/PLE if FfpackSlabRecursive, PLUQ if FfpackTileRecursive)
*/
template <class Field>
void
getReducedEchelonForm (const Field& F, const FFLAS::FFLAS_UPLO Uplo,
const size_t M, const size_t N, const size_t R, const size_t* P,
typename Field::Element_ptr A, const size_t lda,
const FFPACK_LU_TAG LuTag = FfpackSlabRecursive);
/** Extracts a transformation matrix to echelon form from a compact storage A=L\U
* of rank R obtained by RowEchelonForm or ColumnEchelonForm.
* If Uplo == FflasLower:
* T is N x N (already allocated) such that A T = C is a transformation of A in
* Column echelon form
* Else
* T is M x M (already allocated) such that T A = E is a transformation of A in
* Row Echelon form
* @param F: base field
* @param UpLo: selects Col or Row Echelon Form
* @param diag: selects if the echelon matrix has unit pivots
* @param M: row dimension of A
* @param N: column dimension of A
* @param R: rank of the triangular matrix
* @param P: permutation matrix
* @param A: input matrix
* @param lda: leading dimension of A
* @param T: output matrix
* @param ldt: leading dimension of T
* @param LuTag: which factorized form (CUP/PLE if FfpackSlabRecursive, PLUQ if FfpackTileRecursive)
*/
template <class Field>
void
getReducedEchelonTransform (const Field& F, const FFLAS::FFLAS_UPLO Uplo,
const size_t M, const size_t N, const size_t R, const size_t* P, const size_t* Q,
typename Field::ConstElement_ptr A, const size_t lda,
typename Field::Element_ptr T, const size_t ldt,
const FFPACK_LU_TAG LuTag = FfpackSlabRecursive);
/** Auxiliary routine: determines the permutation that changes a PLUQ decomposition
* into a echelon form revealing PLUQ decomposition
*/
void
PLUQtoEchelonPermutation (const size_t N, const size_t R, const size_t * P, size_t * outPerm);
} // FFPACK
// #include "ffpack.inl"
namespace FFPACK { /* not used */
/** 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_ptr
LQUPtoInverseOfFullRankMinor( const Field& F, const size_t rank,
typename Field::Element_ptr A_factors, const size_t lda,
const size_t* QtPointer,
typename Field::Element_ptr X, const size_t ldx);
} // FFPACK
// include precompiled instantiation headers (avoiding to recompile them)
#ifdef FFPACK_COMPILED
#include "fflas-ffpack/interfaces/libs/ffpack_inst.h"
#endif
//---------------------------------------------------------------------
// Checkers
#include "fflas-ffpack/checkers/checkers_ffpack.h"
//---------------------------------------------------------------------
#include "ffpack_fgesv.inl"
#include "ffpack_fgetrs.inl"
#include "ffpack_ftrtr.inl"
//---------------------------------------------------------------------
// Checkers
#include "fflas-ffpack/checkers/checkers_ffpack.inl"
//---------------------------------------------------------------------
#include "ffpack_pluq.inl"
#include "ffpack_pluq_mp.inl"
#include "ffpack_ppluq.inl"
#include "ffpack_ludivine.inl"
#include "ffpack_ludivine_mp.inl"
#include "ffpack_echelonforms.inl"
#include "ffpack_invert.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_frobenius.inl"
#include "ffpack_minpoly.inl"
#include "ffpack_krylovelim.inl"
#include "ffpack_permutation.inl"
#include "ffpack_rankprofiles.inl"
#include "ffpack.inl"
#endif // __FFLASFFPACK_ffpack_H
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