Templated DMat<R> linear algebra: LU, rank, determinant, solve, inverse, nullSpace. More...

#include "util.hpp"
#include "exceptions.hpp"
#include "dmat.hpp"
#include "aring-RR.hpp"
#include "aring-CC.hpp"
#include "aring-RRR.hpp"
#include "aring-CCC.hpp"
#include "aring-zzp.hpp"
#include "aring-m2-gf.hpp"
#include "aring-zzp-ffpack.hpp"
#include "aring-zz-gmp.hpp"
#include "aring-qq.hpp"
#include "aring-zz-flint.hpp"
#include "aring-zzp-flint.hpp"
#include "aring-gf-flint-big.hpp"
#include "aring-gf-flint.hpp"
#include "lapack.hpp"
#include "mat-arith.hpp"
#include "dmat-lu.hpp"
#include "dmat-qq-interface-flint.hpp"
#include "eigen.hpp"
#include <M2/gc-include.h>
#include <flint/flint.h>
#include <flint/fmpq_mat.h>
#include <flint/fmpz.h>
#include <flint/fmpz_mat.h>
#include <flint/fq_nmod_mat.h>
#include <flint/nmod_mat.h>
#include <iostream>
#include <algorithm>

Namespaces
namespace	MatrixOps

Macros
#define	DMatZZpFFPACK DMat<ZZpFFPACK>

Typedefs
typedef DMat< M2::ARingZZp >	DMatZZp
typedef DMat< M2::ARingGFM2 >	DMatGFM2
typedef M2::ARingZZpFFPACK	ZZpFFPACK
typedef DMat< M2::ARingZZGMP >	DMatZZGMP
typedef DMat< M2::ARingZZ >	DMatZZ
typedef DMat< M2::ARingQQ >	DMatQQ
typedef DMat< M2::ARingQQFlint >	DMatQQFlint
typedef DMat< M2::ARingZZpFlint >	DMatZZpFlint
typedef DMat< M2::ARingGFFlintBig >	DMatGFFlintBig
typedef DMat< M2::ARingGFFlint >	DMatGFFlint
typedef DMat< M2::ARingRRR >	DMatRRR
typedef DMat< M2::ARingCCC >	DMatCCC
typedef DMat< M2::ARingRR >	DMatRR
typedef DMat< M2::ARingCC >	DMatCC

Functions
template<typename Mat>
size_t	MatrixOps::rank (const Mat &A)
	the rank of a matrix
template<typename Mat>
void	MatrixOps::determinant (const Mat &A, typename Mat::ElementType &result_det)
	the determinant of a square matrix
template<typename Mat>
bool	MatrixOps::inverse (const Mat &A, Mat &result_inv)
	the inverse of a square matrix
template<typename Mat>
size_t	MatrixOps::rowReducedEchelonForm (const Mat &A, Mat &result_rref)
	the row reduced echelon form of a matrix over a field, or ZZ.
template<typename Mat>
void	MatrixOps::mult (const Mat &A, const Mat &B, Mat &result_product)
	the product of two matrices
template<typename Mat>
size_t	MatrixOps::nullSpace (const Mat &A, Mat &result_nullspace)
	the null space of a matrix
template<typename Mat>
bool	MatrixOps::solveLinear (const Mat &A, const Mat &B, Mat &X)
	solve AX=B, return true if the system has a solution.
template<typename Mat>
bool	MatrixOps::solveInvertible (const Mat &A, const Mat &B, Mat &X)
	solve AX=B, where A is a square (invertible) matrix.
template<typename Mat>
M2_arrayintOrNull	MatrixOps::rankProfile (const Mat &A, bool row_profile)
	Returns either the row or column rank profile of A.
template<typename Mat>
void	MatrixOps::addMultipleTo (Mat &C, const Mat &A, const Mat &B)
	Set C += A*B.
template<typename Mat>
void	MatrixOps::subtractMultipleTo (Mat &C, const Mat &A, const Mat &B)
	Set C -= A*B.
template<typename Mat>
M2_arrayintOrNull	MatrixOps::LU (const Mat &A, Mat &L, Mat &U)
template<typename Mat>
M2_arrayintOrNull	MatrixOps::LUincremental (std::vector< size_t > &P, Mat &LU, const Mat &v, int i)
template<typename Mat>
void	MatrixOps::triangularSolve (Mat &Lv, Mat &x, int m, int strategy)
template<typename Mat, typename Mat2>
bool	MatrixOps::eigenvalues (const Mat &A, Mat2 &eigenvals)
template<typename Mat, typename Mat2>
bool	MatrixOps::eigenvaluesHermitian (const Mat &A, Mat2 &eigenvals)
template<typename Mat, typename Mat2, typename Mat3>
bool	MatrixOps::eigenvectors (const Mat &A, Mat2 &eigenvals, Mat3 &eigenvecs)
template<typename Mat, typename Mat2, typename Mat3>
bool	MatrixOps::eigenvectorsHermitian (const Mat &A, Mat2 &eigenvals, Mat3 &eigenvecs)
template<typename Mat>
bool	MatrixOps::leastSquares (const Mat &A, const Mat &B, Mat &X, bool assume_full_rank)
template<typename Mat, typename Mat2>
bool	MatrixOps::SVD (const Mat &A, Mat2 &Sigma, Mat &U, Mat &Vt, int strategy)
template<typename Mat, typename Mat2, typename Mat3>
bool	MatrixOps::QR (const Mat &A, Mat2 &Q, Mat3 &R, bool return_QR)
template<typename T>
void	MatrixOps::clean (gmp_RR epsilon, T &mat)
template<typename T>
void	MatrixOps::increase_norm (gmp_RRmutable nm, const T &mat)
template<typename RT>
void	MatrixOps::mult (const DMat< RT > &A, const DMat< RT > &B, DMat< RT > &result_product)
template<typename RT>
void	MatrixOps::addMultipleTo (DMat< RT > &C, const DMat< RT > &A, const DMat< RT > &B)
template<typename RT>
void	MatrixOps::subtractMultipleTo (DMat< RT > &C, const DMat< RT > &A, const DMat< RT > &B)
template<typename RT>
void	MatrixOps::determinant (const DMat< RT > &A, typename RT::ElementType &result)
template<typename RT>
M2_arrayintOrNull	MatrixOps::LU (const DMat< RT > &A, DMat< RT > &L, DMat< RT > &U)
template<typename RT>
void	MatrixOps::triangularSolve (DMat< RT > &Lv, DMat< RT > &x, int m, int strategy)
template<typename RT>
M2_arrayintOrNull	MatrixOps::LUincremental (std::vector< size_t > &P, DMat< RT > &LU, const DMat< RT > &v, int m)
template<typename RT>
size_t	MatrixOps::rank (const DMat< RT > &A)
template<typename RT>
M2_arrayintOrNull	MatrixOps::rankProfile (const DMat< RT > &A, bool row_profile)
template<typename RT>
bool	MatrixOps::inverse (const DMat< RT > &A, DMat< RT > &result_inv)
template<typename RT>
size_t	MatrixOps::nullSpace (const DMat< RT > &A, DMat< RT > &result_nullspace)
template<typename RT>
bool	MatrixOps::solveLinear (const DMat< RT > &A, const DMat< RT > &B, DMat< RT > &X)
template<typename RT>
bool	MatrixOps::solveInvertible (const DMat< RT > &A, const DMat< RT > &B, DMat< RT > &X)
void	MatrixOps::mult (const DMatZZpFFPACK &A, const DMatZZpFFPACK &B, DMatZZpFFPACK &C)
void	MatrixOps::addMultipleTo (DMatZZpFFPACK &C, const DMatZZpFFPACK &A, const DMatZZpFFPACK &B)
void	MatrixOps::subtractMultipleTo (DMatZZpFFPACK &C, const DMatZZpFFPACK &A, const DMatZZpFFPACK &B)
M2_arrayintOrNull	MatrixOps::LU (const DMatZZGMP &A, DMatZZGMP &L, DMatZZGMP &U)
M2_arrayintOrNull	MatrixOps::rankProfile (const DMatZZGMP &A, bool row_profile)
bool	MatrixOps::inverse (const DMatZZGMP &A, DMatZZGMP &result_inv)
size_t	MatrixOps::nullSpace (const DMatZZGMP &A, DMatZZGMP &result_nullspace)
bool	MatrixOps::solveLinear (const DMatZZGMP &A, const DMatZZGMP &B, DMatZZGMP &X)
bool	MatrixOps::solveInvertible (const DMatZZGMP &A, const DMatZZGMP &B, DMatZZGMP &X)
void	MatrixOps::mult (const DMatZZGMP &A, const DMatZZGMP &B, DMatZZGMP &result_product)
void	MatrixOps::addMultipleTo (DMatZZGMP &C, const DMatZZGMP &A, const DMatZZGMP &B)
void	MatrixOps::subtractMultipleTo (DMatZZGMP &C, const DMatZZGMP &A, const DMatZZGMP &B)
size_t	MatrixOps::rank (const DMatZZGMP &A)
void	MatrixOps::determinant (const DMatZZGMP &A, M2::ARingZZGMP::ElementType &result_det)
size_t	MatrixOps::rank (const DMatZZ &A)
void	MatrixOps::determinant (const DMatZZ &A, M2::ARingZZ::ElementType &result_det)
bool	MatrixOps::inverse (const DMatZZ &A, DMatZZ &result_inv)
void	MatrixOps::mult (const DMatZZ &A, const DMatZZ &B, DMatZZ &result_product)
size_t	MatrixOps::nullSpace (const DMatZZ &A, DMatZZ &result_nullspace)
bool	MatrixOps::solveLinear (const DMatZZ &A, const DMatZZ &B, DMatZZ &X)
M2_arrayintOrNull	MatrixOps::rankProfile (const DMatZZ &A, bool row_profile)
void	MatrixOps::addMultipleTo (DMatZZ &C, const DMatZZ &A, const DMatZZ &B)
void	MatrixOps::subtractMultipleTo (DMatZZ &C, const DMatZZ &A, const DMatZZ &B)
void	MatrixOps::addMultipleTo (DMatZZpFlint &C, const DMatZZpFlint &A, const DMatZZpFlint &B)
void	MatrixOps::subtractMultipleTo (DMatZZpFlint &C, const DMatZZpFlint &A, const DMatZZpFlint &B)
void	MatrixOps::mult (const DMatZZpFlint &A, const DMatZZpFlint &B, DMatZZpFlint &result_product)
size_t	MatrixOps::rowReducedEchelonForm (const DMatZZpFlint &A, DMatZZpFlint &result_rref)
void	MatrixOps::addMultipleTo (DMatGFFlintBig &C, const DMatGFFlintBig &A, const DMatGFFlintBig &B)
void	MatrixOps::subtractMultipleTo (DMatGFFlintBig &C, const DMatGFFlintBig &A, const DMatGFFlintBig &B)
void	MatrixOps::mult (const DMatGFFlintBig &A, const DMatGFFlintBig &B, DMatGFFlintBig &result_product)
size_t	MatrixOps::rowReducedEchelonForm (const DMatGFFlintBig &A, DMatGFFlintBig &result_rref)
void	MatrixOps::addMultipleTo (DMatGFFlint &C, const DMatGFFlint &A, const DMatGFFlint &B)
void	MatrixOps::subtractMultipleTo (DMatGFFlint &C, const DMatGFFlint &A, const DMatGFFlint &B)
void	MatrixOps::mult (const DMatGFFlint &A, const DMatGFFlint &B, DMatGFFlint &result_product)
size_t	MatrixOps::rowReducedEchelonForm (const DMatGFFlint &A, DMatGFFlint &result_rref)
void	MatrixOps::mult (const DMatQQ &A, const DMatQQ &B, DMatQQ &result_product)
void	MatrixOps::addMultipleTo (DMatQQ &C, const DMatQQ &A, const DMatQQ &B)
void	MatrixOps::subtractMultipleTo (DMatQQ &C, const DMatQQ &A, const DMatQQ &B)
size_t	MatrixOps::rowReducedEchelonForm (const DMatQQ &A, DMatQQ &result_rref)
size_t	MatrixOps::rank (const DMatQQFlint &A)
void	MatrixOps::determinant (const DMatQQFlint &A, M2::ARingQQFlint::ElementType &result_det)
bool	MatrixOps::inverse (const DMatQQFlint &A, DMatQQFlint &result_inv)
size_t	MatrixOps::rowReducedEchelonForm (const DMatQQFlint &A, DMatQQFlint &result_rref)
size_t	MatrixOps::nullSpace (const DMatQQFlint &A, DMatQQFlint &result_nullspace)
bool	MatrixOps::solveLinear (const DMatQQFlint &A, const DMatQQFlint &B, DMatQQFlint &X)
M2_arrayintOrNull	MatrixOps::rankProfile (const DMatQQFlint &A, bool row_profile)
void	MatrixOps::addMultipleTo (DMatQQFlint &C, const DMatQQFlint &A, const DMatQQFlint &B)
void	MatrixOps::subtractMultipleTo (DMatQQFlint &C, const DMatQQFlint &A, const DMatQQFlint &B)
void	MatrixOps::mult (const DMatQQFlint &A, const DMatQQFlint &B, DMatQQFlint &result_product)
bool	MatrixOps::eigenvaluesHermitian (const DMatRR &A, DMatRR &eigenvals)
bool	MatrixOps::eigenvalues (const DMatRR &A, DMatCC &eigenvals)
bool	MatrixOps::eigenvectorsHermitian (const DMatRR &A, DMatRR &eigenvals, DMatRR &eigenvecs)
bool	MatrixOps::eigenvectors (const DMatRR &A, DMatCC &eigenvals, DMatCC &eigenvecs)
bool	MatrixOps::leastSquares (const DMatRR &A, const DMatRR &B, DMatRR &X, bool assume_full_rank)
bool	MatrixOps::SVD (const DMatRR &A, DMatRR &Sigma, DMatRR &U, DMatRR &Vt, int strategy)
bool	MatrixOps::QR (const DMatRR &A, DMatRR &Q, DMatRR &R, bool return_QR)
bool	MatrixOps::QR (const DMatCC &A, DMatCC &Q, DMatCC &R, bool return_QR)
void	MatrixOps::clean (gmp_RR epsilon, DMatRR &mat)
void	MatrixOps::increase_norm (gmp_RRmutable norm, const DMatRR &mat)
bool	MatrixOps::eigenvaluesHermitian (const DMatCC &A, DMatRR &eigenvals)
bool	MatrixOps::eigenvalues (const DMatCC &A, DMatCC &eigenvals)
bool	MatrixOps::eigenvectorsHermitian (const DMatCC &A, DMatRR &eigenvals, DMatCC &eigenvecs)
bool	MatrixOps::eigenvectors (const DMatCC &A, DMatCC &eigenvals, DMatCC &eigenvecs)
bool	MatrixOps::leastSquares (const DMatCC &A, const DMatCC &B, DMatCC &X, bool assume_full_rank)
bool	MatrixOps::SVD (const DMatCC &A, DMatRR &Sigma, DMatCC &U, DMatCC &Vt, int strategy)
void	MatrixOps::clean (gmp_RR epsilon, DMatCC &mat)
void	MatrixOps::increase_norm (gmp_RRmutable norm, const DMatCC &mat)
bool	MatrixOps::eigenvaluesHermitian (const DMatRRR &A, DMatRRR &eigenvals)
bool	MatrixOps::eigenvalues (const DMatRRR &A, DMatCCC &eigenvals)
bool	MatrixOps::eigenvectorsHermitian (const DMatRRR &A, DMatRRR &eigenvals, DMatRRR &eigenvecs)
bool	MatrixOps::eigenvectors (const DMatRRR &A, DMatCCC &eigenvals, DMatCCC &eigenvecs)
bool	MatrixOps::leastSquares (const DMatRRR &A, const DMatRRR &B, DMatRRR &X, bool assume_full_rank)
bool	MatrixOps::SVD (const DMatRRR &A, DMatRRR &Sigma, DMatRRR &U, DMatRRR &Vt, int strategy)
void	MatrixOps::clean (gmp_RR epsilon, DMatRRR &mat)
void	MatrixOps::increase_norm (gmp_RRmutable norm, const DMatRRR &mat)
bool	MatrixOps::eigenvaluesHermitian (const DMatCCC &A, DMatRRR &eigenvals)
bool	MatrixOps::eigenvalues (const DMatCCC &A, DMatCCC &eigenvals)
bool	MatrixOps::eigenvectorsHermitian (const DMatCCC &A, DMatRRR &eigenvals, DMatCCC &eigenvecs)
bool	MatrixOps::eigenvectors (const DMatCCC &A, DMatCCC &eigenvals, DMatCCC &eigenvecs)
bool	MatrixOps::leastSquares (const DMatCCC &A, const DMatCCC &B, DMatCCC &X, bool assume_full_rank)
bool	MatrixOps::SVD (const DMatCCC &A, DMatRRR &Sigma, DMatCCC &U, DMatCCC &Vt, int strategy)
void	MatrixOps::clean (gmp_RR epsilon, DMatCCC &mat)
void	MatrixOps::increase_norm (gmp_RRmutable norm, const DMatCCC &mat)

Detailed Description

Templated DMat<R> linear algebra: LU, rank, determinant, solve, inverse, nullSpace.

Note: AI-generated documentation. Verify against the source before relying on it.

Declares the dense linear-algebra surface as free function templates over RT (the aring coefficient type) so a single generic body covers every supported ring; specialisations route to the fastest available back end at compile time. FFLAS-FFPACK handles small Z/p (DMatZZpFFPACK); FLINT covers everything else over finite, integer, and rational coefficients — nmod_mat for DMatZZpFlint, fmpz_mat for DMatZZ / DMatZZGMP, fmpq_mat for DMatQQFlint, and fq_zech_mat / fq_nmod_mat for DMatGFFlint / DMatGFFlintBig. LAPACK is the default path for hardware- precision RR / CC (DMatRR / DMatCC via lapack.hpp), with the Eigen3-backed EigenM2 namespace in eigen.hpp as the #ifdef NO_LAPACK fallback. Arbitrary-precision RR / CC (DMatRRR / DMatCCC) always go through Eigen3 with <unsupported/Eigen/MPRealSupport>. Z/p with small p (DMatZZp) and the M2-native GF (DMatGFM2) ride the generic templates. The header pulls in every aring it specialises on and defines the full DMat* alias inventory (DMatZZp, DMatZZpFFPACK, DMatZZpFlint, DMatGFM2, DMatGFFlint, DMatGFFlintBig, DMatZZGMP, DMatZZ, DMatQQ, DMatQQFlint, DMatRR, DMatCC, DMatRRR, DMatCCC).

This file is what MutableMatrix (mat.hpp) forwards into for the heavy operations and is the natural complement to the arithmetic primitives in mat-arith.hpp and the elementary row / column operations in mat-elem-ops.hpp.

See also: dmat.hpp; mat-arith.hpp; mat-elem-ops.hpp; lapack.hpp; eigen.hpp

Definition in file mat-linalg.hpp.

Namespaces

Macros

Typedefs

Functions

Detailed Description