Matrix algorithms: Gauss elimination, orthonormalization, QR decomposition etc. More...

Classes
struct	CAPDIntMatrixAlgorithms

class	CAPDSmithForm

class	Gershgorin

class	Gershgorin< MatrixType, false >
	Specialization for non-rigorous types. More...

class	Gershgorin< MatrixType, true >
	Specialization for rigorous (interval) types. More...

struct	GetExpRemainder

struct	GetExpRemainder< T, true >

struct	GetExpRemainder< T, false >

struct	IntMatrixAlgorithmsFactory

struct	IInvert

class	Invert

class	PARIInterface

struct	PARIIntMatrixAlgorithms

struct	GetPARISmithFormTraits

class	PARISmithForm

class	QuotientBaseMatrix

struct	SmithFormTraits

class	SmithForm

class	SmithFormFactory

class	SolveLinearEquation

struct	_CreateSmithForm_

struct	_CreateSmithForm_< Matrix< Scalar, 0, 0 > >

struct	_CreateSmithForm_< Matrix< short, 0, 0 > >

struct	_CreateSmithForm_< Matrix< int, 0, 0 > >

struct	_CreateSmithForm_< Matrix< long, 0, 0 > >

struct	_CreateSmithForm_< Matrix< llong, 0, 0 > >

Typedefs
typedef CAPDIntMatrixAlgorithms	DefaultIntMatrixAlgorithms

typedef long long	llong

typedef Matrix< short,0, 0 >	XXXXshort

typedef Matrix< int,0, 0 >	XXXXint

typedef Matrix< long,0, 0 >	XXXXlong

typedef Matrix< llong,0, 0 >	XXXXllong

typedef Matrix< Zp,0, 0 >	XXXXZp

typedef Matrix< Z2,0, 0 >	XXXXZ2

typedef Matrix< MpInt,0, 0 >	XXXXMpInt

Functions
template<typename MatrixType , typename ResultType >
void	gauss (MatrixType a, ResultType b, ResultType &result)

template<typename MatrixType , typename VectorType >
VectorType	gauss (const MatrixType &A, const VectorType &b)
	Gauss elimination. More...

template<typename MatrixType >
void	orthonormalize (MatrixType &Q)

template<typename MatrixType >
void	orthonormalize (MatrixType &Q, const typename MatrixType::RowVectorType &v)

template<typename MatrixType >
void	orthonormalize (MatrixType &Q, const MatrixType &R)

template<typename MatrixType >
void	QR_decompose (const MatrixType &A, MatrixType &Q, MatrixType &R)

template<typename MatrixType >
int	symMatrixDiagonalize (const MatrixType &A, MatrixType &D, typename MatrixType::ScalarType diagonalizingRelTolerance=capd::TypeTraits< typename MatrixType::ScalarType >::epsilon())

template<typename MatrixType >
MatrixType::ScalarType	spectralRadiusOfSymMatrix (const MatrixType &A, typename MatrixType::ScalarType diagonalizingRelTolerance=capd::TypeTraits< typename MatrixType::ScalarType >::epsilon())
	this function computes bound for spectral radius of a symmetric matrix first it computes matrix which has the same eigenvalues and which is close to diagonal, next bound is computed from Gerschgorin theorem More...

template<typename MatrixType >
MatrixType::ScalarType	maxEigenValueOfSymMatrix (const MatrixType &A, typename MatrixType::ScalarType diagonalizingRelTolerance=capd::TypeTraits< typename MatrixType::ScalarType >::epsilon())
	this function computes bound for maximal eigenvalue of a symmetric matrix first it computes matrix which has the same eigenvalues and which is close to diagonal, next bound is computed from Gerschgorin theorem More...

template<typename MatrixType >
MatrixType	matrixExp (const MatrixType &M, typename MatrixType::ScalarType tolerance=capd::TypeTraits< typename MatrixType::ScalarType >::epsilon())

template<typename MatrixType >
void	croutDecomposition (const MatrixType &A, MatrixType &D, MatrixType &G)
	Crout Decomposition of a matrix As a result matrix D is a lower triangle and G is an upper triangle with 1 on diagonal. More...

template<typename MatrixType >
MatrixType	invLowerTriangleMatrix (const MatrixType &A)

template<typename MatrixType >
MatrixType	invUpperTriangleMatrix (const MatrixType &A)

template<typename MatrixType >
MatrixType	inverseMatrix (const MatrixType &A)

template<typename MatrixType >
MatrixType	gaussInverseMatrix (const MatrixType &A)

template<typename MatrixType >
MatrixType::ScalarType	det (MatrixType A)
	Computes determinant of a matrix A by LU decomposition. More...

template<typename MatrixType , typename VectorType , typename IntVectorType >
void	computeSortingPermutation (const MatrixType &Q, const VectorType &v, IntVectorType &permutation)
	computes sorting permutation according to sizes of the set in the direction of the base vectors More...

template<typename MatrixType , typename IntVectorType >
void	QRdecomposeWithPivoting (const MatrixType &A, const typename MatrixType::RowVectorType &v, MatrixType &Q, MatrixType &R, typename MatrixType::RowVectorType &sizes, IntVectorType &p)

template<class matrix >
void	kernelImage (matrix &B, matrix &kernel, matrix &image)

template<class matrix , class sqMatrix1 , class sqMatrix2 >
void	smithForm (matrix &B, sqMatrix1 &Q, sqMatrix1 &Qinv, sqMatrix2 &R, sqMatrix2 &Rinv, int &s, int &t)

template<class matrix , class vector , class colVector >
bool	solveLinearEquation (const matrix &A, const colVector &b, vector &x)

template<class matrix >
bool	invert (const matrix &A, matrix &Ainv)

template<class matrix , class IntVector >
void	quotientBaseMatrix (const matrix &A_W, const matrix &A_V, matrix &A_U, IntVector &A_orders)

template<class matrix >
void	copy (const matrix &A, matrix &result, int row, int col)

template<class matrix >
void	spaceIntersection (const matrix &A, const matrix &B, matrix &C)

template<typename intType >
bool	isDivisible (intType a, intType b)

bool	isDivisible (double, double)

template<typename intType >
bool	isInvertible (intType a)

template<typename intType >
intType	inverse (intType a)

template<typename matrix >
capd::vectalg::Matrix< typename matrix::ScalarType, 0, 0 >	emptyMatrix ()

template<class matrix >
void	rowExchange (matrix &A, int i, int j)

template<class matrix >
void	rowMultiply (matrix &A, int i, typename matrix::ScalarType s)

template<class matrix >
void	rowAdd (matrix &A, int i, int j, typename matrix::ScalarType s)

template<class matrix >
void	columnExchange (matrix &A, int i, int j)

template<class matrix >
void	columnMultiply (matrix &A, int j, typename matrix::ScalarType s)

template<class matrix >
void	columnAdd (matrix &A, int i, int j, typename matrix::ScalarType s)

template<class matrix , class sqMatrix >
void	rowAdd (matrix &B, sqMatrix &Q, int i, int j, typename matrix::ScalarType q)

template<class matrix , class sqMatrix >
void	columnExchange (matrix &B, sqMatrix &R, int i, int j)

template<class matrix , class sqMatrix >
void	columnMultiply (matrix &B, sqMatrix &R, int i, typename matrix::ScalarType q)

template<class matrix , class sqMatrix >
void	columnAdd (matrix &B, sqMatrix &R, int i, int j, typename matrix::ScalarType q)

template<class matrix , class sqMatrix >
void	rowExchange (matrix &B, sqMatrix &Q, sqMatrix &Qinv, int i, int j)

template<class matrix , class sqMatrix >
void	rowMultiply (matrix &B, sqMatrix &Q, sqMatrix &Qinv, int i, typename matrix::ScalarType q)

template<class matrix , class sqMatrix >
void	rowAdd (matrix &B, sqMatrix &Q, sqMatrix &Qinv, int i, int j, typename matrix::ScalarType q)

template<class matrix , class sqMatrix >
void	columnExchange (matrix &B, sqMatrix &R, sqMatrix &Rinv, int i, int j)

template<class matrix , class sqMatrix >
void	columnMultiply (matrix &B, sqMatrix &R, sqMatrix &Rinv, int i, typename matrix::ScalarType q)

template<class matrix , class sqMatrix >
void	columnAdd (matrix &B, sqMatrix &R, sqMatrix &Rinv, int i, int j, typename matrix::ScalarType q)

template<class matrix , class sqMatrix >
void	partRowReduce (matrix &B, sqMatrix &Q, sqMatrix &Qinv, int k, int l)

template<class matrix , class sqMatrix >
void	partColumnReduce (matrix &B, sqMatrix &R, sqMatrix &Rinv, int k, int l)

template<class matrix >
void	smallestNonZero (const matrix &A, typename matrix::ScalarType &s, int &iOpt, int &jOpt)

template<class matrix >
bool	nonZero (const matrix &A)

template<class matrix , class sqMatrix >
void	rowPrepare (matrix &B, sqMatrix &Q, sqMatrix &Qinv, int k, int l)

template<class matrix , class sqMatrix >
void	rowReduce (matrix &B, sqMatrix &Q, sqMatrix &Qinv, int k, int l)

template<class matrix , class sqMatrix >
void	rowEchelon (matrix &B, sqMatrix &Q, sqMatrix &Qinv, int &k)

template<class matrix , class sqMatrix >
void	columnPrepare (matrix &B, sqMatrix &R, sqMatrix &Rinv, int k, int l)

template<class matrix , class sqMatrix >
void	columnReduce (matrix &B, sqMatrix &R, sqMatrix &Rinv, int k, int l)

template<class matrix , class sqMatrix >
void	columnEchelon (matrix &B, sqMatrix &R, sqMatrix &Rinv, int &l)

template<class MatrixType >
void	krawczykCorrection (const MatrixType &A, MatrixType &invA)

template<class MatrixType >
MatrixType	krawczykInverse (const MatrixType &A)

template void	rowExchange< capd::vectalg::Matrix< int, 0, 0 > > (capd::vectalg::Matrix< int, 0, 0 > &A, int i, int j)

template void	rowMultiply< capd::vectalg::Matrix< int, 0, 0 > > (capd::vectalg::Matrix< int, 0, 0 > &A, int i, capd::vectalg::Matrix< int, 0, 0 >::ScalarType s)

template void	rowAdd< capd::vectalg::Matrix< int, 0, 0 > > (capd::vectalg::Matrix< int, 0, 0 > &A, int i, int j, capd::vectalg::Matrix< int, 0, 0 >::ScalarType s)

template void	columnExchange< capd::vectalg::Matrix< int, 0, 0 > > (capd::vectalg::Matrix< int, 0, 0 > &A, int i, int j)

template void	columnMultiply< capd::vectalg::Matrix< int, 0, 0 > > (capd::vectalg::Matrix< int, 0, 0 > &A, int j, capd::vectalg::Matrix< int, 0, 0 >::ScalarType s)

template void	columnAdd< capd::vectalg::Matrix< int, 0, 0 > > (capd::vectalg::Matrix< int, 0, 0 > &A, int i, int j, capd::vectalg::Matrix< int, 0, 0 >::ScalarType s)

template void	smallestNonZero< capd::vectalg::Matrix< int, 0, 0 > > (const capd::vectalg::Matrix< int, 0, 0 > &A, capd::vectalg::Matrix< int, 0, 0 >::ScalarType &s, int &iOpt, int &jOpt)

template bool	nonZero< capd::vectalg::Matrix< int, 0, 0 > > (const capd::vectalg::Matrix< int, 0, 0 > &A)

template vectalg::Matrix< capd::DInterval, 0, 0 >	krawczykInverse< vectalg::Matrix< capd::DInterval, 0, 0 > > (const vectalg::Matrix< capd::DInterval, 0, 0 > &)

Detailed Description

Matrix algorithms: Gauss elimination, orthonormalization, QR decomposition etc.

Typedef Documentation

◆ DefaultIntMatrixAlgorithms

typedef CAPDIntMatrixAlgorithms capd::matrixAlgorithms::DefaultIntMatrixAlgorithms

◆ llong

typedef long long capd::matrixAlgorithms::llong

◆ XXXXint

typedef Matrix< int ,0,0> capd::matrixAlgorithms::XXXXint

◆ XXXXllong

typedef Matrix< llong ,0,0> capd::matrixAlgorithms::XXXXllong

◆ XXXXlong

typedef Matrix< long ,0,0> capd::matrixAlgorithms::XXXXlong

◆ XXXXMpInt

typedef Matrix< MpInt ,0,0> capd::matrixAlgorithms::XXXXMpInt

◆ XXXXshort

typedef Matrix< short ,0,0> capd::matrixAlgorithms::XXXXshort

◆ XXXXZ2

typedef Matrix< Z2 ,0,0> capd::matrixAlgorithms::XXXXZ2

◆ XXXXZp

typedef Matrix< Zp ,0,0> capd::matrixAlgorithms::XXXXZp

Function Documentation

◆ columnAdd() [1/3]

template<class matrix >

void capd::matrixAlgorithms::columnAdd	(	matrix &	A,
		int	i,
		int	j,
		typename matrix::ScalarType	s
	)

◆ columnAdd() [2/3]

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::columnAdd	(	matrix &	B,
		sqMatrix &	R,
		int	i,
		int	j,
		typename matrix::ScalarType	q
	)

◆ columnAdd() [3/3]

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::columnAdd	(	matrix &	B,
		sqMatrix &	R,
		sqMatrix &	Rinv,
		int	i,
		int	j,
		typename matrix::ScalarType	q
	)

◆ columnAdd< capd::vectalg::Matrix< int, 0, 0 > >()

template void capd::matrixAlgorithms::columnAdd< capd::vectalg::Matrix< int, 0, 0 > >	(	capd::vectalg::Matrix< int, 0, 0 > &	A,
		int	i,
		int	j,
		capd::vectalg::Matrix< int, 0, 0 >::ScalarType	s
	)

◆ columnEchelon()

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::columnEchelon	(	matrix &	B,
		sqMatrix &	R,
		sqMatrix &	Rinv,
		int &	l
	)

◆ columnExchange() [1/3]

template<class matrix >

void capd::matrixAlgorithms::columnExchange	(	matrix &	A,
		int	i,
		int	j
	)

◆ columnExchange() [2/3]

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::columnExchange	(	matrix &	B,
		sqMatrix &	R,
		int	i,
		int	j
	)

◆ columnExchange() [3/3]

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::columnExchange	(	matrix &	B,
		sqMatrix &	R,
		sqMatrix &	Rinv,
		int	i,
		int	j
	)

◆ columnExchange< capd::vectalg::Matrix< int, 0, 0 > >()

template void capd::matrixAlgorithms::columnExchange< capd::vectalg::Matrix< int, 0, 0 > >	(	capd::vectalg::Matrix< int, 0, 0 > &	A,
		int	i,
		int	j
	)

◆ columnMultiply() [1/3]

template<class matrix >

void capd::matrixAlgorithms::columnMultiply	(	matrix &	A,
		int	j,
		typename matrix::ScalarType	s
	)

◆ columnMultiply() [2/3]

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::columnMultiply	(	matrix &	B,
		sqMatrix &	R,
		int	i,
		typename matrix::ScalarType	q
	)

◆ columnMultiply() [3/3]

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::columnMultiply	(	matrix &	B,
		sqMatrix &	R,
		sqMatrix &	Rinv,
		int	i,
		typename matrix::ScalarType	q
	)

◆ columnMultiply< capd::vectalg::Matrix< int, 0, 0 > >()

template void capd::matrixAlgorithms::columnMultiply< capd::vectalg::Matrix< int, 0, 0 > >	(	capd::vectalg::Matrix< int, 0, 0 > &	A,
		int	j,
		capd::vectalg::Matrix< int, 0, 0 >::ScalarType	s
	)

◆ columnPrepare()

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::columnPrepare	(	matrix &	B,
		sqMatrix &	R,
		sqMatrix &	Rinv,
		int	k,
		int	l
	)

◆ columnReduce()

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::columnReduce	(	matrix &	B,
		sqMatrix &	R,
		sqMatrix &	Rinv,
		int	k,
		int	l
	)

◆ computeSortingPermutation()

template<typename MatrixType , typename VectorType , typename IntVectorType >

void capd::matrixAlgorithms::computeSortingPermutation	(	const MatrixType &	Q,
		const VectorType &	v,
		IntVectorType &	permutation
	)

computes sorting permutation according to sizes of the set in the direction of the base vectors

Parameters

	Q
	Q	matrix of base vectors (in columns)
	v
	v	set in given coordinates
	permutation
[out]	permutation

◆ copy()

template<class matrix >

void capd::matrixAlgorithms::copy	(	const matrix &	A,
		matrix &	result,
		int	row,
		int	col
	)

◆ croutDecomposition()

template<typename MatrixType >

void capd::matrixAlgorithms::croutDecomposition	(	const MatrixType &	A,
		MatrixType &	D,
		MatrixType &	G
	)

Crout Decomposition of a matrix As a result matrix D is a lower triangle and G is an upper triangle with 1 on diagonal.

Crout Decomposition of a positive definite matrix As a result matrix D is a lower triangle and G is an upper triangle with 1 on diagonal.

◆ det()

template<typename MatrixType >

MatrixType::ScalarType capd::matrixAlgorithms::det ( MatrixType A )

Computes determinant of a matrix A by LU decomposition.

Remarks: Singular matrix case is not fully implemented.

Parameters

A	given matrix

Returns: determinant of A

◆ emptyMatrix()

template<typename matrix >

capd::vectalg::Matrix<typename matrix::ScalarType, 0, 0> capd::matrixAlgorithms::emptyMatrix ( )

◆ gauss() [1/2]

template<typename MatrixType , typename VectorType >

VectorType capd::matrixAlgorithms::gauss	(	const MatrixType &	A,
		const VectorType &	b
	)

Gauss elimination.

this function solves equaiton A*x=b for x where A is nonsingular matrix

◆ gauss() [2/2]

template<typename MatrixType , typename ResultType >

void capd::matrixAlgorithms::gauss	(	MatrixType	a,
		ResultType	b,
		ResultType &	result
	)

◆ gaussInverseMatrix()

template<typename MatrixType >

MatrixType capd::matrixAlgorithms::gaussInverseMatrix ( const MatrixType & A )

◆ inverse()

template<typename intType >

intType capd::matrixAlgorithms::inverse ( intType a )

inline

◆ inverseMatrix()

template<typename MatrixType >

MatrixType capd::matrixAlgorithms::inverseMatrix ( const MatrixType & A )

◆ invert()

template<class matrix >

bool capd::matrixAlgorithms::invert	(	const matrix &	A,
		matrix &	Ainv
	)

◆ invLowerTriangleMatrix()

template<typename MatrixType >

MatrixType capd::matrixAlgorithms::invLowerTriangleMatrix ( const MatrixType & A )

◆ invUpperTriangleMatrix()

template<typename MatrixType >

MatrixType capd::matrixAlgorithms::invUpperTriangleMatrix ( const MatrixType & A )

◆ isDivisible() [1/2]

bool capd::matrixAlgorithms::isDivisible	(	double	,
		double
	)

inline

◆ isDivisible() [2/2]

template<typename intType >

bool capd::matrixAlgorithms::isDivisible	(	intType	a,
		intType	b
	)

inline

◆ isInvertible()

template<typename intType >

bool capd::matrixAlgorithms::isInvertible ( intType a )

inline

◆ kernelImage()

template<class matrix >

void capd::matrixAlgorithms::kernelImage	(	matrix &	B,
		matrix &	kernel,
		matrix &	image
	)

◆ krawczykCorrection()

template<class MatrixType >

void capd::matrixAlgorithms::krawczykCorrection	(	const MatrixType &	A,
		MatrixType &	invA
	)

◆ krawczykInverse()

template<class MatrixType >

MatrixType capd::matrixAlgorithms::krawczykInverse ( const MatrixType & A )

◆ krawczykInverse< vectalg::Matrix< capd::DInterval, 0, 0 > >()

template vectalg::Matrix<capd::DInterval, 0 , 0 > capd::matrixAlgorithms::krawczykInverse< vectalg::Matrix< capd::DInterval, 0, 0 > > ( const vectalg::Matrix< capd::DInterval, 0, 0 > & )

◆ matrixExp()

template<typename MatrixType >

MatrixType capd::matrixAlgorithms::matrixExp	(	const MatrixType &	M,
		typename MatrixType::ScalarType	tolerance = `capd::TypeTraits<typename MatrixType::ScalarType>::epsilon()`
	)

◆ maxEigenValueOfSymMatrix()

template<typename MatrixType >

MatrixType::ScalarType capd::matrixAlgorithms::maxEigenValueOfSymMatrix	(	const MatrixType &	A,
		typename MatrixType::ScalarType	diagonalizingRelTolerance = `capd::TypeTraits<typename MatrixType::ScalarType>::epsilon()`
	)

this function computes bound for maximal eigenvalue of a symmetric matrix first it computes matrix which has the same eigenvalues and which is close to diagonal, next bound is computed from Gerschgorin theorem

this function computes upper bound for maximal eigenvalue of a symmetric matrix first it computes matrix which has the same eigenvalues and which is close to diagonal, next upper bound is computed from Gerschgorin theorem

◆ nonZero()

template<class matrix >

bool capd::matrixAlgorithms::nonZero ( const matrix & A )

◆ nonZero< capd::vectalg::Matrix< int, 0, 0 > >()

template bool capd::matrixAlgorithms::nonZero< capd::vectalg::Matrix< int, 0, 0 > > ( const capd::vectalg::Matrix< int, 0, 0 > & A )

◆ orthonormalize() [1/3]

template<typename MatrixType >

void capd::matrixAlgorithms::orthonormalize ( MatrixType & Q )

◆ orthonormalize() [2/3]

template<typename MatrixType >

void capd::matrixAlgorithms::orthonormalize	(	MatrixType &	Q,
		const MatrixType &	R
	)

TODO: check and replace with the above

◆ orthonormalize() [3/3]

template<typename MatrixType >

void capd::matrixAlgorithms::orthonormalize	(	MatrixType &	Q,
		const typename MatrixType::RowVectorType &	v
	)

◆ partColumnReduce()

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::partColumnReduce	(	matrix &	B,
		sqMatrix &	R,
		sqMatrix &	Rinv,
		int	k,
		int	l
	)

◆ partRowReduce()

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::partRowReduce	(	matrix &	B,
		sqMatrix &	Q,
		sqMatrix &	Qinv,
		int	k,
		int	l
	)

◆ QR_decompose()

template<typename MatrixType >

void capd::matrixAlgorithms::QR_decompose	(	const MatrixType &	A,
		MatrixType &	Q,
		MatrixType &	R
	)

◆ QRdecomposeWithPivoting()

template<typename MatrixType , typename IntVectorType >

void capd::matrixAlgorithms::QRdecomposeWithPivoting	(	const MatrixType &	A,
		const typename MatrixType::RowVectorType &	v,
		MatrixType &	Q,
		MatrixType &	R,
		typename MatrixType::RowVectorType &	sizes,
		IntVectorType &	p
	)

◆ quotientBaseMatrix()

template<class matrix , class IntVector >

void capd::matrixAlgorithms::quotientBaseMatrix	(	const matrix &	A_W,
		const matrix &	A_V,
		matrix &	A_U,
		IntVector &	A_orders
	)

◆ rowAdd() [1/3]

template<class matrix >

void capd::matrixAlgorithms::rowAdd	(	matrix &	A,
		int	i,
		int	j,
		typename matrix::ScalarType	s
	)

◆ rowAdd() [2/3]

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::rowAdd	(	matrix &	B,
		sqMatrix &	Q,
		int	i,
		int	j,
		typename matrix::ScalarType	q
	)

◆ rowAdd() [3/3]

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::rowAdd	(	matrix &	B,
		sqMatrix &	Q,
		sqMatrix &	Qinv,
		int	i,
		int	j,
		typename matrix::ScalarType	q
	)

◆ rowAdd< capd::vectalg::Matrix< int, 0, 0 > >()

template void capd::matrixAlgorithms::rowAdd< capd::vectalg::Matrix< int, 0, 0 > >	(	capd::vectalg::Matrix< int, 0, 0 > &	A,
		int	i,
		int	j,
		capd::vectalg::Matrix< int, 0, 0 >::ScalarType	s
	)

◆ rowEchelon()

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::rowEchelon	(	matrix &	B,
		sqMatrix &	Q,
		sqMatrix &	Qinv,
		int &	k
	)

◆ rowExchange() [1/2]

template<class matrix >

void capd::matrixAlgorithms::rowExchange	(	matrix &	A,
		int	i,
		int	j
	)

◆ rowExchange() [2/2]

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::rowExchange	(	matrix &	B,
		sqMatrix &	Q,
		sqMatrix &	Qinv,
		int	i,
		int	j
	)

◆ rowExchange< capd::vectalg::Matrix< int, 0, 0 > >()

template void capd::matrixAlgorithms::rowExchange< capd::vectalg::Matrix< int, 0, 0 > >	(	capd::vectalg::Matrix< int, 0, 0 > &	A,
		int	i,
		int	j
	)

◆ rowMultiply() [1/2]

template<class matrix >

void capd::matrixAlgorithms::rowMultiply	(	matrix &	A,
		int	i,
		typename matrix::ScalarType	s
	)

◆ rowMultiply() [2/2]

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::rowMultiply	(	matrix &	B,
		sqMatrix &	Q,
		sqMatrix &	Qinv,
		int	i,
		typename matrix::ScalarType	q
	)

◆ rowMultiply< capd::vectalg::Matrix< int, 0, 0 > >()

template void capd::matrixAlgorithms::rowMultiply< capd::vectalg::Matrix< int, 0, 0 > >	(	capd::vectalg::Matrix< int, 0, 0 > &	A,
		int	i,
		capd::vectalg::Matrix< int, 0, 0 >::ScalarType	s
	)

◆ rowPrepare()

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::rowPrepare	(	matrix &	B,
		sqMatrix &	Q,
		sqMatrix &	Qinv,
		int	k,
		int	l
	)

◆ rowReduce()

template<class matrix , class sqMatrix >

void capd::matrixAlgorithms::rowReduce	(	matrix &	B,
		sqMatrix &	Q,
		sqMatrix &	Qinv,
		int	k,
		int	l
	)

◆ smallestNonZero()

template<class matrix >

void capd::matrixAlgorithms::smallestNonZero	(	const matrix &	A,
		typename matrix::ScalarType &	s,
		int &	iOpt,
		int &	jOpt
	)

◆ smallestNonZero< capd::vectalg::Matrix< int, 0, 0 > >()

template void capd::matrixAlgorithms::smallestNonZero< capd::vectalg::Matrix< int, 0, 0 > >	(	const capd::vectalg::Matrix< int, 0, 0 > &	A,
		capd::vectalg::Matrix< int, 0, 0 >::ScalarType &	s,
		int &	iOpt,
		int &	jOpt
	)

◆ smithForm()

template<class matrix , class sqMatrix1 , class sqMatrix2 >

void capd::matrixAlgorithms::smithForm	(	matrix &	B,
		sqMatrix1 &	Q,
		sqMatrix1 &	Qinv,
		sqMatrix2 &	R,
		sqMatrix2 &	Rinv,
		int &	s,
		int &	t
	)

◆ solveLinearEquation()

template<class matrix , class vector , class colVector >

bool capd::matrixAlgorithms::solveLinearEquation	(	const matrix &	A,
		const colVector &	b,
		vector &	x
	)

◆ spaceIntersection()

template<class matrix >

void capd::matrixAlgorithms::spaceIntersection	(	const matrix &	A,
		const matrix &	B,
		matrix &	C
	)

◆ spectralRadiusOfSymMatrix()

template<typename MatrixType >

MatrixType::ScalarType capd::matrixAlgorithms::spectralRadiusOfSymMatrix	(	const MatrixType &	A,
		typename MatrixType::ScalarType	diagonalizingRelTolerance = `capd::TypeTraits<typename MatrixType::ScalarType>::epsilon()`
	)

this function computes bound for spectral radius of a symmetric matrix first it computes matrix which has the same eigenvalues and which is close to diagonal, next bound is computed from Gerschgorin theorem

◆ symMatrixDiagonalize()

template<typename MatrixType >

int capd::matrixAlgorithms::symMatrixDiagonalize	(	const MatrixType &	A,
		MatrixType &	D,
		typename MatrixType::ScalarType	diagonalizingRelTolerance = `capd::TypeTraits<typename MatrixType::ScalarType>::epsilon()`
	)

Classes

Typedefs

Functions

Detailed Description

Typedef Documentation

◆ DefaultIntMatrixAlgorithms

◆ llong

◆ XXXXint

◆ XXXXllong

◆ XXXXlong

◆ XXXXMpInt

◆ XXXXshort

◆ XXXXZ2

◆ XXXXZp

Function Documentation

◆ columnAdd() [1/3]

◆ columnAdd() [2/3]

◆ columnAdd() [3/3]

◆ columnAdd< capd::vectalg::Matrix< int, 0, 0 > >()

◆ columnEchelon()

◆ columnExchange() [1/3]

◆ columnExchange() [2/3]

◆ columnExchange() [3/3]

◆ columnExchange< capd::vectalg::Matrix< int, 0, 0 > >()

◆ columnMultiply() [1/3]

◆ columnMultiply() [2/3]

◆ columnMultiply() [3/3]

◆ columnMultiply< capd::vectalg::Matrix< int, 0, 0 > >()

◆ columnPrepare()

◆ columnReduce()

◆ computeSortingPermutation()

◆ copy()

◆ croutDecomposition()

◆ det()

◆ emptyMatrix()

◆ gauss() [1/2]

◆ gauss() [2/2]

◆ gaussInverseMatrix()

◆ inverse()

◆ inverseMatrix()

◆ invert()

◆ invLowerTriangleMatrix()

◆ invUpperTriangleMatrix()

◆ isDivisible() [1/2]

◆ isDivisible() [2/2]

◆ isInvertible()

◆ kernelImage()

◆ krawczykCorrection()

◆ krawczykInverse()

◆ krawczykInverse< vectalg::Matrix< capd::DInterval, 0, 0 > >()

◆ matrixExp()

◆ maxEigenValueOfSymMatrix()

◆ nonZero()

◆ nonZero< capd::vectalg::Matrix< int, 0, 0 > >()

◆ orthonormalize() [1/3]

◆ orthonormalize() [2/3]

◆ orthonormalize() [3/3]

◆ partColumnReduce()

◆ partRowReduce()

◆ QR_decompose()

◆ QRdecomposeWithPivoting()

◆ quotientBaseMatrix()

◆ rowAdd() [1/3]

◆ rowAdd() [2/3]

◆ rowAdd() [3/3]

◆ rowAdd< capd::vectalg::Matrix< int, 0, 0 > >()

◆ rowEchelon()

◆ rowExchange() [1/2]

◆ rowExchange() [2/2]

◆ rowExchange< capd::vectalg::Matrix< int, 0, 0 > >()

◆ rowMultiply() [1/2]

◆ rowMultiply() [2/2]

◆ rowMultiply< capd::vectalg::Matrix< int, 0, 0 > >()

◆ rowPrepare()

◆ rowReduce()

◆ smallestNonZero()

◆ smallestNonZero< capd::vectalg::Matrix< int, 0, 0 > >()

◆ smithForm()

◆ solveLinearEquation()

◆ spaceIntersection()