Typedefs
template<typename TReal>
using	CholeskyRowMajor = Eigen::Matrix<TReal, Eigen::Dynamic, Eigen::Dynamic, Eigen::RowMajor>

template<typename TReal>
using	CholeskyVector = Eigen::Matrix<TReal, Eigen::Dynamic, 1>

Functions
template<unsigned int VDim, typename TReal>
TReal	DeterminantEigen (const TReal *inData)

template<typename TVector>
auto	DeterminantMagnitude (const TVector &W) -> std::decay_t< decltype(W[0])>

template<typename TReal>
TReal	DynamicDeterminantEigen (const TReal *inData, unsigned int n)

template<typename TReal>
void	DynamicSquareSVDEigen (const TReal inData, unsigned int n, TReal uData, TReal wData, TReal vData)

template void	DynamicSquareSVDEigen< double > (const double , unsigned int, double , double , double )

template void	DynamicSquareSVDEigen< float > (const float , unsigned int, float , float , float )

template<typename TReal>
void	InverseSymmetricLDLTEigen (const TReal aData, unsigned int n, TReal invData)

template<unsigned int VRows, unsigned int VColumns, typename TReal>
void	MatrixExponentialEigen (const TReal inData, TReal outData)

template<typename TReal>
void	MatrixExponentialEigen (const TReal inData, TReal outData, unsigned int n)

template<typename TVector, typename TReal>
unsigned int	NumericalRank (const TVector &W, TReal rcond)

template<typename TMatrixU, typename TVector, typename TMatrixV, typename TReal>
auto	PseudoInverse (const TMatrixU &U, const TVector &W, const TMatrixV &V, TReal rcond)

template<typename TMatrixU, typename TVector, typename TMatrixV, typename TReal>
auto	Recompose (const TMatrixU &U, const TVector &W, const TMatrixV &V, TReal rcond)

template<typename TMatrixU, typename TVector, typename TMatrixV>
auto	RecomposeWith (const TMatrixU &U, const TVector &modifiedW, const TMatrixV &V)

template<typename TReal>
void	RectangularSVDEigen (const TReal inData, unsigned int rows, unsigned int cols, TReal uData, TReal wData, TReal vData)

template void	RectangularSVDEigen< double > (const double , unsigned int, unsigned int, double , double , double )

template void	RectangularSVDEigen< float > (const float , unsigned int, unsigned int, float , float , float )

template<typename TReal>
TReal	ResolveRcond (TReal rcond, unsigned int n)

template<typename TMatrixU, typename TVector, typename TMatrixV, typename TVectorB, typename TReal>
auto	SolveLinear (const TMatrixU &U, const TVector &W, const TMatrixV &V, const TVectorB &b, TReal rcond)

template<typename TReal>
void	SolveSymmetricLDLTEigen (const TReal aData, const TReal bData, unsigned int n, TReal *xData)

template<typename TReal>
void	SolveSymmetricMatrixLDLTEigen (const TReal aData, const TReal bData, unsigned int n, unsigned int m, TReal *xData)

template<unsigned int VDim, typename TReal>
void	SquareSVDEigen (const TReal inData, TReal uData, TReal wData, TReal vData)

template<typename TVector>
auto	WellCondition (const TVector &W) -> std::decay_t< decltype(W[0])>

Variables
constexpr unsigned int	kFixedSVDMaxDim = 16

constexpr unsigned int	kJacobiMaxDim = 6

Detailed Description

Cholesky-based linear algebra for symmetric matrices, backed by Eigen.

Matrix exponential exp(A) for a real square matrix, backed by Eigen.

SolveSymmetricPositiveDefinite() uses Eigen's robust LDL^T (pivoted) factor, which handles near-semidefinite regularized Gram matrices. CholeskyLowerTriangle() returns the L of A = L L^T via Eigen's LLT.

Replacement for the removed vnl_matrix_exp. Delegates to Eigen's MatrixBase::exp(), which uses scaling-and-squaring with a degree-13 Pade approximant (Higham, SIAM J. Matrix Anal. Appl. 26(4), 2005) – more robust than the truncated Taylor series formerly provided by vnl_matrix_exp.

Typedef Documentation

◆ CholeskyRowMajor

template<typename TReal>

using itk::Math::detail::CholeskyRowMajor = Eigen::Matrix<TReal, Eigen::Dynamic, Eigen::Dynamic, Eigen::RowMajor>

Definition at line 44 of file itkCholeskySolve.h.

◆ CholeskyVector

template<typename TReal>

using itk::Math::detail::CholeskyVector = Eigen::Matrix<TReal, Eigen::Dynamic, 1>

Definition at line 46 of file itkCholeskySolve.h.

Function Documentation

◆ DeterminantEigen()

template<unsigned int VDim, typename TReal>

TReal itk::Math::detail::DeterminantEigen ( const TReal * inData )

Definition at line 44 of file itkMathDeterminant.h.

Referenced by itk::Math::Determinant(), and DynamicDeterminantEigen().

◆ DeterminantMagnitude()

template<typename TVector>

auto itk::Math::detail::DeterminantMagnitude ( const TVector & W ) -> std::decay_t<decltype(W[0])>

Definition at line 161 of file itkMathSVD.h.

Referenced by itk::Math::FixedRectangularSVDResult< TReal, VRows, VCols >::DeterminantMagnitude(), itk::Math::FixedSquareSVDResult< TReal, VDim >::DeterminantMagnitude(), and itk::Math::SVDResult< TReal >::DeterminantMagnitude().

◆ DynamicDeterminantEigen()

template<typename TReal>

TReal itk::Math::detail::DynamicDeterminantEigen	(	const TReal *	inData,
		unsigned int	n )

Definition at line 52 of file itkMathDeterminant.h.

References DeterminantEigen().

Referenced by itk::Math::Determinant().

◆ DynamicSquareSVDEigen()

template<typename TReal>

void itk::Math::detail::DynamicSquareSVDEigen	(	const TReal *	inData,
		unsigned int	n,
		TReal *	uData,
		TReal *	wData,
		TReal *	vData )

Definition at line 400 of file itkMathSVD.h.

References kJacobiMaxDim.

Referenced by SquareSVDEigen(), and itk::Math::SVD().

◆ DynamicSquareSVDEigen< double >()

template void itk::Math::detail::DynamicSquareSVDEigen< double >	(	const double *	,
		unsigned int	,
		double *	,
		double *	,
		double *	)

extern

References ITKCommon_EXPORT_EXPLICIT.

◆ DynamicSquareSVDEigen< float >()

template void itk::Math::detail::DynamicSquareSVDEigen< float >	(	const float *	,
		unsigned int	,
		float *	,
		float *	,
		float *	)

extern

References ITKCommon_EXPORT_EXPLICIT.

◆ InverseSymmetricLDLTEigen()

template<typename TReal>

void itk::Math::detail::InverseSymmetricLDLTEigen	(	const TReal *	aData,
		unsigned int	n,
		TReal *	invData )

Definition at line 105 of file itkMathLDLT.h.

Referenced by itk::Math::InverseSymmetric(), and itk::Math::InverseSymmetric().

◆ MatrixExponentialEigen() [1/2]

template<unsigned int VRows, unsigned int VColumns, typename TReal>

void itk::Math::detail::MatrixExponentialEigen	(	const TReal *	inData,
		TReal *	outData )

Definition at line 45 of file itkMatrixExponential.h.

Referenced by itk::Math::MatrixExponential(), itk::Math::MatrixExponential(), and itk::Math::MatrixExponential().

◆ MatrixExponentialEigen() [2/2]

template<typename TReal>

void itk::Math::detail::MatrixExponentialEigen	(	const TReal *	inData,
		TReal *	outData,
		unsigned int	n )

Definition at line 55 of file itkMatrixExponential.h.

◆ NumericalRank()

template<typename TVector, typename TReal>

unsigned int itk::Math::detail::NumericalRank	(	const TVector &	W,
		TReal	rcond )

Definition at line 93 of file itkMathSVD.h.

References ResolveRcond().

◆ PseudoInverse()

template<typename TMatrixU, typename TVector, typename TMatrixV, typename TReal>

auto itk::Math::detail::PseudoInverse	(	const TMatrixU &	U,
		const TVector &	W,
		const TMatrixV &	V,
		TReal	rcond )

Definition at line 59 of file itkMathSVD.h.

References ResolveRcond().

◆ Recompose()

template<typename TMatrixU, typename TVector, typename TMatrixV, typename TReal>

auto itk::Math::detail::Recompose	(	const TMatrixU &	U,
		const TVector &	W,
		const TMatrixV &	V,
		TReal	rcond )

Definition at line 112 of file itkMathSVD.h.

References ResolveRcond().

◆ RecomposeWith()

template<typename TMatrixU, typename TVector, typename TMatrixV>

auto itk::Math::detail::RecomposeWith	(	const TMatrixU &	U,
		const TVector &	modifiedW,
		const TMatrixV &	V )

Definition at line 132 of file itkMathSVD.h.

Referenced by itk::Math::FixedRectangularSVDResult< TReal, VRows, VCols >::RecomposeWith(), itk::Math::FixedSquareSVDResult< TReal, VDim >::RecomposeWith(), and itk::Math::SVDResult< TReal >::RecomposeWith().

◆ RectangularSVDEigen()

template<typename TReal>

void itk::Math::detail::RectangularSVDEigen	(	const TReal *	inData,
		unsigned int	rows,
		unsigned int	cols,
		TReal *	uData,
		TReal *	wData,
		TReal *	vData )

Definition at line 475 of file itkMathSVD.h.

Referenced by itk::Math::SVD(), and itk::Math::SVD().

◆ RectangularSVDEigen< double >()

template void itk::Math::detail::RectangularSVDEigen< double >	(	const double *	,
		unsigned int	,
		unsigned int	,
		double *	,
		double *	,
		double *	)

extern

References ITKCommon_EXPORT_EXPLICIT.

◆ RectangularSVDEigen< float >()

template void itk::Math::detail::RectangularSVDEigen< float >	(	const float *	,
		unsigned int	,
		unsigned int	,
		float *	,
		float *	,
		float *	)

extern

References ITKCommon_EXPORT_EXPLICIT.

◆ ResolveRcond()

template<typename TReal>

TReal itk::Math::detail::ResolveRcond	(	TReal	rcond,
		unsigned int	n )

Definition at line 50 of file itkMathSVD.h.

Referenced by NumericalRank(), PseudoInverse(), Recompose(), and SolveLinear().

◆ SolveLinear()

template<typename TMatrixU, typename TVector, typename TMatrixV, typename TVectorB, typename TReal>

auto itk::Math::detail::SolveLinear	(	const TMatrixU &	U,
		const TVector &	W,
		const TMatrixV &	V,
		const TVectorB &	b,
		TReal	rcond )

Definition at line 79 of file itkMathSVD.h.

References ResolveRcond().

◆ SolveSymmetricLDLTEigen()

template<typename TReal>

void itk::Math::detail::SolveSymmetricLDLTEigen	(	const TReal *	aData,
		const TReal *	bData,
		unsigned int	n,
		TReal *	xData )

Definition at line 63 of file itkMathLDLT.h.

Referenced by itk::Math::SolveSymmetric(), itk::Math::SolveSymmetric(), itk::Math::SolveSymmetric(), and itk::Math::SolveSymmetric().

◆ SolveSymmetricMatrixLDLTEigen()

template<typename TReal>

void itk::Math::detail::SolveSymmetricMatrixLDLTEigen	(	const TReal *	aData,
		const TReal *	bData,
		unsigned int	n,
		unsigned int	m,
		TReal *	xData )

Definition at line 85 of file itkMathLDLT.h.

Referenced by itk::Math::SolveSymmetric(), and itk::Math::SolveSymmetric().

◆ SquareSVDEigen()

template<unsigned int VDim, typename TReal>

void itk::Math::detail::SquareSVDEigen	(	const TReal *	inData,
		TReal *	uData,
		TReal *	wData,
		TReal *	vData )

Definition at line 437 of file itkMathSVD.h.

References DynamicSquareSVDEigen(), and kFixedSVDMaxDim.

Referenced by itk::Math::SVD().

◆ WellCondition()

template<typename TVector>

auto itk::Math::detail::WellCondition ( const TVector & W ) -> std::decay_t<decltype(W[0])>

Definition at line 148 of file itkMathSVD.h.

Referenced by itk::Math::FixedRectangularSVDResult< TReal, VRows, VCols >::WellCondition(), itk::Math::FixedSquareSVDResult< TReal, VDim >::WellCondition(), and itk::Math::SVDResult< TReal >::WellCondition().

Variable Documentation

◆ kFixedSVDMaxDim

unsigned int itk::Math::detail::kFixedSVDMaxDim = 16

constexpr

Definition at line 390 of file itkMathSVD.h.

Referenced by SquareSVDEigen().

◆ kJacobiMaxDim

unsigned int itk::Math::detail::kJacobiMaxDim = 6

constexpr

Definition at line 394 of file itkMathSVD.h.

Referenced by DynamicSquareSVDEigen().

Typedefs

Functions

Variables

Detailed Description

Typedef Documentation

◆ CholeskyRowMajor

◆ CholeskyVector

Function Documentation

◆ DeterminantEigen()

◆ DeterminantMagnitude()

◆ DynamicDeterminantEigen()

◆ DynamicSquareSVDEigen()

◆ DynamicSquareSVDEigen< double >()

◆ DynamicSquareSVDEigen< float >()

◆ InverseSymmetricLDLTEigen()

◆ MatrixExponentialEigen() [1/2]

◆ MatrixExponentialEigen() [2/2]

◆ NumericalRank()

◆ PseudoInverse()

◆ Recompose()

◆ RecomposeWith()

◆ RectangularSVDEigen()

◆ RectangularSVDEigen< double >()

◆ RectangularSVDEigen< float >()

◆ ResolveRcond()

◆ SolveLinear()

◆ SolveSymmetricLDLTEigen()

◆ SolveSymmetricMatrixLDLTEigen()

◆ SquareSVDEigen()

◆ WellCondition()

Variable Documentation

◆ kFixedSVDMaxDim

◆ kJacobiMaxDim