Two-sided Jacobi SVD decomposition of a rectangular matrix. More...

#include <JacobiSVD.h>

Inheritance diagram for Eigen::JacobiSVD< _MatrixType, QRPreconditioner >:

Public Types
enum	{ RowsAtCompileTime = MatrixType::RowsAtCompileTime , ColsAtCompileTime = MatrixType::ColsAtCompileTime , DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime) , MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime , MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime , MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime) , MatrixOptions = MatrixType::Options }

typedef _MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

typedef NumTraits< typenameMatrixType::Scalar >::Real	RealScalar

typedef Base::MatrixUType	MatrixUType

typedef Base::MatrixVType	MatrixVType

typedef Base::SingularValuesType	SingularValuesType

typedef internal::plain_row_type< MatrixType >::type	RowType

typedef internal::plain_col_type< MatrixType >::type	ColType

typedef Matrix< Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime >	WorkMatrixType

Public Types inherited from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >
enum

typedef internal::traits< JacobiSVD< _MatrixType, QRPreconditioner > >::MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

typedef NumTraits< typenameMatrixType::Scalar >::Real	RealScalar

typedef Eigen::internal::traits< SVDBase >::StorageIndex	StorageIndex

typedef Eigen::Index	Index

typedef Matrix< Scalar, RowsAtCompileTime, RowsAtCompileTime, MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime >	MatrixUType

typedef Matrix< Scalar, ColsAtCompileTime, ColsAtCompileTime, MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime >	MatrixVType

typedef internal::plain_diag_type< MatrixType, RealScalar >::type	SingularValuesType

Public Types inherited from Eigen::SolverBase< Derived >
enum	{ RowsAtCompileTime = internal::traits<Derived>::RowsAtCompileTime , ColsAtCompileTime = internal::traits<Derived>::ColsAtCompileTime , SizeAtCompileTime , MaxRowsAtCompileTime = internal::traits<Derived>::MaxRowsAtCompileTime , MaxColsAtCompileTime = internal::traits<Derived>::MaxColsAtCompileTime , MaxSizeAtCompileTime , IsVectorAtCompileTime , NumDimensions = int(MaxSizeAtCompileTime) == 1 ? 0 : bool(IsVectorAtCompileTime) ? 1 : 2 }

typedef EigenBase< Derived >	Base

typedef internal::traits< Derived >::Scalar	Scalar

typedef Scalar	CoeffReturnType

typedef Transpose< const Derived >	ConstTransposeReturnType

typedef internal::conditional< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, constConstTransposeReturnType >, constConstTransposeReturnType >::type	AdjointReturnType

Public Types inherited from Eigen::EigenBase< Derived >
typedef Eigen::Index	Index
	The interface type of indices.

typedef internal::traits< Derived >::StorageKind	StorageKind

Public Member Functions
	JacobiSVD ()
	Default Constructor.

	JacobiSVD (Index rows, Index cols, unsigned int computationOptions=0)
	Default Constructor with memory preallocation.

	JacobiSVD (const MatrixType &matrix, unsigned int computationOptions=0)
	Constructor performing the decomposition of given matrix.

JacobiSVD &	compute (const MatrixType &matrix, unsigned int computationOptions)
	Method performing the decomposition of given matrix using custom options.

JacobiSVD &	compute (const MatrixType &matrix)
	Method performing the decomposition of given matrix using current options.

bool	computeU () const

bool	computeV () const

Index	rows () const

Index	cols () const

Index	rank () const

Public Member Functions inherited from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >
JacobiSVD< _MatrixType, QRPreconditioner > &	derived ()

const JacobiSVD< _MatrixType, QRPreconditioner > &	derived () const

const MatrixUType &	matrixU () const

const MatrixVType &	matrixV () const

const SingularValuesType &	singularValues () const

Index	nonzeroSingularValues () const

Index	rank () const

JacobiSVD< _MatrixType, QRPreconditioner > &	setThreshold (const RealScalar &threshold)
	Allows to prescribe a threshold to be used by certain methods, such as rank() and solve(), which need to determine when singular values are to be considered nonzero.

JacobiSVD< _MatrixType, QRPreconditioner > &	setThreshold (Default_t)
	Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.

RealScalar	threshold () const
	Returns the threshold that will be used by certain methods such as rank().

bool	computeU () const

bool	computeV () const

Index	rows () const

Index	cols () const

EIGEN_DEVICE_FUNC ComputationInfo	info () const
	Reports whether previous computation was successful.

void	_solve_impl (const RhsType &rhs, DstType &dst) const

void	_solve_impl_transposed (const RhsType &rhs, DstType &dst) const

Public Member Functions inherited from Eigen::SolverBase< Derived >
	SolverBase ()
	Default constructor.

template<typename Rhs >
const Solve< Derived, Rhs >	solve (const MatrixBase< Rhs > &b) const

const ConstTransposeReturnType	transpose () const

const AdjointReturnType	adjoint () const

EIGEN_DEVICE_FUNC Derived &	derived ()

EIGEN_DEVICE_FUNC const Derived &	derived () const

Public Member Functions inherited from Eigen::EigenBase< Derived >
EIGEN_DEVICE_FUNC Derived &	derived ()

EIGEN_DEVICE_FUNC const Derived &	derived () const

EIGEN_DEVICE_FUNC Derived &	const_cast_derived () const

EIGEN_DEVICE_FUNC const Derived &	const_derived () const

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

template<typename Dest >
EIGEN_DEVICE_FUNC void	evalTo (Dest &dst) const

template<typename Dest >
EIGEN_DEVICE_FUNC void	addTo (Dest &dst) const

template<typename Dest >
EIGEN_DEVICE_FUNC void	subTo (Dest &dst) const

template<typename Dest >
EIGEN_DEVICE_FUNC void	applyThisOnTheRight (Dest &dst) const

template<typename Dest >
EIGEN_DEVICE_FUNC void	applyThisOnTheLeft (Dest &dst) const

Protected Attributes
WorkMatrixType	m_workMatrix

internal::qr_preconditioner_impl< MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows >	m_qr_precond_morecols

internal::qr_preconditioner_impl< MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols >	m_qr_precond_morerows

MatrixType	m_scaledMatrix

MatrixUType	m_matrixU

MatrixVType	m_matrixV

SingularValuesType	m_singularValues

ComputationInfo	m_info

bool	m_isInitialized

bool	m_isAllocated

bool	m_usePrescribedThreshold

bool	m_computeFullU

bool	m_computeThinU

bool	m_computeFullV

bool	m_computeThinV

unsigned int	m_computationOptions

Index	m_nonzeroSingularValues

Index	m_rows

Index	m_cols

Index	m_diagSize

RealScalar	m_prescribedThreshold

Protected Attributes inherited from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >
MatrixUType	m_matrixU

MatrixVType	m_matrixV

SingularValuesType	m_singularValues

ComputationInfo	m_info

bool	m_isInitialized

bool	m_isAllocated

bool	m_usePrescribedThreshold

bool	m_computeFullU

bool	m_computeThinU

bool	m_computeFullV

bool	m_computeThinV

unsigned int	m_computationOptions

Index	m_nonzeroSingularValues

Index	m_rows

Index	m_cols

Index	m_diagSize

RealScalar	m_prescribedThreshold

Friends
template<typename __MatrixType , int _QRPreconditioner, bool _IsComplex>
struct	internal::svd_precondition_2x2_block_to_be_real

template<typename __MatrixType , int _QRPreconditioner, int _Case, bool _DoAnything>
struct	internal::qr_preconditioner_impl

Additional Inherited Members
Protected Member Functions inherited from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >
void	_check_compute_assertions () const

void	_check_solve_assertion (const Rhs &b) const

bool	allocate (Index rows, Index cols, unsigned int computationOptions)

	SVDBase ()
	Default Constructor.

Protected Member Functions inherited from Eigen::SolverBase< Derived >
template<bool Transpose_, typename Rhs >
void	_check_solve_assertion (const Rhs &b) const

Static Protected Member Functions inherited from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >
static void	check_template_parameters ()

Detailed Description

template<typename _MatrixType, int QRPreconditioner>
class Eigen::JacobiSVD< _MatrixType, QRPreconditioner >

Two-sided Jacobi SVD decomposition of a rectangular matrix.

Template Parameters

_MatrixType	the type of the matrix of which we are computing the SVD decomposition
QRPreconditioner	this optional parameter allows to specify the type of QR decomposition that will be used internally for the R-SVD step for non-square matrices. See discussion of possible values below.

SVD decomposition consists in decomposing any n-by-p matrix A as a product

$A = U S V^*$

where U is a n-by-n unitary, V is a p-by-p unitary, and S is a n-by-p real positive matrix which is zero outside of its main diagonal; the diagonal entries of S are known as the singular values of A and the columns of U and V are known as the left and right singular vectors of A respectively.

Singular values are always sorted in decreasing order.

This JacobiSVD decomposition computes only the singular values by default. If you want U or V, you need to ask for them explicitly.

You can ask for only thin U or V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting m be the smaller value among n and p, there are only m singular vectors; the remaining columns of U and V do not correspond to actual singular vectors. Asking for thin U or V means asking for only their m first columns to be formed. So U is then a n-by-m matrix, and V is then a p-by-m matrix. Notice that thin U and V are all you need for (least squares) solving.

Here's an example demonstrating basic usage:

Output:

This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than bidiagonalizing SVD algorithms for large square matrices; however its complexity is still $ O(n^2p) $ where n is the smaller dimension and p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms. In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.

If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to terminate in finite (and reasonable) time.

The possible values for QRPreconditioner are:

ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR. Contrary to other QRs, it doesn't allow computing thin unitaries.
HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR. This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive process is more reliable than the optimized bidiagonal SVD iterations.
NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking if QR preconditioning is needed before applying it anyway.

See also: MatrixBase::jacobiSvd()

Constructor & Destructor Documentation

◆ JacobiSVD() [1/3]

template<typename _MatrixType , int QRPreconditioner>

Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD ( )

inline

Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via JacobiSVD::compute(const MatrixType&).

◆ JacobiSVD() [2/3]

template<typename _MatrixType , int QRPreconditioner>

Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD	(	Index	rows,
		Index	cols,
		unsigned int	computationOptions = `0`
	)

inline

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also: JacobiSVD()

◆ JacobiSVD() [3/3]

template<typename _MatrixType , int QRPreconditioner>

Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD	(	const MatrixType &	matrix,
		unsigned int	computationOptions = `0`
	)

inlineexplicit

Constructor performing the decomposition of given matrix.

Parameters

matrix	the matrix to decompose
computationOptions	optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. By default, none is computed. This is a bit-field, the possible bits are ComputeFullU, ComputeThinU, ComputeFullV, ComputeThinV.

Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not available with the (non-default) FullPivHouseholderQR preconditioner.

Member Function Documentation

◆ compute() [1/2]

template<typename _MatrixType , int QRPreconditioner>

JacobiSVD & Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::compute ( const MatrixType & matrix )

inline

Method performing the decomposition of given matrix using current options.

Parameters

matrix the matrix to decompose

This method uses the current computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).

◆ compute() [2/2]

template<typename MatrixType , int QRPreconditioner>

JacobiSVD< MatrixType, QRPreconditioner > & Eigen::JacobiSVD< MatrixType, QRPreconditioner >::compute	(	const MatrixType &	matrix,
		unsigned int	computationOptions
	)

Method performing the decomposition of given matrix using custom options.

Parameters

matrix	the matrix to decompose
computationOptions	optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. By default, none is computed. This is a bit-field, the possible bits are ComputeFullU, ComputeThinU, ComputeFullV, ComputeThinV.

Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not available with the (non-default) FullPivHouseholderQR preconditioner.

◆ computeU()

template<typename _MatrixType , int QRPreconditioner>

bool Eigen::SVDBase< Derived >::computeU ( ) const

inline

Returns: true if U (full or thin) is asked for in this SVD decomposition

◆ computeV()

template<typename _MatrixType , int QRPreconditioner>

bool Eigen::SVDBase< Derived >::computeV ( ) const

inline

Returns: true if V (full or thin) is asked for in this SVD decomposition

◆ rank()

template<typename _MatrixType , int QRPreconditioner>

Index Eigen::SVDBase< Derived >::rank ( ) const

inline

Returns: the rank of the matrix of which *this is the SVD.

Note: This method has to determine which singular values should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

The documentation for this class was generated from the following files:

External/Eigen/src/Core/util/ForwardDeclarations.h
External/Eigen/src/SVD/JacobiSVD.h

Public Types

Public Member Functions

Protected Attributes

Friends

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ JacobiSVD() [1/3]

◆ JacobiSVD() [2/3]

◆ JacobiSVD() [3/3]

Member Function Documentation

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeU()

◆ computeV()

◆ rank()