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

#include <ForwardDeclarations.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 internal::plain_col_type< MatrixType >::type	ColType

typedef _MatrixType	MatrixType

typedef Base::MatrixUType	MatrixUType

typedef Base::MatrixVType	MatrixVType

typedef NumTraits< typename MatrixType::Scalar >::Real	RealScalar

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

typedef MatrixType::Scalar	Scalar

typedef Base::SingularValuesType	SingularValuesType

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

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

typedef Eigen::Index	Index

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

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

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

typedef NumTraits< typename MatrixType::Scalar >::Real	RealScalar

typedef MatrixType::Scalar	Scalar

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

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

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

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

typedef EigenBase< SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > > >	Base

typedef Scalar	CoeffReturnType

typedef internal::add_const< Transpose< const SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > > > >::type	ConstTransposeReturnType

typedef internal::traits< SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > > >::Scalar	Scalar

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

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

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

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

	JacobiSVD ()
	Default Constructor. More...

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

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

Public Member Functions inherited from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >
void	_solve_impl (const RhsType &rhs, DstType &dst) const

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

Index	cols () const

bool	computeU () const

bool	computeV () const

JacobiSVD< _MatrixType, QRPreconditioner > &	derived ()

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

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

const MatrixUType &	matrixU () const

const MatrixVType &	matrixV () const

Index	nonzeroSingularValues () const

Index	rank () const

Index	rows () const

JacobiSVD< _MatrixType, QRPreconditioner > &	setThreshold (const RealScalar &threshold)

JacobiSVD< _MatrixType, QRPreconditioner > &	setThreshold (Default_t)

const SingularValuesType &	singularValues () const

RealScalar	threshold () const

Public Member Functions inherited from Eigen::SolverBase< SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > > >
AdjointReturnType	adjoint () const

const Solve< SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >, Rhs >	solve (const MatrixBase< Rhs > &b) const

	SolverBase ()

ConstTransposeReturnType	transpose () const

	~SolverBase ()

Public Member Functions inherited from Eigen::EigenBase< Derived >
template<typename Dest >
EIGEN_DEVICE_FUNC void	addTo (Dest &dst) const

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

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

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC Derived &	const_cast_derived () const

EIGEN_DEVICE_FUNC const Derived &	const_derived () const

EIGEN_DEVICE_FUNC Derived &	derived ()

EIGEN_DEVICE_FUNC const Derived &	derived () const

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

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

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

Protected Attributes
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

WorkMatrixType	m_workMatrix

Protected Attributes inherited from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >
Index	m_cols

unsigned int	m_computationOptions

bool	m_computeFullU

bool	m_computeFullV

bool	m_computeThinU

bool	m_computeThinV

Index	m_diagSize

ComputationInfo	m_info

bool	m_isAllocated

bool	m_isInitialized

MatrixUType	m_matrixU

MatrixVType	m_matrixV

Index	m_nonzeroSingularValues

RealScalar	m_prescribedThreshold

Index	m_rows

SingularValuesType	m_singularValues

bool	m_usePrescribedThreshold

Private Types
typedef SVDBase< JacobiSVD >	Base

Private Member Functions
void	allocate (Index rows, Index cols, unsigned int computationOptions)

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

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

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. More...

Protected Member Functions inherited from Eigen::SolverBase< SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > > >
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:

MatrixXf m = MatrixXf::Random(3,2);
cout << "Here is the matrix m:" << endl << m << endl;
JacobiSVD<MatrixXf> svd(m, ComputeThinU | ComputeThinV);
cout << "Its singular values are:" << endl << svd.singularValues() << endl;
cout << "Its left singular vectors are the columns of the thin U matrix:" << endl << svd.matrixU() << endl;
cout << "Its right singular vectors are the columns of the thin V matrix:" << endl << svd.matrixV() << endl;
Vector3f rhs(1, 0, 0);
cout << "Now consider this rhs vector:" << endl << rhs << endl;
cout << "A least-squares solution of m*x = rhs is:" << endl << svd.solve(rhs) << endl;

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()

Definition at line 278 of file ForwardDeclarations.h.

Member Typedef Documentation

◆ Base

template<typename _MatrixType, int QRPreconditioner>

typedef SVDBase<JacobiSVD> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::Base

private

Definition at line 491 of file JacobiSVD.h.

◆ ColType

template<typename _MatrixType, int QRPreconditioner>

typedef internal::plain_col_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::ColType

Definition at line 512 of file JacobiSVD.h.

◆ MatrixType

template<typename _MatrixType, int QRPreconditioner>

typedef _MatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixType

Definition at line 494 of file JacobiSVD.h.

◆ MatrixUType

template<typename _MatrixType, int QRPreconditioner>

typedef Base::MatrixUType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixUType

Definition at line 507 of file JacobiSVD.h.

◆ MatrixVType

template<typename _MatrixType, int QRPreconditioner>

typedef Base::MatrixVType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixVType

Definition at line 508 of file JacobiSVD.h.

◆ RealScalar

template<typename _MatrixType, int QRPreconditioner>

typedef NumTraits<typename MatrixType::Scalar>::Real Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RealScalar

Definition at line 496 of file JacobiSVD.h.

◆ RowType

template<typename _MatrixType, int QRPreconditioner>

typedef internal::plain_row_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RowType

Definition at line 511 of file JacobiSVD.h.

◆ Scalar

template<typename _MatrixType, int QRPreconditioner>

typedef MatrixType::Scalar Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::Scalar

Definition at line 495 of file JacobiSVD.h.

◆ SingularValuesType

template<typename _MatrixType, int QRPreconditioner>

typedef Base::SingularValuesType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::SingularValuesType

Definition at line 509 of file JacobiSVD.h.

◆ WorkMatrixType

template<typename _MatrixType, int QRPreconditioner>

typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::WorkMatrixType

Definition at line 515 of file JacobiSVD.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType, int QRPreconditioner>

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
DiagSizeAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime
MaxDiagSizeAtCompileTime
MatrixOptions

Definition at line 497 of file JacobiSVD.h.

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&).

Definition at line 522 of file JacobiSVD.h.

◆ 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()

Definition at line 532 of file JacobiSVD.h.

◆ 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.

Definition at line 547 of file JacobiSVD.h.

Member Function Documentation

◆ allocate()

template<typename MatrixType , int QRPreconditioner>

void Eigen::JacobiSVD< MatrixType, QRPreconditioner >::allocate	(	Eigen::Index	rows,
		Eigen::Index	cols,
		unsigned int	computationOptions
	)

private

Definition at line 615 of file JacobiSVD.h.

◆ compute() [1/2]

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.

Definition at line 666 of file JacobiSVD.h.

◆ compute() [2/2]

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).

Definition at line 570 of file JacobiSVD.h.

Friends And Related Function Documentation

◆ internal::qr_preconditioner_impl

template<typename _MatrixType, int QRPreconditioner>

template<typename __MatrixType , int _QRPreconditioner, int _Case, bool _DoAnything>

friend struct internal::qr_preconditioner_impl

friend

Definition at line 607 of file JacobiSVD.h.

◆ internal::svd_precondition_2x2_block_to_be_real

template<typename _MatrixType, int QRPreconditioner>

template<typename __MatrixType , int _QRPreconditioner, bool _IsComplex>

friend struct internal::svd_precondition_2x2_block_to_be_real

friend

Definition at line 605 of file JacobiSVD.h.

Member Data Documentation

◆ m_qr_precond_morecols

template<typename _MatrixType, int QRPreconditioner>

internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_qr_precond_morecols

protected

Definition at line 609 of file JacobiSVD.h.

◆ m_qr_precond_morerows

template<typename _MatrixType, int QRPreconditioner>

internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_qr_precond_morerows

protected

Definition at line 610 of file JacobiSVD.h.

◆ m_scaledMatrix

template<typename _MatrixType, int QRPreconditioner>

MatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_scaledMatrix

protected

Definition at line 611 of file JacobiSVD.h.

◆ m_workMatrix

template<typename _MatrixType, int QRPreconditioner>

WorkMatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_workMatrix

protected

Definition at line 602 of file JacobiSVD.h.

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

Public Types

Public Member Functions

Protected Attributes

Private Types

Private Member Functions

Friends

Additional Inherited Members

Detailed Description

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

Member Typedef Documentation

◆ Base

◆ ColType

◆ MatrixType

◆ MatrixUType

◆ MatrixVType

◆ RealScalar

◆ RowType

◆ Scalar

◆ SingularValuesType

◆ WorkMatrixType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ JacobiSVD() [1/3]

◆ JacobiSVD() [2/3]

◆ JacobiSVD() [3/3]

Member Function Documentation

◆ allocate()

◆ compute() [1/2]

◆ compute() [2/2]

Friends And Related Function Documentation

◆ internal::qr_preconditioner_impl

◆ internal::svd_precondition_2x2_block_to_be_real

Member Data Documentation

◆ m_qr_precond_morecols

◆ m_qr_precond_morerows

◆ m_scaledMatrix

◆ m_workMatrix

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