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

#include <ForwardDeclarations.h>

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 Member Functions
Index	cols () const

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

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

bool	computeU () const

bool	computeV () const

	JacobiSVD ()
	Default Constructor. More...

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

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

Index	rank () const

Index	rows () const

Protected Attributes
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

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

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

Index	m_rows

MatrixType	m_scaledMatrix

SingularValuesType	m_singularValues

bool	m_usePrescribedThreshold

WorkMatrixType	m_workMatrix

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

Detailed Description

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

typedef SVDBase<JacobiSVD> Eigen::JacobiSVD::Base

private

Definition at line 491 of file JacobiSVD.h.

◆ ColType

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

Definition at line 512 of file JacobiSVD.h.

◆ MatrixType

typedef _MatrixType Eigen::JacobiSVD::MatrixType

Definition at line 494 of file JacobiSVD.h.

◆ MatrixUType

typedef Base::MatrixUType Eigen::JacobiSVD::MatrixUType

Definition at line 507 of file JacobiSVD.h.

◆ MatrixVType

typedef Base::MatrixVType Eigen::JacobiSVD::MatrixVType

Definition at line 508 of file JacobiSVD.h.

◆ RealScalar

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

Definition at line 496 of file JacobiSVD.h.

◆ RowType

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

Definition at line 511 of file JacobiSVD.h.

◆ Scalar

typedef MatrixType::Scalar Eigen::JacobiSVD::Scalar

Definition at line 495 of file JacobiSVD.h.

◆ SingularValuesType

typedef Base::SingularValuesType Eigen::JacobiSVD::SingularValuesType

Definition at line 509 of file JacobiSVD.h.

◆ WorkMatrixType

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

Definition at line 515 of file JacobiSVD.h.

Member Enumeration Documentation

◆ anonymous enum

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
DiagSizeAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime
MaxDiagSizeAtCompileTime
MatrixOptions

Definition at line 497 of file JacobiSVD.h.

Constructor & Destructor Documentation

◆ JacobiSVD() [1/3]

Eigen::JacobiSVD::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]

Eigen::JacobiSVD::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]

Eigen::JacobiSVD::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()

void Eigen::JacobiSVD::allocate	(	Eigen::Index	rows,
		Eigen::Index	cols,
		unsigned int	computationOptions
	)

private

Definition at line 615 of file JacobiSVD.h.

◆ cols()

Index Eigen::SVDBase::cols

inline

Definition at line 213 of file SVDBase.h.

◆ compute() [1/2]

JacobiSVD& Eigen::JacobiSVD::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.

◆ compute() [2/2]

JacobiSVD< MatrixType, QRPreconditioner > & Eigen::JacobiSVD::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.

◆ computeU()

bool Eigen::SVDBase::computeU

inline

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

Definition at line 208 of file SVDBase.h.

◆ computeV()

bool Eigen::SVDBase::computeV

inline

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

Definition at line 210 of file SVDBase.h.

◆ rank()

Index Eigen::SVDBase::rank

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&).

Definition at line 148 of file SVDBase.h.

◆ rows()

Index Eigen::SVDBase::rows

inline

Definition at line 212 of file SVDBase.h.

Friends And Related Function Documentation

◆ internal::qr_preconditioner_impl

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, 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_cols

Index Eigen::SVDBase::m_cols

protected

Definition at line 280 of file SVDBase.h.

◆ m_computationOptions

unsigned int Eigen::SVDBase::m_computationOptions

protected

Definition at line 279 of file SVDBase.h.

◆ m_computeFullU

bool Eigen::SVDBase::m_computeFullU

protected

Definition at line 277 of file SVDBase.h.

◆ m_computeFullV

bool Eigen::SVDBase::m_computeFullV

protected

Definition at line 278 of file SVDBase.h.

◆ m_computeThinU

bool Eigen::SVDBase::m_computeThinU

protected

Definition at line 277 of file SVDBase.h.

◆ m_computeThinV

bool Eigen::SVDBase::m_computeThinV

protected

Definition at line 278 of file SVDBase.h.

◆ m_diagSize

Index Eigen::SVDBase::m_diagSize

protected

Definition at line 280 of file SVDBase.h.

◆ m_info

ComputationInfo Eigen::SVDBase::m_info

protected

Definition at line 275 of file SVDBase.h.

◆ m_isAllocated

bool Eigen::SVDBase::m_isAllocated

protected

Definition at line 276 of file SVDBase.h.

◆ m_isInitialized

bool Eigen::SVDBase::m_isInitialized

protected

Definition at line 276 of file SVDBase.h.

◆ m_matrixU

MatrixUType Eigen::SVDBase::m_matrixU

protected

Definition at line 272 of file SVDBase.h.

◆ m_matrixV

MatrixVType Eigen::SVDBase::m_matrixV

protected

Definition at line 273 of file SVDBase.h.

◆ m_nonzeroSingularValues

Index Eigen::SVDBase::m_nonzeroSingularValues

protected

Definition at line 280 of file SVDBase.h.

◆ m_prescribedThreshold

RealScalar Eigen::SVDBase::m_prescribedThreshold

protected

Definition at line 281 of file SVDBase.h.

◆ m_qr_precond_morecols

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

protected

Definition at line 609 of file JacobiSVD.h.

◆ m_qr_precond_morerows

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

protected

Definition at line 610 of file JacobiSVD.h.

◆ m_rows

Index Eigen::SVDBase::m_rows

protected

Definition at line 280 of file SVDBase.h.

◆ m_scaledMatrix

MatrixType Eigen::JacobiSVD::m_scaledMatrix

protected

Definition at line 611 of file JacobiSVD.h.

◆ m_singularValues

SingularValuesType Eigen::SVDBase::m_singularValues

protected

Definition at line 274 of file SVDBase.h.

◆ m_usePrescribedThreshold

bool Eigen::SVDBase::m_usePrescribedThreshold

protected

Definition at line 276 of file SVDBase.h.

◆ m_workMatrix

WorkMatrixType Eigen::JacobiSVD::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

Detailed Description

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()

◆ cols()

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeU()

◆ computeV()

◆ rank()

◆ rows()

Friends And Related Function Documentation

◆ internal::qr_preconditioner_impl

◆ internal::svd_precondition_2x2_block_to_be_real

Member Data Documentation

◆ m_cols

◆ m_computationOptions

◆ m_computeFullU

◆ m_computeFullV

◆ m_computeThinU

◆ m_computeThinV

◆ m_diagSize

◆ m_info

◆ m_isAllocated

◆ m_isInitialized

◆ m_matrixU

◆ m_matrixV

◆ m_nonzeroSingularValues

◆ m_prescribedThreshold

◆ m_qr_precond_morecols

◆ m_qr_precond_morerows

◆ m_rows

◆ m_scaledMatrix

◆ m_singularValues

◆ m_usePrescribedThreshold

◆ m_workMatrix