Sparse left-looking QR factorization with numerical column pivoting. More...

#include <SparseQR.h>

Inheritance diagram for Eigen::SparseQR< _MatrixType, _OrderingType >:

Public Types
enum	{ ColsAtCompileTime = MatrixType::ColsAtCompileTime , MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }

typedef _MatrixType	MatrixType

typedef _OrderingType	OrderingType

typedef MatrixType::Scalar	Scalar

typedef MatrixType::RealScalar	RealScalar

typedef MatrixType::StorageIndex	StorageIndex

typedef SparseMatrix< Scalar, ColMajor, StorageIndex >	QRMatrixType

typedef Matrix< StorageIndex, Dynamic, 1 >	IndexVector

typedef Matrix< Scalar, Dynamic, 1 >	ScalarVector

typedef PermutationMatrix< Dynamic, Dynamic, StorageIndex >	PermutationType

Public Member Functions
	SparseQR (const MatrixType &mat)
	Construct a QR factorization of the matrix mat.

void	compute (const MatrixType &mat)
	Computes the QR factorization of the sparse matrix mat.

void	analyzePattern (const MatrixType &mat)
	Preprocessing step of a QR factorization.

void	factorize (const MatrixType &mat)
	Performs the numerical QR factorization of the input matrix.

Index	rows () const

Index	cols () const

const QRMatrixType &	matrixR () const

Index	rank () const

SparseQRMatrixQReturnType< SparseQR >	matrixQ () const

const PermutationType &	colsPermutation () const

std::string	lastErrorMessage () const

template<typename Rhs , typename Dest >
bool	_solve_impl (const MatrixBase< Rhs > &B, MatrixBase< Dest > &dest) const

void	setPivotThreshold (const RealScalar &threshold)
	Sets the threshold that is used to determine linearly dependent columns during the factorization.

template<typename Rhs >
const Solve< SparseQR, Rhs >	solve (const MatrixBase< Rhs > &B) const

template<typename Rhs >
const Solve< SparseQR, Rhs >	solve (const SparseMatrixBase< Rhs > &B) const

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

void	_sort_matrix_Q ()

Public Member Functions inherited from Eigen::SparseSolverBase< SparseQR< _MatrixType, _OrderingType > >
	SparseSolverBase ()
	Default constructor.

SparseQR< _MatrixType, _OrderingType > &	derived ()

const SparseQR< _MatrixType, _OrderingType > &	derived () const

const Solve< SparseQR< _MatrixType, _OrderingType >, Rhs >	solve (const MatrixBase< Rhs > &b) const

const Solve< SparseQR< _MatrixType, _OrderingType >, Rhs >	solve (const SparseMatrixBase< Rhs > &b) const

void	_solve_impl (const SparseMatrixBase< Rhs > &b, SparseMatrixBase< Dest > &dest) const

Protected Types
typedef SparseSolverBase< SparseQR< _MatrixType, _OrderingType > >	Base

Protected Attributes
bool	m_analysisIsok

bool	m_factorizationIsok

ComputationInfo	m_info

std::string	m_lastError

QRMatrixType	m_pmat

QRMatrixType	m_R

QRMatrixType	m_Q

ScalarVector	m_hcoeffs

PermutationType	m_perm_c

PermutationType	m_pivotperm

PermutationType	m_outputPerm_c

RealScalar	m_threshold

bool	m_useDefaultThreshold

Index	m_nonzeropivots

IndexVector	m_etree

IndexVector	m_firstRowElt

bool	m_isQSorted

bool	m_isEtreeOk

Protected Attributes inherited from Eigen::SparseSolverBase< SparseQR< _MatrixType, _OrderingType > >
bool	m_isInitialized

Friends
template<typename , typename >
struct	SparseQR_QProduct

Detailed Description

template<typename _MatrixType, typename _OrderingType>
class Eigen::SparseQR< _MatrixType, _OrderingType >

Sparse left-looking QR factorization with numerical column pivoting.

This class implements a left-looking QR decomposition of sparse matrices with numerical column pivoting. When a column has a norm less than a given tolerance it is implicitly permuted to the end. The QR factorization thus obtained is given by A*P = Q*R where R is upper triangular or trapezoidal.

P is the column permutation which is the product of the fill-reducing and the numerical permutations. Use colsPermutation() to get it.

Q is the orthogonal matrix represented as products of Householder reflectors. Use matrixQ() to get an expression and matrixQ().adjoint() to get the adjoint. You can then apply it to a vector.

R is the sparse triangular or trapezoidal matrix. The later occurs when A is rank-deficient. matrixR().topLeftCorner(rank(), rank()) always returns a triangular factor of full rank.

Template Parameters

_MatrixType	The type of the sparse matrix A, must be a column-major SparseMatrix<>
_OrderingType	The fill-reducing ordering method. See the OrderingMethods module for the list of built-in and external ordering methods.

\implsparsesolverconcept

The numerical pivoting strategy and default threshold are the same as in SuiteSparse QR, and detailed in the following paper: Tim Davis, "Algorithm 915, SuiteSparseQR: Multifrontal Multithreaded Rank-Revealing Sparse QR Factorization", ACM Trans. on Math. Soft. 38(1), 2011. Even though it is qualified as "rank-revealing", this strategy might fail for some rank deficient problems. When this class is used to solve linear or least-square problems it is thus strongly recommended to check the accuracy of the computed solution. If it failed, it usually helps to increase the threshold with setPivotThreshold.

Warning: The input sparse matrix A must be in compressed mode (see SparseMatrix::makeCompressed()).; For complex matrices matrixQ().transpose() will actually return the adjoint matrix.

Constructor & Destructor Documentation

◆ SparseQR()

template<typename _MatrixType , typename _OrderingType >

Eigen::SparseQR< _MatrixType, _OrderingType >::SparseQR ( const MatrixType & mat )

inlineexplicit

Construct a QR factorization of the matrix mat.

Warning: The matrix mat must be in compressed mode (see SparseMatrix::makeCompressed()).

See also: compute()

Member Function Documentation

◆ analyzePattern()

template<typename MatrixType , typename OrderingType >

void Eigen::SparseQR< MatrixType, OrderingType >::analyzePattern ( const MatrixType & mat )

Preprocessing step of a QR factorization.

Warning: The matrix mat must be in compressed mode (see SparseMatrix::makeCompressed()).

In this step, the fill-reducing permutation is computed and applied to the columns of A and the column elimination tree is computed as well. Only the sparsity pattern of mat is exploited.

Note: In this step it is assumed that there is no empty row in the matrix mat.

◆ cols()

template<typename _MatrixType , typename _OrderingType >

Index Eigen::SparseQR< _MatrixType, _OrderingType >::cols ( ) const

inline

Returns: the number of columns of the represented matrix.

◆ colsPermutation()

template<typename _MatrixType , typename _OrderingType >

const PermutationType & Eigen::SparseQR< _MatrixType, _OrderingType >::colsPermutation ( ) const

inline

Returns: a const reference to the column permutation P that was applied to A such that A*P = Q*R It is the combination of the fill-in reducing permutation and numerical column pivoting.

◆ compute()

template<typename _MatrixType , typename _OrderingType >

void Eigen::SparseQR< _MatrixType, _OrderingType >::compute ( const MatrixType & mat )

inline

Computes the QR factorization of the sparse matrix mat.

Warning: The matrix mat must be in compressed mode (see SparseMatrix::makeCompressed()).

See also: analyzePattern(), factorize()

◆ factorize()

template<typename MatrixType , typename OrderingType >

void Eigen::SparseQR< MatrixType, OrderingType >::factorize ( const MatrixType & mat )

Performs the numerical QR factorization of the input matrix.

The function SparseQR::analyzePattern(const MatrixType&) must have been called beforehand with a matrix having the same sparsity pattern than mat.

Parameters

mat	The sparse column-major matrix

◆ info()

template<typename _MatrixType , typename _OrderingType >

ComputationInfo Eigen::SparseQR< _MatrixType, _OrderingType >::info ( ) const

inline

Reports whether previous computation was successful.

Returns: Success if computation was successful, NumericalIssue if the QR factorization reports a numerical problem InvalidInput if the input matrix is invalid

See also: iparm()

◆ lastErrorMessage()

template<typename _MatrixType , typename _OrderingType >

std::string Eigen::SparseQR< _MatrixType, _OrderingType >::lastErrorMessage ( ) const

inline

Returns: A string describing the type of error. This method is provided to ease debugging, not to handle errors.

◆ matrixQ()

template<typename _MatrixType , typename _OrderingType >

SparseQRMatrixQReturnType< SparseQR > Eigen::SparseQR< _MatrixType, _OrderingType >::matrixQ ( ) const

inline

Returns: an expression of the matrix Q as products of sparse Householder reflectors. The common usage of this function is to apply it to a dense matrix or vector
VectorXd B1, B2;

// Initialize B1

B2 = matrixQ() * B1;

Eigen::MatrixBase
Base class for all dense matrices, vectors, and expressions.
Definition MatrixBase.h:50

Eigen::SparseQR::matrixQ
SparseQRMatrixQReturnType< SparseQR > matrixQ() const
Definition SparseQR.h:186

To get a plain SparseMatrix representation of Q:

SparseMatrix<double> Q;

Q = SparseQR<SparseMatrix<double> >(A).matrixQ();

Internally, this call simply performs a sparse product between the matrix Q and a sparse identity matrix. However, due to the fact that the sparse reflectors are stored unsorted, two transpositions are needed to sort them before performing the product.

◆ matrixR()

template<typename _MatrixType , typename _OrderingType >

const QRMatrixType & Eigen::SparseQR< _MatrixType, _OrderingType >::matrixR ( ) const

inline

Returns: a const reference to the sparse upper triangular matrix R of the QR factorization.

Warning: The entries of the returned matrix are not sorted. This means that using it in algorithms expecting sorted entries will fail. This include random coefficient accesses (SpaseMatrix::coeff()), and coefficient-wise operations. Matrix products and triangular solves are fine though.

To sort the entries, you can assign it to a row-major matrix, and if a column-major matrix is required, you can copy it again:

SparseMatrix<double>          R  = qr.matrixR();  // column-major, not sorted!
SparseMatrix<double,RowMajor> Rr = qr.matrixR();  // row-major, sorted
SparseMatrix<double>          Rc = Rr;            // column-major, sorted

◆ rank()

template<typename _MatrixType , typename _OrderingType >

Index Eigen::SparseQR< _MatrixType, _OrderingType >::rank ( ) const

inline

Returns: the number of non linearly dependent columns as determined by the pivoting threshold.

See also: setPivotThreshold()

◆ rows()

template<typename _MatrixType , typename _OrderingType >

Index Eigen::SparseQR< _MatrixType, _OrderingType >::rows ( ) const

inline

Returns: the number of rows of the represented matrix.

◆ setPivotThreshold()

template<typename _MatrixType , typename _OrderingType >

void Eigen::SparseQR< _MatrixType, _OrderingType >::setPivotThreshold ( const RealScalar & threshold )

inline

Sets the threshold that is used to determine linearly dependent columns during the factorization.

In practice, if during the factorization the norm of the column that has to be eliminated is below this threshold, then the entire column is treated as zero, and it is moved at the end.

◆ solve()

template<typename _MatrixType , typename _OrderingType >

template<typename Rhs >

const Solve< SparseQR, Rhs > Eigen::SparseQR< _MatrixType, _OrderingType >::solve ( const MatrixBase< Rhs > & B ) const

inline

Returns: the solution X of using the current decomposition of A.

See also: compute()

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

External/Eigen/src/SparseQR/SparseQR.h

Public Types

Public Member Functions

Protected Types

Protected Attributes

Friends

Detailed Description

Constructor & Destructor Documentation

◆ SparseQR()

Member Function Documentation

◆ analyzePattern()

◆ cols()

◆ colsPermutation()

◆ compute()

◆ factorize()

◆ info()

◆ lastErrorMessage()

◆ matrixQ()

◆ matrixR()

◆ rank()

◆ rows()

◆ setPivotThreshold()

◆ solve()