Eigen::SparseQR< _MatrixType, _OrderingType > Class Template Reference

Detailed Description

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

Sparse left-looking rank-revealing QR factorization.

This class implements a left-looking rank-revealing QR decomposition of sparse matrices. 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 rank-revealing 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.

This class follows the sparse solver concept .

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.

Public Member Functions
void	analyzePattern (const MatrixType &mat)
	Preprocessing step of a QR factorization. More...
Index	cols () const
const PermutationType &	colsPermutation () const
void	compute (const MatrixType &mat)
void	factorize (const MatrixType &mat)
	Performs the numerical QR factorization of the input matrix. More...
ComputationInfo	info () const
	Reports whether previous computation was successful. More...
std::string	lastErrorMessage () const
SparseQRMatrixQReturnType< SparseQR >	matrixQ () const
const QRMatrixType &	matrixR () const
Index	rank () const
Index	rows () const
void	setPivotThreshold (const RealScalar &threshold)
template<typename Rhs >
const Solve< SparseQR, Rhs >	solve (const MatrixBase< Rhs > &B) const
	SparseQR (const MatrixType &mat)

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 ( void  ) 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 ( void  ) 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;

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 ( void  ) 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 \( A X = B \) using the current decomposition of A.

See also

compute()

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

• SparseQR.h