Eigen  3.3.3
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().transpose() to get the transpose. 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:
_MatrixTypeThe type of the sparse matrix A, must be a column-major SparseMatrix<>
_OrderingTypeThe 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()).
+ Inheritance diagram for Eigen::SparseQR< _MatrixType, _OrderingType >:

List of all members.

Public Member Functions

void analyzePattern (const MatrixType &mat)
 Preprocessing step of a QR factorization.
Index cols () const
const PermutationTypecolsPermutation () const
void compute (const MatrixType &mat)
void factorize (const MatrixType &mat)
 Performs the numerical QR factorization of the input matrix.
ComputationInfo info () const
 Reports whether previous computation was successful.
std::string lastErrorMessage () const
SparseQRMatrixQReturnType
< SparseQR
matrixQ () const
const QRMatrixTypematrixR () 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
template<typename Rhs >
const Solve< SparseQR, Rhs > solve (const SparseMatrixBase< Rhs > &B) const
 SparseQR (const MatrixType &mat)

Constructor & Destructor Documentation

template<typename _MatrixType , typename _OrderingType >
Eigen::SparseQR< _MatrixType, _OrderingType >::SparseQR ( const MatrixType &  mat) [inline, explicit]

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

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.
template<typename _MatrixType , typename _OrderingType >
Index Eigen::SparseQR< _MatrixType, _OrderingType >::cols ( void  ) const [inline]
Returns:
the number of columns of the represented matrix.
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.
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()
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:
matThe sparse column-major matrix
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()
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.
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.

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
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()
template<typename _MatrixType , typename _OrderingType >
Index Eigen::SparseQR< _MatrixType, _OrderingType >::rows ( void  ) const [inline]
Returns:
the number of rows of the represented matrix.
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.

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()

Reimplemented from Eigen::SparseSolverBase< SparseQR< _MatrixType, _OrderingType > >.

template<typename _MatrixType , typename _OrderingType >
template<typename Rhs >
const Solve<SparseQR, Rhs> Eigen::SparseQR< _MatrixType, _OrderingType >::solve ( const SparseMatrixBase< Rhs > &  b) const [inline]
Returns:
an expression of the solution x of $ A x = b $ using the current decomposition of A.
See also:
compute()

Reimplemented from Eigen::SparseSolverBase< SparseQR< _MatrixType, _OrderingType > >.


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