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
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_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. |
- Warning
- The input sparse matrix A must be in compressed mode (see SparseMatrix::makeCompressed()).
template<typename MatrixType , typename OrderingType >
void Eigen::SparseQR< MatrixType, OrderingType >::analyzePattern |
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const MatrixType & |
mat | ) |
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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.
Referenced by Eigen::SparseQR< _MatrixType, _OrderingType >::compute().
template<typename _MatrixType , typename _OrderingType >
- 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
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 >
void Eigen::SparseQR< _MatrixType, _OrderingType >::setPivotThreshold |
( |
const RealScalar & |
threshold | ) |
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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.