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. |
This class follows the sparse solver concept .
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.
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
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 >
- 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 >
| 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.
Public Member Functions inherited from Eigen::SparseSolverBase< SparseQR< _MatrixType, _OrderingType > >