Eigen
3.4.90 (git rev 67eeba6e720c5745abc77ae6c92ce0a44aa7b7ae)
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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.
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.
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) | |
Public Member Functions inherited from Eigen::SparseSolverBase< SparseQR< MatrixType_, OrderingType_ > > | |
const Solve< SparseQR< MatrixType_, OrderingType_ >, Rhs > | solve (const MatrixBase< Rhs > &b) const |
const Solve< SparseQR< MatrixType_, OrderingType_ >, Rhs > | solve (const SparseMatrixBase< Rhs > &b) const |
SparseSolverBase () | |
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Construct a QR factorization of the matrix mat.
void Eigen::SparseQR< MatrixType, OrderingType >::analyzePattern | ( | const MatrixType & | mat | ) |
Preprocessing step of a QR factorization.
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.
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Computes the QR factorization of the sparse matrix mat.
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.
mat | The sparse column-major matrix |
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Reports whether previous computation was successful.
Success
if computation was successful, NumericalIssue
if the QR factorization reports a numerical problem InvalidInput
if the input matrix is invalid
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To get a plain SparseMatrix representation of Q:
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.
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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:
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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.
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