Linear solve¶
Solves \(AX=B\) for \(X\) given a general square nonsingular matrix \(A\) and right-hand side matrix \(B\). In all cases, on exit, the following routines overwrite \(B\) with \(\text{inv}(A) B\).
Dense algorithm¶
For dense matrices, the solution is computed through Gaussian elimination with partial pivoting.
Sparse-direct algorithm¶
For sparse matrices, the original problem is embedded into the augmented system
where \(\alpha \approx \sigma_{\text{min}}(A)\) is known to be near-optimal with respect to minimizing the condition number of the augmented system. A priori regularization is added to the linear system, a sparse-direct Cholesky-like factorization is performed, and the factorization is used as a preconditioner for Flexible GMRES.
It is worth noting that this is a special case of LeastSquares(),
and that, unlike cases where \(A\) is not invertible, it is possible for
LinearSolve() to choose \(\alpha=0\).
Python API¶
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LinearSolve(A, B[, ctrl=None])¶ - Parameters
A – Dense or sparse matrix to solve against
B – Right-hand sides (which will be overwritten with the solution).
ctrl – (optional) sparse-direct Least Squares control structure
Type of \(A\) |
Type of B |
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C++ API¶
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void
LinearSolve(const AbstractDistMatrix<F> &A, AbstractDistMatrix<F> &B)¶
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void
LinearSolve(const SparseMatrix<F> &A, Matrix<F> &B, const LeastSquaresCtrl<Base<F>> &ctrl = LeastSquaresCtrl<Base<F>>())¶
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void
LinearSolve(const DistSparseMatrix<F> &A, DistMultiVec<F> &B, const LeastSquaresCtrl<Base<F>> &ctrl = LeastSquaresCtrl<Base<F>>())¶
Dense in-place alternatives¶
The following routines factor \(A\) in place.
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lin_solve::Overwrite(AbstractDistMatrix<F> &A, AbstractDistMatrix<F> &B)¶