gcrotmk(A, b, x0=None, tol=1e-05, maxiter=1000, M=None, callback=None, m=20, k=None, CU=None, discard_C=False, truncate='oldest', atol=None)
The real or complex N-by-N matrix of the linear system. Alternatively, A
can be a linear operator which can produce Ax
using, e.g., scipy.sparse.linalg.LinearOperator
.
Right hand side of the linear system. Has shape (N,) or (N,1).
Starting guess for the solution.
Tolerances for convergence, norm(residual) <= max(tol*norm(b), atol)
. The default for atol
is :None:None:`tol`.
The default value for :None:None:`atol` will be changed in a future release. For future compatibility, specify :None:None:`atol` explicitly.
Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.
Preconditioner for A. The preconditioner should approximate the inverse of A. gcrotmk is a 'flexible' algorithm and the preconditioner can vary from iteration to iteration. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance.
User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.
Number of inner FGMRES iterations per each outer iteration. Default: 20
Number of vectors to carry between inner FGMRES iterations. According to , good values are around m. Default: m
List of tuples (c, u)
which contain the columns of the matrices C and U in the GCROT(m,k) algorithm. For details, see . The list given and vectors contained in it are modified in-place. If not given, start from empty matrices. The c
elements in the tuples can be None
, in which case the vectors are recomputed via c = A u
on start and orthogonalized as described in .
Discard the C-vectors at the end. Useful if recycling Krylov subspaces for different linear systems.
Truncation scheme to use. Drop: oldest vectors, or vectors with smallest singular values using the scheme discussed in [1,2]. See for detailed comparison. Default: 'oldest'
Solve a matrix equation using flexible GCROT(m,k) algorithm.
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