scipy.sparse.linalg.gcrotmk#
- scipy.sparse.linalg.gcrotmk(A, b, x0=None, *, tol=<object object>, maxiter=1000, M=None, callback=None, m=20, k=None, CU=None, discard_C=False, truncate='oldest', atol=None, rtol=1e-05)[source]#
Solve a matrix equation using flexible GCROT(m,k) algorithm.
- Parameters:
- A{sparse matrix, ndarray, LinearOperator}
The real or complex N-by-N matrix of the linear system. Alternatively,
A
can be a linear operator which can produceAx
using, e.g.,scipy.sparse.linalg.LinearOperator
.- bndarray
Right hand side of the linear system. Has shape (N,) or (N,1).
- x0ndarray
Starting guess for the solution.
- rtol, atolfloat, optional
Parameters for the convergence test. For convergence,
norm(b - A @ x) <= max(rtol*norm(b), atol)
should be satisfied. The default isrtol=1e-5
, the default foratol
isrtol
.Warning
The default value for
atol
will be changed to0.0
in SciPy 1.14.0.- maxiterint, optional
Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.
- M{sparse matrix, ndarray, LinearOperator}, optional
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.
- callbackfunction, optional
User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.
- mint, optional
Number of inner FGMRES iterations per each outer iteration. Default: 20
- kint, optional
Number of vectors to carry between inner FGMRES iterations. According to [2], good values are around m. Default: m
- CUlist of tuples, optional
List of tuples
(c, u)
which contain the columns of the matrices C and U in the GCROT(m,k) algorithm. For details, see [2]. The list given and vectors contained in it are modified in-place. If not given, start from empty matrices. Thec
elements in the tuples can beNone
, in which case the vectors are recomputed viac = A u
on start and orthogonalized as described in [3].- discard_Cbool, optional
Discard the C-vectors at the end. Useful if recycling Krylov subspaces for different linear systems.
- truncate{‘oldest’, ‘smallest’}, optional
Truncation scheme to use. Drop: oldest vectors, or vectors with smallest singular values using the scheme discussed in [1,2]. See [2] for detailed comparison. Default: ‘oldest’
- tolfloat, optional, deprecated
Deprecated since version 1.12.0:
gcrotmk
keyword argumenttol
is deprecated in favor ofrtol
and will be removed in SciPy 1.14.0.
- Returns:
- xndarray
The solution found.
- infoint
Provides convergence information:
0 : successful exit
>0 : convergence to tolerance not achieved, number of iterations
References
[1]E. de Sturler, ‘’Truncation strategies for optimal Krylov subspace methods’’, SIAM J. Numer. Anal. 36, 864 (1999).
[2] (1,2,3)J.E. Hicken and D.W. Zingg, ‘’A simplified and flexible variant of GCROT for solving nonsymmetric linear systems’’, SIAM J. Sci. Comput. 32, 172 (2010).
[3]M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti, ‘’Recycling Krylov subspaces for sequences of linear systems’’, SIAM J. Sci. Comput. 28, 1651 (2006).
Examples
>>> import numpy as np >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import gcrotmk >>> R = np.random.randn(5, 5) >>> A = csc_matrix(R) >>> b = np.random.randn(5) >>> x, exit_code = gcrotmk(A, b, atol=1e-5) >>> print(exit_code) 0 >>> np.allclose(A.dot(x), b) True