scipy.sparse.linalg.

gcrotmk#

scipy.sparse.linalg.gcrotmk(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=1000, M=None, callback=None, m=20, k=None, CU=None, discard_C=False, truncate='oldest')[source]#

Solve Ax = b with the flexible GCROT(m,k) algorithm.

Parameters:

A{sparse array, ndarray, LinearOperator}: 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., 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 is rtol=1e-5 and atol=0.0.
maxiterint, optional: Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. The default is 1000.
M{sparse array, 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. 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 [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’

Returns:

xndarray

The solution found.

infoint

Provides convergence information:

0 : successful exit
>0 : convergence to tolerance not achieved, number of iterations

Notes

Array API Standard Support

gcrotmk has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	⛔	⛔
JAX	⛔	⛔
Dask	⛔	n/a

See Support for the array API standard for more information.

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_array
>>> from scipy.sparse.linalg import gcrotmk
>>> R = np.random.randn(5, 5)
>>> A = csc_array(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