scipy.sparse.linalg.gcrotmk¶

scipy.sparse.linalg.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)[source]¶

Solve a matrix equation using flexible GCROT(m,k) algorithm.

Parameters:

Parameters:	A : {sparse matrix, dense matrix, LinearOperator} The real or complex N-by-N matrix of the linear system. b : {array, matrix} Right hand side of the linear system. Has shape (N,) or (N,1). x0 : {array, matrix} Starting guess for the solution. tol, atol : float, optional Tolerances for convergence, `norm(residual) <= max(tolnorm(b), atol)`. The default for `atol` is tol. Warning The default value for atol* will be changed in a future release. For future compatibility, specify atol explicitly. maxiter : int, optional Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : {sparse matrix, dense matrix, 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. callback : function, optional User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector. m : int, optional Number of inner FGMRES iterations per each outer iteration. Default: 20 k : int, optional Number of vectors to carry between inner FGMRES iterations. According to [2], good values are around m. Default: m CU : list 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_C : bool, 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:	x : array or matrix The solution found. info : int Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations

A : {sparse matrix, dense matrix, LinearOperator}: The real or complex N-by-N matrix of the linear system.
b : {array, matrix}: Right hand side of the linear system. Has shape (N,) or (N,1).
x0 : {array, matrix}: Starting guess for the solution.
tol, atol : float, optional: Tolerances for convergence, norm(residual) <= max(tol*norm(b), atol). The default for atol is tol.

Warning

The default value for atol will be changed in a future release. For future compatibility, specify atol explicitly.
maxiter : int, optional: Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.
M : {sparse matrix, dense matrix, 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.
callback : function, optional: User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.
m : int, optional: Number of inner FGMRES iterations per each outer iteration. Default: 20
k : int, optional: Number of vectors to carry between inner FGMRES iterations. According to [2], good values are around m. Default: m
CU : list 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_C : bool, 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:

x : array or matrix

The solution found.

info : int

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, 4) 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]	(1, 2) 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).

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