# 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')[source]

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

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 : float, optional Tolerance to achieve. The algorithm terminates when either the relative or the absolute residual is below tol. 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 [R26], 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 [R26]. 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 [R27]. 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 [R26] for detailed comparison. Default: ‘oldest’ 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

 [R25] E. de Sturler, ‘’Truncation strategies for optimal Krylov subspace methods’‘, SIAM J. Numer. Anal. 36, 864 (1999).
 [R26] (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).
 [R27] (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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