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.

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.


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’

x : array or matrix

The solution found.

info : int

Provides convergence information:

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


[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).