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 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 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 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. 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 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

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
```