gmres(A, b, x0=None, tol=1e-05, restart=None, maxiter=None, M=None, callback=None, restrt=None, atol=None, callback_type=None)

Notes

A preconditioner, P, is chosen such that P is close to A but easy to solve for. The preconditioner parameter required by this routine is M = P^-1 . The inverse should preferably not be calculated explicitly. Rather, use the following template to produce M:

# Construct a linear operator that computes P^-1 @ x.
import scipy.sparse.linalg as spla
M_x = lambda x: spla.spsolve(P, x)
M = spla.LinearOperator((n, n), M_x)

Other Parameters

x0 : ndarray

Starting guess for the solution (a vector of zeros by default).

tol, atol : float, optional

Tolerances for convergence, norm(residual) <= max(tol*norm(b), atol) . The default for atol is 'legacy' , which emulates a different legacy behavior.

warning

The default value for :None:None:`atol` will be changed in a future release. For future compatibility, specify :None:None:`atol` explicitly.

restart : int, optional

Number of iterations between restarts. Larger values increase iteration cost, but may be necessary for convergence. Default is 20.

maxiter : int, optional

Maximum number of iterations (restart cycles). Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.

M : {sparse matrix, ndarray, LinearOperator}

Inverse of the preconditioner of A. M should approximate the inverse of A and be easy to solve for (see Notes). Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. By default, no preconditioner is used.

callback : function

User-supplied function to call after each iteration. It is called as :None:None:`callback(args)`, where :None:None:`args` are selected by :None:None:`callback_type`.

callback_type : {'x', 'pr_norm', 'legacy'}, optional

Callback function argument requested:

• x : current iterate (ndarray), called on every restart

• pr_norm : relative (preconditioned) residual norm (float), called on every inner iteration

• legacy (default): same as pr_norm , but also changes the meaning of 'maxiter' to count inner iterations instead of restart cycles.

restrt : int, optional

DEPRECATED - use restart instead.

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., scipy.sparse.linalg.LinearOperator .

b : ndarray

Right hand side of the linear system. Has shape (N,) or (N,1).

Returns

x : ndarray

The converged solution.

info : int

Use Generalized Minimal RESidual iteration to solve Ax = b .

See Also

LinearOperator

Examples

>>> from scipy.sparse import csc_matrix
... from scipy.sparse.linalg import gmres
... A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
... b = np.array([2, 4, -1], dtype=float)
... x, exitCode = gmres(A, b)
... print(exitCode)            # 0 indicates successful convergence
0

>>> np.allclose(A.dot(x), b)
True

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.optimize._nonlin.newton_krylov scipy.optimize._root._root_krylov_doc scipy.optimize._nonlin.KrylovJacobian

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