scipy.sparse.linalg.

gmres#

scipy.sparse.linalg.gmres(A, b, x0=None, *, rtol=1e-05, atol=0.0, restart=None, maxiter=None, M=None, callback=None, callback_type=None)[source]#

Solve Ax = b with the Generalized Minimal RESidual method.

Parameters:
A{sparse array, 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.

bndarray

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

x0ndarray

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

rtol, atolfloat

Parameters for the convergence test. For convergence, norm(b - A @ x) <= max(rtol*norm(b), atol) should be satisfied. The default is atol=0. and rtol=1e-5.

restartint, optional

Number of iterations between restarts. Larger values increase iteration cost, but may be necessary for convergence. If omitted, min(20, n) is used.

maxiterint, optional

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

M{sparse array, 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. In this implementation, left preconditioning is used, and the preconditioned residual is minimized. However, the final convergence is tested with respect to the b - A @ x residual.

callbackfunction

User-supplied function to call after each iteration. It is called as callback(args), where args are selected by 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.

This keyword has no effect if callback is not set.

Returns:
xndarray

The converged solution.

infoint
Provides convergence information:

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

See also

LinearOperator

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)

Array API Standard Support

gmres has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library

CPU

GPU

NumPy

✅

n/a

CuPy

n/a

⛔

PyTorch

⛔

⛔

JAX

⛔

⛔

Dask

⛔

n/a

See Support for the array API standard for more information.

Examples

>>> import numpy as np
>>> from scipy.sparse import csc_array
>>> from scipy.sparse.linalg import gmres
>>> A = csc_array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> b = np.array([2, 4, -1], dtype=float)
>>> x, exitCode = gmres(A, b, atol=1e-5)
>>> print(exitCode)            # 0 indicates successful convergence
0
>>> np.allclose(A.dot(x), b)
True
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