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

Use Generalized Minimal RESidual iteration to solve Ax = b.

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.


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


The converged solution.

Provides convergence information:
  • 0 : successful exit

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

  • <0 : illegal input or breakdown

Other Parameters:

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

tol, atolfloat, optional

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


The default value for atol will be changed in a future release. For future compatibility, specify atol explicitly.

restartint, optional

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

maxiterint, 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. In this implementation, left preconditioning is used, and the preconditioned residual is minimized.


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.

restrtint, optional, deprecated

Deprecated since version 0.11.0: gmres keyword argument restrt is deprecated infavour of restart and will be removed in SciPy 1.12.0.

See also



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)


>>> import numpy as np
>>> 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
>>> np.allclose(, b)