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

Use Generalized Minimal RESidual iteration to solve Ax = b.

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).

x : {array, matrix}

The converged solution.

info : int
Provides convergence information:
  • 0 : successful exit
  • >0 : convergence to tolerance not achieved, number of iterations
  • <0 : illegal input or breakdown
Other Parameters:
x0 : {array, matrix}

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.


The default value for atol will be changed in a future release. For future compatibility, specify 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, dense matrix, 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 callback(rk), where rk is the current residual vector.

restrt : int, optional

DEPRECATED - use restart instead.

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)


>>> 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)