# scipy.sparse.linalg.gmres¶

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

Use Generalized Minimal RESidual iteration to solve Ax = b.

Parameters: Returns: 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 x0 : {array, matrix} Starting guess for the solution (a vector of zeros by default). tol : float Tolerance to achieve. The algorithm terminates when either the relative or the absolute residual is below tol. 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. xtype : {‘f’,’d’,’F’,’D’} This parameter is DEPRECATED — avoid using it. The type of the result. If None, then it will be determined from A.dtype.char and b. If A does not have a typecode method then it will compute A.matvec(x0) to get a typecode. To save the extra computation when A does not have a typecode attribute use xtype=0 for the same type as b or use xtype=’f’,’d’,’F’,or ‘D’. This parameter has been superseded by LinearOperator. 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.

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)


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


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