scipy.sparse.linalg.gmres¶

scipy.sparse.linalg.
gmres
(A, b, x0=None, tol=1e05, restart=None, maxiter=None, M=None, callback=None, restrt=None)[source]¶ Use Generalized Minimal RESidual iteration to solve
Ax = b
.Parameters: A : {sparse matrix, dense matrix, LinearOperator}
The real or complex NbyN matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
Returns: 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 : 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.
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
Usersupplied 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
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