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: 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).
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.
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 This parameter has been superseded by LinearOperator.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.
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)