"""Generalized Minimum Residual Method (GMRES) Krylov solver."""
import warnings
from ._gmres_mgs import gmres_mgs
from ._gmres_householder import gmres_householder
[docs]
def gmres(A, b, x0=None, tol=1e-5, restart=None, maxiter=None,
M=None, callback=None, residuals=None, orthog='householder', restrt=None,
**kwargs):
"""Generalized Minimum Residual Method (GMRES).
GMRES iteratively refines the initial solution guess to the
system Ax = b. Left preconditioned. Residuals are preconditioned residuals.
Parameters
----------
A : array, matrix, sparse matrix, LinearOperator
n x n, linear system to solve
b : array, matrix
right hand side, shape is (n,) or (n,1)
x0 : array, matrix
initial guess, default is a vector of zeros
tol : float
Tolerance for stopping criteria, let r=r_k
||M r|| < tol ||M b||
if ||b||=0, then set ||M b||=1 for these tests.
restart : None, int
- if int, restart is max number of inner iterations
and maxiter is the max number of outer iterations
- if None, do not restart GMRES, and max number of inner iterations
is maxiter
maxiter : None, int
- if restart is None, maxiter is the max number of inner iterations
and GMRES does not restart
- if restart is int, maxiter is the max number of outer iterations,
and restart is the max number of inner iterations
- defaults to min(n,40) if restart=None
M : array, matrix, sparse matrix, LinearOperator
n x n, inverted preconditioner, i.e. solve M A x = M b.
callback : function
User-supplied function is called after each iteration as
callback(xk), where xk is the current solution vector
residuals : list
preconditioned residual history in the 2-norm, including the initial residual
orthog : string
'householder' calls _gmres_householder which uses Householder
reflections to find the orthogonal basis for the Krylov space.
'mgs' calls _gmres_mgs which uses modified Gram-Schmidt to find the
orthogonal basis for the Krylov space
restrt : None, int
Deprecated. See restart.
Returns
-------
(xk, info)
xk : an updated guess after k iterations to the solution of Ax = b
info : halting status
== =======================================
0 successful exit
>0 convergence to tolerance not achieved,
return iteration count instead.
<0 numerical breakdown, or illegal input
== =======================================
Notes
-----
The LinearOperator class is in scipy.sparse.linalg.
Use this class if you prefer to define A or M as a mat-vec routine
as opposed to explicitly constructing the matrix.
The orthogonalization method, orthog='householder', is more robust
than orthog='mgs', however for the majority of problems your
problem will converge before 'mgs' loses orthogonality in your basis.
orthog='householder' has been more rigorously tested, and is
therefore currently the default
The residual is the *preconditioned* residual.
Examples
--------
>>> from pyamg.krylov import gmres
>>> from pyamg.util.linalg import norm
>>> import numpy as np
>>> from pyamg.gallery import poisson
>>> A = poisson((10,10))
>>> b = np.ones((A.shape[0],))
>>> (x,flag) = gmres(A,b, maxiter=2, tol=1e-8)
>>> print(f'{norm(b - A*x):.6}')
6.54282
References
----------
.. [1] Yousef Saad, "Iterative Methods for Sparse Linear Systems,
Second Edition", SIAM, pp. 151-172, pp. 272-275, 2003
http://www-users.cs.umn.edu/~saad/books.html
"""
if restrt is not None:
if restart is not None:
raise ValueError('Only use restart, not restrt (deprecated).')
restart = restrt
msg = ('The keyword argument "restrt" is deprecated and will '
'be removed in 2024. Use "restart" instead.')
warnings.warn(msg, DeprecationWarning, stacklevel=1)
# pass along **kwargs
if orthog == 'householder':
(x, flag) = gmres_householder(A, b, x0=x0, tol=tol, restart=restart,
maxiter=maxiter, M=M,
callback=callback, residuals=residuals,
**kwargs)
elif orthog == 'mgs':
(x, flag) = gmres_mgs(A, b, x0=x0, tol=tol, restart=restart,
maxiter=maxiter, M=M,
callback=callback, residuals=residuals, **kwargs)
return (x, flag)
