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# Copyright (C) 2009, Pauli Virtanen <>
# Distributed under the same license as Scipy.
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.linalg import LinAlgError
from scipy._lib.six import xrange
from scipy.linalg import get_blas_funcs, get_lapack_funcs
from .utils import make_system
from ._gcrotmk import _fgmres
__all__ = ['lgmres']
def lgmres(A, b, x0=None, tol=1e-5, maxiter=1000, M=None, callback=None,
inner_m=30, outer_k=3, outer_v=None, store_outer_Av=True,
Solve a matrix equation using the LGMRES algorithm.
The LGMRES algorithm [1]_ [2]_ is designed to avoid some problems
in the convergence in restarted GMRES, and often converges in fewer
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).
x0 : {array, matrix}
Starting guess for the solution.
tol : float, optional
Tolerance to achieve. The algorithm terminates when either the relative
or the absolute residual is below `tol`.
maxiter : int, optional
Maximum number of iterations. Iteration will stop after maxiter
steps even if the specified tolerance has not been achieved.
M : {sparse matrix, dense matrix, LinearOperator}, optional
Preconditioner for A. The preconditioner should approximate the
inverse of A. Effective preconditioning dramatically improves the
rate of convergence, which implies that fewer iterations are needed
to reach a given error tolerance.
callback : function, optional
User-supplied function to call after each iteration. It is called
as callback(xk), where xk is the current solution vector.
inner_m : int, optional
Number of inner GMRES iterations per each outer iteration.
outer_k : int, optional
Number of vectors to carry between inner GMRES iterations.
According to [1]_, good values are in the range of 1...3.
However, note that if you want to use the additional vectors to
accelerate solving multiple similar problems, larger values may
be beneficial.
outer_v : list of tuples, optional
List containing tuples ``(v, Av)`` of vectors and corresponding
matrix-vector products, used to augment the Krylov subspace, and
carried between inner GMRES iterations. The element ``Av`` can
be `None` if the matrix-vector product should be re-evaluated.
This parameter is modified in-place by `lgmres`, and can be used
to pass "guess" vectors in and out of the algorithm when solving
similar problems.
store_outer_Av : bool, optional
Whether LGMRES should store also A*v in addition to vectors `v`
in the `outer_v` list. Default is True.
prepend_outer_v : bool, optional
Whether to put outer_v augmentation vectors before Krylov iterates.
In standard LGMRES, prepend_outer_v=False.
x : array or 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
The LGMRES algorithm [1]_ [2]_ is designed to avoid the
slowing of convergence in restarted GMRES, due to alternating
residual vectors. Typically, it often outperforms GMRES(m) of
comparable memory requirements by some measure, or at least is not
much worse.
Another advantage in this algorithm is that you can supply it with
'guess' vectors in the `outer_v` argument that augment the Krylov
subspace. If the solution lies close to the span of these vectors,
the algorithm converges faster. This can be useful if several very
similar matrices need to be inverted one after another, such as in
Newton-Krylov iteration where the Jacobian matrix often changes
little in the nonlinear steps.
.. [1] A.H. Baker and E.R. Jessup and T. Manteuffel, "A Technique for
Accelerating the Convergence of Restarted GMRES", SIAM J. Matrix
Anal. Appl. 26, 962 (2005).
.. [2] A.H. Baker, "On Improving the Performance of the Linear Solver
restarted GMRES", PhD thesis, University of Colorado (2003).
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import lgmres
>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> b = np.array([2, 4, -1], dtype=float)
>>> x, exitCode = lgmres(A, b)
>>> print(exitCode) # 0 indicates successful convergence
>>> np.allclose(, b)
A,M,x,b,postprocess = make_system(A,M,x0,b)
if not np.isfinite(b).all():
raise ValueError("RHS must contain only finite numbers")
matvec = A.matvec
psolve = M.matvec
if outer_v is None:
outer_v = []
axpy, dot, scal = None, None, None
nrm2 = get_blas_funcs('nrm2', [b])
b_norm = nrm2(b)
if b_norm == 0:
b_norm = 1
for k_outer in xrange(maxiter):
r_outer = matvec(x) - b
# -- callback
if callback is not None:
# -- determine input type routines
if axpy is None:
if np.iscomplexobj(r_outer) and not np.iscomplexobj(x):
x = x.astype(r_outer.dtype)
axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'],
(x, r_outer))
trtrs = get_lapack_funcs('trtrs', (x, r_outer))
# -- check stopping condition
r_norm = nrm2(r_outer)
if r_norm <= tol * b_norm or r_norm <= tol:
# -- inner LGMRES iteration
v0 = -psolve(r_outer)
inner_res_0 = nrm2(v0)
if inner_res_0 == 0:
rnorm = nrm2(r_outer)
raise RuntimeError("Preconditioner returned a zero vector; "
"|v| ~ %.1g, |M v| = 0" % rnorm)
v0 = scal(1.0/inner_res_0, v0)
Q, R, B, vs, zs, y = _fgmres(matvec,
y *= inner_res_0
if not np.isfinite(y).all():
# Overflow etc. in computation. There's no way to
# recover from this, so we have to bail out.
raise LinAlgError()
except LinAlgError:
# Floating point over/underflow, non-finite result from
# matmul etc. -- report failure.
return postprocess(x), k_outer + 1
# -- GMRES terminated: eval solution
dx = zs[0]*y[0]
for w, yc in zip(zs[1:], y[1:]):
dx = axpy(w, dx, dx.shape[0], yc) # dx += w*yc
# -- Store LGMRES augmentation vectors
nx = nrm2(dx)
if nx > 0:
if store_outer_Av:
q =
ax = vs[0]*q[0]
for v, qc in zip(vs[1:], q[1:]):
ax = axpy(v, ax, ax.shape[0], qc)
outer_v.append((dx/nx, ax/nx))
outer_v.append((dx/nx, None))
# -- Retain only a finite number of augmentation vectors
while len(outer_v) > outer_k:
del outer_v[0]
# -- Apply step
x += dx
# didn't converge ...
return postprocess(x), maxiter
return postprocess(x), 0