from numpy import array, zeros, sqrt, ravel, abs, max, conjugate, real,\
iscomplexobj
from scipy.sparse.linalg.isolve.utils import make_system
from scipy.sparse.sputils import upcast
from scipy.linalg import get_blas_funcs
from warnings import warn
import scipy
__docformat__ = "restructuredtext en"
__all__ = ['gmres_mgs']
def apply_givens(Q, v, k):
'''
Apply the first k Givens rotations in Q to v
Parameters
----------
Q : {list}
list of consecutive 2x2 Givens rotations
v : {array}
vector to apply the rotations to
k : {int}
number of rotations to apply.
Returns
-------
v is changed in place
Notes
-----
This routine is specialized for GMRES. It assumes that the first Givens
rotation is for dofs 0 and 1, the second Givens rotation is for
dofs 1 and 2, and so on.
'''
for j in xrange(k):
Qloc = Q[j]
v[j:j+2] = scipy.dot(Qloc, v[j:j+2])
[docs]def gmres_mgs(A, b, x0=None, tol=1e-5, restrt=None, maxiter=None, xtype=None,
M=None, callback=None, residuals=None, reorth=False):
'''
Generalized Minimum Residual Method (GMRES)
GMRES iteratively refines the initial solution guess to the system
Ax = b
Modified Gram-Schmidt version
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
relative convergence tolerance, i.e. tol is scaled by the norm
of the initial preconditioned residual
restrt : {None, int}
- if int, restrt 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 restrt is None, maxiter is the max number of inner iterations
and GMRES does not restart
- if restrt is int, maxiter is the max number of outer iterations,
and restrt is the max number of inner iterations
xtype : type
dtype for the solution, default is automatic type detection
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( ||rk||_2 ), where rk is the current preconditioned residual
vector
residuals : list
residuals contains the preconditioned residual norm history,
including the initial residual.
reorth : boolean
If True, then a check is made whether to re-orthogonalize the Krylov
space each GMRES iteration
Returns
-------
(xNew, info)
xNew : an updated guess to the solution of Ax = b
info : halting status of gmres
== =============================================
0 successful exit
>0 convergence to tolerance not achieved,
return iteration count instead. This value
is precisely the order of the Krylov space.
<0 numerical breakdown, or illegal input
== =============================================
Notes
-----
- The LinearOperator class is in scipy.sparse.linalg.interface.
Use this class if you prefer to define A or M as a mat-vec routine
as opposed to explicitly constructing the matrix. A.psolve(..) is
still supported as a legacy.
- For robustness, modified Gram-Schmidt is used to orthogonalize the
Krylov Space Givens Rotations are used to provide the residual norm
each iteration
Examples
--------
>>> from pyamg.krylov import gmres
>>> from pyamg.util.linalg import norm
>>> import numpy
>>> from pyamg.gallery import poisson
>>> A = poisson((10,10))
>>> b = numpy.ones((A.shape[0],))
>>> (x,flag) = gmres(A,b, maxiter=2, tol=1e-8, orthog='mgs')
>>> print norm(b - A*x)
>>> 6.5428213057
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
.. [2] C. T. Kelley, http://www4.ncsu.edu/~ctk/matlab_roots.html
'''
# Convert inputs to linear system, with error checking
A, M, x, b, postprocess = make_system(A, M, x0, b, xtype)
dimen = A.shape[0]
# Ensure that warnings are always reissued from this function
import warnings
warnings.filterwarnings('always', module='pyamg\.krylov\._gmres_mgs')
# Choose type
if not hasattr(A, 'dtype'):
Atype = upcast(x.dtype, b.dtype)
else:
Atype = A.dtype
if not hasattr(M, 'dtype'):
Mtype = upcast(x.dtype, b.dtype)
else:
Mtype = M.dtype
xtype = upcast(Atype, x.dtype, b.dtype, Mtype)
if restrt is not None:
restrt = int(restrt)
if maxiter is not None:
maxiter = int(maxiter)
# Get fast access to underlying BLAS routines
# dotc is the conjugate dot, dotu does no conjugation
if iscomplexobj(zeros((1,), dtype=xtype)):
[axpy, dotu, dotc, scal, rotg] =\
get_blas_funcs(['axpy', 'dotu', 'dotc', 'scal', 'rotg'], [x])
else:
# real type
[axpy, dotu, dotc, scal, rotg] =\
get_blas_funcs(['axpy', 'dot', 'dot', 'scal', 'rotg'], [x])
# Make full use of direct access to BLAS by defining own norm
def norm(z):
return sqrt(real(dotc(z, z)))
# Should norm(r) be kept
if residuals == []:
keep_r = True
else:
keep_r = False
# Set number of outer and inner iterations
if restrt:
if maxiter:
max_outer = maxiter
else:
max_outer = 1
if restrt > dimen:
warn('Setting number of inner iterations (restrt) to maximum\
allowed, which is A.shape[0] ')
restrt = dimen
max_inner = restrt
else:
max_outer = 1
if maxiter > dimen:
warn('Setting number of inner iterations (maxiter) to maximum\
allowed, which is A.shape[0] ')
maxiter = dimen
elif maxiter is None:
maxiter = min(dimen, 40)
max_inner = maxiter
# Is this a one dimensional matrix?
if dimen == 1:
entry = ravel(A*array([1.0], dtype=xtype))
return (postprocess(b/entry), 0)
# Prep for method
r = b - ravel(A*x)
# Apply preconditioner
r = ravel(M*r)
normr = norm(r)
if keep_r:
residuals.append(normr)
# Check for nan, inf
# if isnan(r).any() or isinf(r).any():
# warn('inf or nan after application of preconditioner')
# return(postprocess(x), -1)
# Check initial guess ( scaling by b, if b != 0,
# must account for case when norm(b) is very small)
normb = norm(b)
if normb == 0.0:
normb = 1.0
if normr < tol*normb:
if callback is not None:
callback(norm(r))
return (postprocess(x), 0)
# Scale tol by ||r_0||_2, we use the preconditioned residual
# because this is left preconditioned GMRES.
if normr != 0.0:
tol = tol*normr
# Use separate variable to track iterations. If convergence fails, we
# cannot simply report niter = (outer-1)*max_outer + inner. Numerical
# error could cause the inner loop to halt while the actual ||r|| > tol.
niter = 0
# Begin GMRES
for outer in xrange(max_outer):
# Preallocate for Givens Rotations, Hessenberg matrix and Krylov Space
# Space required is O(dimen*max_inner).
# NOTE: We are dealing with row-major matrices, so we traverse in a
# row-major fashion,
# i.e., H and V's transpose is what we store.
Q = [] # Givens Rotations
# Upper Hessenberg matrix, which is then
# converted to upper tri with Givens Rots
H = zeros((max_inner+1, max_inner+1), dtype=xtype)
V = zeros((max_inner+1, dimen), dtype=xtype) # Krylov Space
# vs store the pointers to each column of V.
# This saves a considerable amount of time.
vs = []
# v = r/normr
V[0, :] = scal(1.0/normr, r)
vs.append(V[0, :])
# This is the RHS vector for the problem in the Krylov Space
g = zeros((dimen,), dtype=xtype)
g[0] = normr
for inner in xrange(max_inner):
# New Search Direction
v = V[inner+1, :]
v[:] = ravel(M*(A*vs[-1]))
vs.append(v)
normv_old = norm(v)
# Check for nan, inf
# if isnan(V[inner+1, :]).any() or isinf(V[inner+1, :]).any():
# warn('inf or nan after application of preconditioner')
# return(postprocess(x), -1)
# Modified Gram Schmidt
for k in xrange(inner+1):
vk = vs[k]
alpha = dotc(vk, v)
H[inner, k] = alpha
v[:] = axpy(vk, v, dimen, -alpha)
normv = norm(v)
H[inner, inner+1] = normv
# Re-orthogonalize
if (reorth is True) and (normv_old == normv_old + 0.001*normv):
for k in xrange(inner+1):
vk = vs[k]
alpha = dotc(vk, v)
H[inner, k] = H[inner, k] + alpha
v[:] = axpy(vk, v, dimen, -alpha)
# Check for breakdown
if H[inner, inner+1] != 0.0:
v[:] = scal(1.0/H[inner, inner+1], v)
# Apply previous Givens rotations to H
if inner > 0:
apply_givens(Q, H[inner, :], inner)
# Calculate and apply next complex-valued Givens Rotation
# ==> Note that if max_inner = dimen, then this is unnecessary
# for the last inner
# iteration, when inner = dimen-1.
if inner != dimen-1:
if H[inner, inner+1] != 0:
[c, s] = rotg(H[inner, inner], H[inner, inner+1])
Qblock = array([[c, s], [-conjugate(s), c]], dtype=xtype)
Q.append(Qblock)
# Apply Givens Rotation to g,
# the RHS for the linear system in the Krylov Subspace.
g[inner:inner+2] = scipy.dot(Qblock, g[inner:inner+2])
# Apply effect of Givens Rotation to H
H[inner, inner] = dotu(Qblock[0, :],
H[inner, inner:inner+2])
H[inner, inner+1] = 0.0
niter += 1
# Don't update normr if last inner iteration, because
# normr is calculated directly after this loop ends.
if inner < max_inner-1:
normr = abs(g[inner+1])
if normr < tol:
break
# Allow user access to residual
if callback is not None:
callback(normr)
if keep_r:
residuals.append(normr)
# end inner loop, back to outer loop
# Find best update to x in Krylov Space V. Solve inner x inner system.
y = scipy.linalg.solve(H[0:inner+1, 0:inner+1].T, g[0:inner+1])
update = ravel(scipy.mat(V[:inner+1, :]).T*y.reshape(-1, 1))
x = x + update
r = b - ravel(A*x)
# Apply preconditioner
r = ravel(M*r)
normr = norm(r)
# Check for nan, inf
# if isnan(r).any() or isinf(r).any():
# warn('inf or nan after application of preconditioner')
# return(postprocess(x), -1)
# Allow user access to residual
if callback is not None:
callback(normr)
if keep_r:
residuals.append(normr)
# Has GMRES stagnated?
indices = (x != 0)
if indices.any():
change = max(abs(update[indices] / x[indices]))
if change < 1e-12:
# No change, halt
return (postprocess(x), -1)
# test for convergence
if normr < tol:
return (postprocess(x), 0)
# end outer loop
return (postprocess(x), niter)
if __name__ == '__main__':
# from numpy import diag
# A = random((4,4))
# A = A*A.transpose() + diag([10,10,10,10])
# b = random((4,1))
# x0 = random((4,1))
# %timeit -n 15 (x,flag) = gmres(A,b,x0,tol=1e-8,maxiter=100)
from pyamg.gallery import poisson
from numpy.random import random
from pyamg.util.linalg import norm
A = poisson((125, 125), dtype=float, format='csr')
# A.data = A.data + 0.001j*rand(A.data.shape[0])
b = random((A.shape[0],))
x0 = random((A.shape[0],))
import time
from scipy.sparse.linalg.isolve import gmres as igmres
print '\n\nTesting GMRES with %d x %d 2D Laplace Matrix' %\
(A.shape[0], A.shape[0])
t1 = time.time()
(x, flag) = gmres_mgs(A, b, x0, tol=1e-8, maxiter=500)
t2 = time.time()
print '%s took %0.3f ms' % ('gmres', (t2-t1)*1000.0)
print 'norm = %g' % (norm(b - A*x))
print 'info flag = %d' % (flag)
t1 = time.time()
# DON"T Enforce a maxiter as scipy gmres can't handle it correctly
(y, flag) = igmres(A, b, x0, tol=1e-8)
t2 = time.time()
print '\n%s took %0.3f ms' % ('linalg gmres', (t2-t1)*1000.0)
print 'norm = %g' % (norm(b - A*y))
print 'info flag = %d' % (flag)