from numpy import array, zeros, ravel, abs, max, dot, conjugate
from scipy.sparse.linalg.isolve.utils import make_system
from scipy.sparse.sputils import upcast
from warnings import warn
from pyamg.util.linalg import norm
from pyamg import amg_core
from scipy.linalg import get_blas_funcs
import scipy
__docformat__ = "restructuredtext en"
__all__ = ['gmres_householder']
def mysign(x):
if x == 0.0:
return 1.0
else:
# return the complex "sign"
return x/abs(x)
[docs]def gmres_householder(A, b, x0=None, tol=1e-5, restrt=None, maxiter=None,
xtype=None, M=None, callback=None, residuals=None):
'''
Generalized Minimum Residual Method (GMRES)
GMRES iteratively refines the initial solution guess to the
system Ax = b
Householder reflections are used for orthogonalization
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.
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, Householder reflections are used to orthonormalize
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='householder')
>>> 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
'''
# 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_householder')
# 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)
# known bug vvvvvv remove when fixed
if Atype == complex:
raise ValueError('[Known Bug] Housholder fails with complex matrices; \
use MGS')
if restrt is not None:
restrt = int(restrt)
if maxiter is not None:
maxiter = int(maxiter)
# 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
# Get fast access to underlying BLAS routines
[rotg] = get_blas_funcs(['rotg'], [x])
# 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 range(max_outer):
# Calculate vector w, which defines the Householder reflector
# Take shortcut in calculating,
# w = r + sign(r[1])*||r||_2*e_1
w = r
beta = mysign(w[0])*normr
w[0] = w[0] + beta
w[:] = w / norm(w)
# Preallocate for Krylov vectors, Householder reflectors and
# Hessenberg matrix
# Space required is O(dimen*max_inner)
# Givens Rotations
Q = zeros((4*max_inner,), dtype=xtype)
# upper Hessenberg matrix (made upper tri with Givens Rotations)
H = zeros((max_inner, max_inner), dtype=xtype)
# Householder reflectors
W = zeros((max_inner+1, dimen), dtype=xtype)
W[0, :] = w
# Multiply r with (I - 2*w*w.T), i.e. apply the Householder reflector
# This is the RHS vector for the problem in the Krylov Space
g = zeros((dimen,), dtype=xtype)
g[0] = -beta
for inner in range(max_inner):
# Calculate Krylov vector in two steps
# (1) Calculate v = P_j = (I - 2*w*w.T)v, where k = inner
v = -2.0*conjugate(w[inner])*w
v[inner] = v[inner] + 1.0
# (2) Calculate the rest, v = P_1*P_2*P_3...P_{j-1}*ej.
# for j in range(inner-1,-1,-1):
# v -= 2.0*dot(conjugate(W[j,:]), v)*W[j,:]
amg_core.apply_householders(v, ravel(W), dimen, inner-1, -1, -1)
# Calculate new search direction
v = ravel(A*v)
# Apply preconditioner
v = ravel(M*v)
# Check for nan, inf
# if isnan(v).any() or isinf(v).any():
# warn('inf or nan after application of preconditioner')
# return(postprocess(x), -1)
# Factor in all Householder orthogonal reflections on new search
# direction
# for j in range(inner+1):
# v -= 2.0*dot(conjugate(W[j,:]), v)*W[j,:]
amg_core.apply_householders(v, ravel(W), dimen, 0, inner+1, 1)
# Calculate next Householder reflector, w
# w = v[inner+1:] + sign(v[inner+1])*||v[inner+1:]||_2*e_{inner+1)
# Note that if max_inner = dimen, then this is unnecessary for the
# last inner iteration, when inner = dimen-1. Here we do not need
# to calculate a Householder reflector or Givens rotation because
# nnz(v) is already the desired length, i.e. we do not need to
# zero anything out.
if inner != dimen-1:
if inner < (max_inner-1):
w = W[inner+1, :]
vslice = v[inner+1:]
alpha = norm(vslice)
if alpha != 0:
alpha = mysign(vslice[0])*alpha
# do not need the final reflector for future calculations
if inner < (max_inner-1):
w[inner+1:] = vslice
w[inner+1] += alpha
w[:] = w / norm(w)
# Apply new reflector to v
# v = v - 2.0*w*(w.T*v)
v[inner+1] = -alpha
v[inner+2:] = 0.0
if inner > 0:
# Apply all previous Givens Rotations to v
amg_core.apply_givens(Q, v, dimen, inner)
# Calculate the next Givens rotation, where j = inner Note that if
# max_inner = dimen, then this is unnecessary for the last inner
# iteration, when inner = dimen-1. Here we do not need to
# calculate a Householder reflector or Givens rotation because
# nnz(v) is already the desired length, i.e. we do not need to zero
# anything out.
if inner != dimen-1:
if v[inner+1] != 0:
[c, s] = rotg(v[inner], v[inner+1])
Qblock = array([[c, s], [-conjugate(s), c]], dtype=xtype)
Q[(inner*4): ((inner+1)*4)] = ravel(Qblock).copy()
# Apply Givens Rotation to g, the RHS for the linear system
# in the Krylov Subspace. Note that this dot does a matrix
# multiply, not an actual dot product where a conjugate
# transpose is taken
g[inner:inner+2] = dot(Qblock, g[inner:inner+2])
# Apply effect of Givens Rotation to v
v[inner] = dot(Qblock[0, :], v[inner:inner+2])
v[inner+1] = 0.0
# Write to upper Hessenberg Matrix,
# the LHS for the linear system in the Krylov Subspace
H[:, inner] = v[0:max_inner]
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+1 x inner+1
# system. Apparently this is the best way to solve a triangular system
# in the magical world of scipy
# piv = arange(inner+1)
# y = lu_solve((H[0:(inner+1), 0:(inner+1)], piv), g[0:(inner+1)],
# trans=0)
y = scipy.linalg.solve(H[0:(inner+1), 0:(inner+1)], g[0:(inner+1)])
# Use Horner like Scheme to map solution, y, back to original space.
# Note that we do not use the last reflector.
update = zeros(x.shape, dtype=xtype)
# for j in range(inner,-1,-1):
# update[j] += y[j]
# # Apply j-th reflector, (I - 2.0*w_j*w_j.T)*upadate
# update -= 2.0*dot(conjugate(W[j,:]), update)*W[j,:]
amg_core.householder_hornerscheme(update, ravel(W), ravel(y), dimen,
inner, -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 numpy.random import random
from pyamg.gallery import poisson
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_householder(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)