from numpy import array, inner, conjugate, ravel
from scipy.sparse.linalg.isolve.utils import make_system
from pyamg.util.linalg import norm
__docformat__ = "restructuredtext en"
__all__ = ['bicgstab']
[docs]def bicgstab(A, b, x0=None, tol=1e-5, maxiter=None, xtype=None, M=None, callback=None, residuals=None):
'''Biconjugate Gradient Algorithm with Stabilization
Solves the linear system Ax = b. Left preconditioning is supported.
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 ||r_0||_2
maxiter : int
maximum number of allowed 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 A.H x = b.
callback : function
User-supplied function is called after each iteration as
callback(xk), where xk is the current solution vector
residuals : list
residuals has the residual norm history,
including the initial residual, appended to it
Returns
-------
(xNew, info)
xNew : an updated guess to the solution of Ax = b
info : halting status of bicgstab
== ======================================
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.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.
Examples
--------
>>> from pyamg.krylov.bicgstab import bicgstab
>>> 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) = bicgstab(A,b, maxiter=2, tol=1e-8)
>>> print norm(b - A*x)
4.68163045309
References
----------
.. [1] Yousef Saad, "Iterative Methods for Sparse Linear Systems,
Second Edition", SIAM, pp. 231-234, 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)
##
# Ensure that warnings are always reissued from this function
import warnings
warnings.filterwarnings('always', module='pyamg\.krylov\._bicgstab')
# Check iteration numbers
if maxiter == None:
maxiter = len(x) + 5
elif maxiter < 1:
raise ValueError('Number of iterations must be positive')
# Prep for method
r = b - A*x
normr = norm(r)
if residuals is not None:
residuals[:] = [normr]
# 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:
return (postprocess(x), 0)
# Scale tol by ||r_0||_2
if normr != 0.0:
tol = tol*normr
# Is this a one dimensional matrix?
if A.shape[0] == 1:
entry = ravel(A*array([1.0], dtype=xtype))
return (postprocess(b/entry), 0)
rstar = r.copy()
p = r.copy()
rrstarOld = inner(rstar.conjugate(), r)
iter = 0
# Begin BiCGStab
while True:
Mp = M*p
AMp = A*Mp
# alpha = (r_j, rstar) / (A*M*p_j, rstar)
alpha = rrstarOld/inner(rstar.conjugate(), AMp)
# s_j = r_j - alpha*A*M*p_j
s = r - alpha*AMp
Ms = M*s
AMs = A*Ms
# omega = (A*M*s_j, s_j)/(A*M*s_j, A*M*s_j)
omega = inner(AMs.conjugate(), s)/inner(AMs.conjugate(), AMs)
# x_{j+1} = x_j + alpha*M*p_j + omega*M*s_j
x = x + alpha*Mp + omega*Ms
# r_{j+1} = s_j - omega*A*M*s
r = s - omega*AMs
# beta_j = (r_{j+1}, rstar)/(r_j, rstar) * (alpha/omega)
rrstarNew = inner(rstar.conjugate(), r)
beta = (rrstarNew / rrstarOld) * (alpha / omega)
rrstarOld = rrstarNew
# p_{j+1} = r_{j+1} + beta*(p_j - omega*A*M*p)
p = r + beta*(p - omega*AMp)
iter += 1
normr = norm(r)
if residuals is not None:
residuals.append(normr)
if callback is not None:
callback(x)
if normr < tol:
return (postprocess(x), 0)
if iter == maxiter:
return (postprocess(x), iter)
#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) = bicgstab(A,b,x0,tol=1e-8,maxiter=100)
# from pyamg.gallery import stencil_grid
# from numpy.random import random
# A = stencil_grid([[0,-1,0],[-1,4,-1],[0,-1,0]],(100,100),dtype=float,format='csr')
# b = random((A.shape[0],))
# x0 = random((A.shape[0],))
#
# import time
# from scipy.sparse.linalg.isolve import bicgstab as ibicgstab
#
# print '\n\nTesting BiCGStab with %d x %d 2D Laplace Matrix'%(A.shape[0],A.shape[0])
# t1=time.time()
# (x,flag) = bicgstab(A,b,x0,tol=1e-8,maxiter=100)
# t2=time.time()
# print '%s took %0.3f ms' % ('bicgstab', (t2-t1)*1000.0)
# print 'norm = %g'%(norm(b - A*x))
# print 'info flag = %d'%(flag)
#
# t1=time.time()
# (y,flag) = ibicgstab(A,b,x0,tol=1e-8,maxiter=100)
# t2=time.time()
# print '\n%s took %0.3f ms' % ('linalg bicgstab', (t2-t1)*1000.0)
# print 'norm = %g'%(norm(b - A*y))
# print 'info flag = %d'%(flag)