def gauss_seidel(A, x, b, iterations=1, sweep='forward'):
"""Perform Gauss-Seidel iteration on the linear system Ax=b
Parameters
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
x : ndarray
Approximate solution (length N)
b : ndarray
Right-hand side (length N)
iterations : int
Number of iterations to perform
sweep : {'forward','backward','symmetric'}
Direction of sweep
Returns
-------
Nothing, x will be modified in place.
Examples
--------
>>> ## Use Gauss-Seidel as a Stand-Alone Solver
>>> from pyamg.relaxation import *
>>> from pyamg.gallery import poisson
>>> from pyamg.util.linalg import norm
>>> import numpy
>>> A = poisson((10,10), format='csr')
>>> x0 = numpy.zeros((A.shape[0],1))
>>> b = numpy.ones((A.shape[0],1))
>>> gauss_seidel(A, x0, b, iterations=10)
>>> print norm(b-A*x0)
4.00733716236
>>> #
>>> ## Use Gauss-Seidel as the Multigrid Smoother
>>> from pyamg import smoothed_aggregation_solver
>>> sa = smoothed_aggregation_solver(A, B=numpy.ones((A.shape[0],1)),
... coarse_solver='pinv2', max_coarse=50,
... presmoother=('gauss_seidel', {'sweep':'symmetric'}),
... postsmoother=('gauss_seidel', {'sweep':'symmetric'}))
>>> x0=numpy.zeros((A.shape[0],1))
>>> residuals=[]
>>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals)
"""
A,x,b = make_system(A, x, b, formats=['csr','bsr'])
if sweep == 'forward':
row_start,row_stop,row_step = 0,len(x),1
elif sweep == 'backward':
row_start,row_stop,row_step = len(x)-1,-1,-1
elif sweep == 'symmetric':
for iter in xrange(iterations):
gauss_seidel(A, x, b, iterations=1, sweep='forward')
gauss_seidel(A, x, b, iterations=1, sweep='backward')
return
else:
raise ValueError("valid sweep directions are 'forward', 'backward', and 'symmetric'")
if sparse.isspmatrix_csr(A):
for iter in xrange(iterations):
amg_core.gauss_seidel(A.indptr, A.indices, A.data, x, b,
row_start, row_stop, row_step)
else:
R,C = A.blocksize
if R != C:
raise ValueError('BSR blocks must be square')
row_start = row_start / R
row_stop = row_stop / R
for iter in xrange(iterations):
amg_core.bsr_gauss_seidel(A.indptr, A.indices, numpy.ravel(A.data),
x, b, row_start, row_stop, row_step, R)
def gauss_seidel(A, x, b, iterations=1, sweep='forward'):
"""Perform Gauss-Seidel iteration on the linear system Ax=b
Parameters
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
x : ndarray
Approximate solution (length N)
b : ndarray
Right-hand side (length N)
iterations : int
Number of iterations to perform
sweep : {'forward','backward','symmetric'}
Direction of sweep
Returns
-------
Nothing, x will be modified in place.
Examples
--------
>>> ## Use Gauss-Seidel as a Stand-Alone Solver
>>> from pyamg.relaxation import *
>>> from pyamg.gallery import poisson
>>> from pyamg.util.linalg import norm
>>> import numpy
>>> A = poisson((10,10), format='csr')
>>> x0 = numpy.zeros((A.shape[0],1))
>>> b = numpy.ones((A.shape[0],1))
>>> gauss_seidel(A, x0, b, iterations=10)
>>> print norm(b-A*x0)
4.00733716236
>>> #
>>> ## Use Gauss-Seidel as the Multigrid Smoother
>>> from pyamg import smoothed_aggregation_solver
>>> sa = smoothed_aggregation_solver(A, B=numpy.ones((A.shape[0],1)),
... coarse_solver='pinv2', max_coarse=50,
... presmoother=('gauss_seidel', {'sweep':'symmetric'}),
... postsmoother=('gauss_seidel', {'sweep':'symmetric'}))
>>> x0=numpy.zeros((A.shape[0],1))
>>> residuals=[]
>>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals)
"""
A, x, b = make_system(A, x, b, formats=['csr', 'bsr'])
if sweep == 'forward':
row_start, row_stop, row_step = 0, len(x), 1
elif sweep == 'backward':
row_start, row_stop, row_step = len(x) - 1, -1, -1
elif sweep == 'symmetric':
for iter in xrange(iterations):
gauss_seidel(A, x, b, iterations=1, sweep='forward')
gauss_seidel(A, x, b, iterations=1, sweep='backward')
return
else:
raise ValueError(
"valid sweep directions are 'forward', 'backward', and 'symmetric'"
)
if sparse.isspmatrix_csr(A):
for iter in xrange(iterations):
amg_core.gauss_seidel(A.indptr, A.indices, A.data, x, b, row_start,
row_stop, row_step)
else:
R, C = A.blocksize
if R != C:
raise ValueError('BSR blocks must be square')
row_start = row_start / R
row_stop = row_stop / R
for iter in xrange(iterations):
amg_core.bsr_gauss_seidel(A.indptr, A.indices, numpy.ravel(A.data),
x, b, row_start, row_stop, row_step, R)