def block_gauss_seidel(A, x, b, iterations=1, sweep='forward', blocksize=1, Dinv=None):
"""Perform block 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
Dinv : array
Array holding block diagonal inverses of A
size (N/blocksize, blocksize, blocksize)
blocksize : int
Desired dimension of blocks
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))
>>> block_gauss_seidel(A, x0, b, iterations=10, blocksize=4, sweep='symmetric')
>>> print norm(b-A*x0)
0.958333817624
>>> #
>>> ## 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=('block_gauss_seidel', {'sweep':'symmetric', 'blocksize' : 4}),
... postsmoother=('block_gauss_seidel', {'sweep':'symmetric', 'blocksize' : 4}))
>>> 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'])
A = A.tobsr(blocksize=(blocksize, blocksize))
if Dinv == None:
Dinv = get_block_diag(A, blocksize=blocksize, inv_flag=True)
elif Dinv.shape[0] != A.shape[0]/blocksize:
raise ValueError('Dinv and A have incompatible dimensions')
elif (Dinv.shape[1] != blocksize) or (Dinv.shape[2] != blocksize):
raise ValueError('Dinv and blocksize are incompatible')
if sweep == 'forward':
row_start,row_stop,row_step = 0,len(x)/blocksize,1
elif sweep == 'backward':
row_start,row_stop,row_step = len(x)/blocksize-1,-1,-1
elif sweep == 'symmetric':
for iter in xrange(iterations):
block_gauss_seidel(A, x, b, iterations=1, sweep='forward', blocksize=blocksize, Dinv=Dinv)
block_gauss_seidel(A, x, b, iterations=1, sweep='backward',blocksize=blocksize, Dinv=Dinv)
return
else:
raise ValueError("valid sweep directions are 'forward', 'backward', and 'symmetric'")
for iter in xrange(iterations):
amg_core.block_gauss_seidel(A.indptr, A.indices, numpy.ravel(A.data),
x, b, numpy.ravel(Dinv),
row_start, row_stop, row_step, blocksize)
def block_gauss_seidel(A,
x,
b,
iterations=1,
sweep='forward',
blocksize=1,
Dinv=None):
"""Perform block 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
Dinv : array
Array holding block diagonal inverses of A
size (N/blocksize, blocksize, blocksize)
blocksize : int
Desired dimension of blocks
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))
>>> block_gauss_seidel(A, x0, b, iterations=10, blocksize=4, sweep='symmetric')
>>> print norm(b-A*x0)
0.958333817624
>>> #
>>> ## 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=('block_gauss_seidel', {'sweep':'symmetric', 'blocksize' : 4}),
... postsmoother=('block_gauss_seidel', {'sweep':'symmetric', 'blocksize' : 4}))
>>> 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'])
A = A.tobsr(blocksize=(blocksize, blocksize))
if Dinv == None:
Dinv = get_block_diag(A, blocksize=blocksize, inv_flag=True)
elif Dinv.shape[0] != A.shape[0] / blocksize:
raise ValueError('Dinv and A have incompatible dimensions')
elif (Dinv.shape[1] != blocksize) or (Dinv.shape[2] != blocksize):
raise ValueError('Dinv and blocksize are incompatible')
if sweep == 'forward':
row_start, row_stop, row_step = 0, len(x) / blocksize, 1
elif sweep == 'backward':
row_start, row_stop, row_step = len(x) / blocksize - 1, -1, -1
elif sweep == 'symmetric':
for iter in xrange(iterations):
block_gauss_seidel(A,
x,
b,
iterations=1,
sweep='forward',
blocksize=blocksize,
Dinv=Dinv)
block_gauss_seidel(A,
x,
b,
iterations=1,
sweep='backward',
blocksize=blocksize,
Dinv=Dinv)
return
else:
raise ValueError(
"valid sweep directions are 'forward', 'backward', and 'symmetric'"
)
for iter in xrange(iterations):
amg_core.block_gauss_seidel(A.indptr,
A.indices, numpy.ravel(A.data), x, b,
numpy.ravel(Dinv), row_start, row_stop,
row_step, blocksize)
