def gauss_seidel_indexed(A, x, b, indices, iterations=1, sweep='forward'):
"""Perform indexed Gauss-Seidel iteration on the linear system Ax=b
In indexed Gauss-Seidel, the sequence in which unknowns are relaxed is
specified explicitly. In contrast, the standard Gauss-Seidel method
always performs complete sweeps of all variables in increasing or
decreasing order. The indexed method may be used to implement
specialized smoothers, like F-smoothing in Classical AMG.
Parameters
----------
A : csr_matrix
Sparse NxN matrix
x : ndarray
Approximate solution (length N)
b : ndarray
Right-hand side (length N)
indices : ndarray
Row indices to relax.
iterations : int
Number of iterations to perform
sweep : {'forward','backward','symmetric'}
Direction of sweep
Returns
-------
Nothing, x will be modified in place.
Examples
--------
>>> from pyamg.gallery import poisson
>>> from pyamg.relaxation import gauss_seidel_indexed
>>> import numpy
>>> A = poisson((4,), format='csr')
>>> x = numpy.array([0.0, 0.0, 0.0, 0.0])
>>> b = numpy.array([0.0, 1.0, 2.0, 3.0])
>>> gauss_seidel_indexed(A, x, b, [0,1,2,3]) #relax all four rows, in order
>>> gauss_seidel_indexed(A, x, b, [0,1]) #relax first two rows
>>> gauss_seidel_indexed(A, x, b, [2,0]) #relax row 2, then row 0
>>> gauss_seidel_indexed(A, x, b, [2,3], sweep='backward') #relax row 3, then row 2
>>> gauss_seidel_indexed(A, x, b, [2,0,2]) #relax row 2, then 0, then 2 again
"""
A,x,b = make_system(A, x, b, formats=['csr'])
indices = numpy.asarray(indices, dtype='intc')
#if indices.min() < 0:
# raise ValueError('row index (%d) is invalid' % indices.min())
#if indices.max() >= A.shape[0]
# raise ValueError('row index (%d) is invalid' % indices.max())
if sweep == 'forward':
row_start,row_stop,row_step = 0,len(indices),1
elif sweep == 'backward':
row_start,row_stop,row_step = len(indices)-1,-1,-1
elif sweep == 'symmetric':
for iter in xrange(iterations):
gauss_seidel_indexed(A, x, b, indices, iterations=1, sweep='forward')
gauss_seidel_indexed(A, x, b, indices, iterations=1, sweep='backward')
return
else:
raise ValueError('valid sweep directions are \'forward\', \'backward\', and \'symmetric\'')
for iter in xrange(iterations):
amg_core.gauss_seidel_indexed(A.indptr, A.indices, A.data,
x, b, indices,
row_start, row_stop, row_step)
def gauss_seidel_indexed(A, x, b, indices, iterations=1, sweep='forward'):
"""Perform indexed Gauss-Seidel iteration on the linear system Ax=b
In indexed Gauss-Seidel, the sequence in which unknowns are relaxed is
specified explicitly. In contrast, the standard Gauss-Seidel method
always performs complete sweeps of all variables in increasing or
decreasing order. The indexed method may be used to implement
specialized smoothers, like F-smoothing in Classical AMG.
Parameters
----------
A : csr_matrix
Sparse NxN matrix
x : ndarray
Approximate solution (length N)
b : ndarray
Right-hand side (length N)
indices : ndarray
Row indices to relax.
iterations : int
Number of iterations to perform
sweep : {'forward','backward','symmetric'}
Direction of sweep
Returns
-------
Nothing, x will be modified in place.
Examples
--------
>>> from pyamg.gallery import poisson
>>> from pyamg.relaxation import gauss_seidel_indexed
>>> import numpy
>>> A = poisson((4,), format='csr')
>>> x = numpy.array([0.0, 0.0, 0.0, 0.0])
>>> b = numpy.array([0.0, 1.0, 2.0, 3.0])
>>> gauss_seidel_indexed(A, x, b, [0,1,2,3]) #relax all four rows, in order
>>> gauss_seidel_indexed(A, x, b, [0,1]) #relax first two rows
>>> gauss_seidel_indexed(A, x, b, [2,0]) #relax row 2, then row 0
>>> gauss_seidel_indexed(A, x, b, [2,3], sweep='backward') #relax row 3, then row 2
>>> gauss_seidel_indexed(A, x, b, [2,0,2]) #relax row 2, then 0, then 2 again
"""
A, x, b = make_system(A, x, b, formats=['csr'])
indices = numpy.asarray(indices, dtype='intc')
#if indices.min() < 0:
# raise ValueError('row index (%d) is invalid' % indices.min())
#if indices.max() >= A.shape[0]
# raise ValueError('row index (%d) is invalid' % indices.max())
if sweep == 'forward':
row_start, row_stop, row_step = 0, len(indices), 1
elif sweep == 'backward':
row_start, row_stop, row_step = len(indices) - 1, -1, -1
elif sweep == 'symmetric':
for iter in xrange(iterations):
gauss_seidel_indexed(A,
x,
b,
indices,
iterations=1,
sweep='forward')
gauss_seidel_indexed(A,
x,
b,
indices,
iterations=1,
sweep='backward')
return
else:
raise ValueError(
'valid sweep directions are \'forward\', \'backward\', and \'symmetric\''
)
for iter in xrange(iterations):
amg_core.gauss_seidel_indexed(A.indptr, A.indices, A.data, x, b,
indices, row_start, row_stop, row_step)