def gauss_seidel_ne(A, x, b, iterations=1, sweep='forward', omega=1.0, Dinv=None):
"""Perform Gauss-Seidel iterations on the linear system A A.H x = b
(Also known as Kaczmarz relaxation)
Parameters
----------
A : csr_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
omega : { float}
Relaxation parameter typically in (0, 2)
if omega != 1.0, then algorithm becomes SOR on A A.H
Dinv : { ndarray}
Inverse of diag(A A.H), (length N)
Returns
-------
Nothing, x will be modified in place.
References
----------
.. [1] Brandt, Ta'asan.
"Multigrid Method For Nearly Singular And Slightly Indefinite Problems."
1985. NASA Technical Report Numbers: ICASE-85-57; NAS 1.26:178026; NASA-CR-178026;
.. [2] Kaczmarz. Angenaeherte Aufloesung von Systemen Linearer Gleichungen.
Bull. Acad. Polon. Sci. Lett. A 35, 355-57. 1937
Examples
--------
>>> ## Use NE 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_ne(A, x0, b, iterations=10, sweep='symmetric')
>>> print norm(b-A*x0)
8.47576806771
>>> #
>>> ## Use NE 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_ne', {'sweep' : 'symmetric'}),
... postsmoother=('gauss_seidel_ne', {'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'])
# Dinv for A*A.H
if Dinv == None:
Dinv = numpy.ravel(get_diagonal(A, norm_eq=2, inv=True))
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_ne(A, x, b, iterations=1, sweep='forward', omega=omega, Dinv=Dinv)
gauss_seidel_ne(A, x, b, iterations=1, sweep='backward', omega=omega, Dinv=Dinv)
return
else:
raise ValueError("valid sweep directions are 'forward', 'backward', and 'symmetric'")
for i in xrange(iterations):
amg_core.gauss_seidel_ne(A.indptr, A.indices, A.data,
x, b, row_start,
row_stop, row_step, Dinv, omega)
def gauss_seidel_ne(A,
x,
b,
iterations=1,
sweep='forward',
omega=1.0,
Dinv=None):
"""Perform Gauss-Seidel iterations on the linear system A A.H x = b
(Also known as Kaczmarz relaxation)
Parameters
----------
A : csr_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
omega : { float}
Relaxation parameter typically in (0, 2)
if omega != 1.0, then algorithm becomes SOR on A A.H
Dinv : { ndarray}
Inverse of diag(A A.H), (length N)
Returns
-------
Nothing, x will be modified in place.
References
----------
.. [1] Brandt, Ta'asan.
"Multigrid Method For Nearly Singular And Slightly Indefinite Problems."
1985. NASA Technical Report Numbers: ICASE-85-57; NAS 1.26:178026; NASA-CR-178026;
.. [2] Kaczmarz. Angenaeherte Aufloesung von Systemen Linearer Gleichungen.
Bull. Acad. Polon. Sci. Lett. A 35, 355-57. 1937
Examples
--------
>>> ## Use NE 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_ne(A, x0, b, iterations=10, sweep='symmetric')
>>> print norm(b-A*x0)
8.47576806771
>>> #
>>> ## Use NE 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_ne', {'sweep' : 'symmetric'}),
... postsmoother=('gauss_seidel_ne', {'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'])
# Dinv for A*A.H
if Dinv == None:
Dinv = numpy.ravel(get_diagonal(A, norm_eq=2, inv=True))
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_ne(A,
x,
b,
iterations=1,
sweep='forward',
omega=omega,
Dinv=Dinv)
gauss_seidel_ne(A,
x,
b,
iterations=1,
sweep='backward',
omega=omega,
Dinv=Dinv)
return
else:
raise ValueError(
"valid sweep directions are 'forward', 'backward', and 'symmetric'"
)
for i in xrange(iterations):
amg_core.gauss_seidel_ne(A.indptr, A.indices, A.data, x, b, row_start,
row_stop, row_step, Dinv, omega)