def jacobi_ne(A, x, b, iterations=1, omega=1.0):
"""Perform Jacobi iterations on the linear system A A.H x = A.H b
(Also known as Cimmino 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
omega : scalar
Damping parameter
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
.. [3] Cimmino. La ricerca scientifica ser. II 1.
Pubbliz. dell'Inst. pre le Appl. del Calculo 34, 326-333, 1938.
Examples
--------
>>> ## Use NE Jacobi as a Stand-Alone Solver
>>> from pyamg.relaxation import jacobi_ne
>>> from pyamg.gallery import poisson
>>> from pyamg.util.linalg import norm
>>> import numpy
>>> A = poisson((50,50), format='csr')
>>> x0 = numpy.zeros((A.shape[0],1))
>>> b = numpy.ones((A.shape[0],1))
>>> jacobi_ne(A, x0, b, iterations=10, omega=2.0/3.0)
>>> print norm(b-A*x0)
49.3886046066
>>> #
>>> ## Use NE Jacobi 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=('jacobi_ne', {'iterations' : 2, 'omega' : 4.0/3.0}),
... postsmoother=('jacobi_ne', {'iterations' : 2, 'omega' : 4.0/3.0}))
>>> 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'])
sweep = slice(None)
(row_start,row_stop,row_step) = sweep.indices(A.shape[0])
temp = numpy.zeros_like(x)
# Dinv for A*A.H
Dinv = get_diagonal(A, norm_eq=2, inv=True)
# Create uniform type, and convert possibly complex scalars to length 1 arrays
[omega] = type_prep(A.dtype, [omega])
for i in range(iterations):
delta = (numpy.ravel(b - A*x)*numpy.ravel(Dinv)).astype(A.dtype)
amg_core.jacobi_ne(A.indptr, A.indices, A.data,
x, b, delta, temp, row_start,
row_stop, row_step, omega)
def jacobi_ne(A, x, b, iterations=1, omega=1.0):
"""Perform Jacobi iterations on the linear system A A.H x = A.H b
(Also known as Cimmino 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
omega : scalar
Damping parameter
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
.. [3] Cimmino. La ricerca scientifica ser. II 1.
Pubbliz. dell'Inst. pre le Appl. del Calculo 34, 326-333, 1938.
Examples
--------
>>> ## Use NE Jacobi as a Stand-Alone Solver
>>> from pyamg.relaxation import jacobi_ne
>>> from pyamg.gallery import poisson
>>> from pyamg.util.linalg import norm
>>> import numpy
>>> A = poisson((50,50), format='csr')
>>> x0 = numpy.zeros((A.shape[0],1))
>>> b = numpy.ones((A.shape[0],1))
>>> jacobi_ne(A, x0, b, iterations=10, omega=2.0/3.0)
>>> print norm(b-A*x0)
49.3886046066
>>> #
>>> ## Use NE Jacobi 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=('jacobi_ne', {'iterations' : 2, 'omega' : 4.0/3.0}),
... postsmoother=('jacobi_ne', {'iterations' : 2, 'omega' : 4.0/3.0}))
>>> 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'])
sweep = slice(None)
(row_start, row_stop, row_step) = sweep.indices(A.shape[0])
temp = numpy.zeros_like(x)
# Dinv for A*A.H
Dinv = get_diagonal(A, norm_eq=2, inv=True)
# Create uniform type, and convert possibly complex scalars to length 1 arrays
[omega] = type_prep(A.dtype, [omega])
for i in range(iterations):
delta = (numpy.ravel(b - A * x) * numpy.ravel(Dinv)).astype(A.dtype)
amg_core.jacobi_ne(A.indptr, A.indices, A.data, x, b, delta, temp,
row_start, row_stop, row_step, omega)