def jacobi(A, x, b, iterations=1, omega=1.0):
"""Perform Jacobi iteration on the linear system Ax=b
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.
Examples
--------
>>> ## Use Jacobi 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))
>>> jacobi(A, x0, b, iterations=10, omega=1.0)
>>> print norm(b-A*x0)
5.83475132751
>>> #
>>> ## Use 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', {'omega': 4.0/3.0, 'iterations' : 2}),
... postsmoother=('jacobi', {'omega': 4.0/3.0, 'iterations' : 2}))
>>> 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'])
sweep = slice(None)
(row_start,row_stop,row_step) = sweep.indices(A.shape[0])
if (row_stop - row_start) * row_step <= 0: #no work to do
return
temp = numpy.empty_like(x)
# Create uniform type, and convert possibly complex scalars to length 1 arrays
[omega] = type_prep(A.dtype, [omega])
if sparse.isspmatrix_csr(A):
for iter in xrange(iterations):
amg_core.jacobi(A.indptr, A.indices, A.data, x, b, temp,
row_start, row_stop, row_step, omega)
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_jacobi(A.indptr, A.indices, numpy.ravel(A.data),
x, b, temp, row_start, row_stop, row_step, R, omega)
def jacobi(A, x, b, iterations=1, omega=1.0):
"""Perform Jacobi iteration on the linear system Ax=b
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.
Examples
--------
>>> ## Use Jacobi 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))
>>> jacobi(A, x0, b, iterations=10, omega=1.0)
>>> print norm(b-A*x0)
5.83475132751
>>> #
>>> ## Use 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', {'omega': 4.0/3.0, 'iterations' : 2}),
... postsmoother=('jacobi', {'omega': 4.0/3.0, 'iterations' : 2}))
>>> 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'])
sweep = slice(None)
(row_start, row_stop, row_step) = sweep.indices(A.shape[0])
if (row_stop - row_start) * row_step <= 0: #no work to do
return
temp = numpy.empty_like(x)
# Create uniform type, and convert possibly complex scalars to length 1 arrays
[omega] = type_prep(A.dtype, [omega])
if sparse.isspmatrix_csr(A):
for iter in xrange(iterations):
amg_core.jacobi(A.indptr, A.indices, A.data, x, b, temp, row_start,
row_stop, row_step, omega)
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_jacobi(A.indptr, A.indices, numpy.ravel(A.data), x, b,
temp, row_start, row_stop, row_step, R, omega)
