def block_jacobi(A, x, b, Dinv=None, blocksize=1, iterations=1, omega=1.0):
"""Perform block Jacobi iteration on the linear system Ax=b
Parameters
----------
A : csr_matrix or bsr_matrix
Sparse NxN matrix
x : ndarray
Approximate solution (length N)
b : ndarray
Right-hand side (length N)
Dinv : array
Array holding block diagonal inverses of A
size (N/blocksize, blocksize, blocksize)
blocksize : int
Desired dimension of blocks
iterations : int
Number of iterations to perform
omega : scalar
Damping parameter
Returns
-------
Nothing, x will be modified in place.
Examples
--------
>>> ## Use block 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))
>>> block_jacobi(A, x0, b, blocksize=4, iterations=10, omega=1.0)
>>> print norm(b-A*x0)
4.66474230129
>>> #
>>> ## Use block 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=('block_jacobi', {'omega': 4.0/3.0, 'iterations' : 2, 'blocksize' : 4}),
... postsmoother=('block_jacobi', {'omega': 4.0/3.0, 'iterations' : 2, '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')
sweep = slice(None)
(row_start,row_stop,row_step) = sweep.indices(A.shape[0]/blocksize)
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])
for iter in xrange(iterations):
amg_core.block_jacobi(A.indptr, A.indices, numpy.ravel(A.data),
x, b, numpy.ravel(Dinv), temp,
row_start, row_stop, row_step,
omega, blocksize)
def block_jacobi(A, x, b, Dinv=None, blocksize=1, iterations=1, omega=1.0):
"""Perform block Jacobi iteration on the linear system Ax=b
Parameters
----------
A : csr_matrix or bsr_matrix
Sparse NxN matrix
x : ndarray
Approximate solution (length N)
b : ndarray
Right-hand side (length N)
Dinv : array
Array holding block diagonal inverses of A
size (N/blocksize, blocksize, blocksize)
blocksize : int
Desired dimension of blocks
iterations : int
Number of iterations to perform
omega : scalar
Damping parameter
Returns
-------
Nothing, x will be modified in place.
Examples
--------
>>> ## Use block 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))
>>> block_jacobi(A, x0, b, blocksize=4, iterations=10, omega=1.0)
>>> print norm(b-A*x0)
4.66474230129
>>> #
>>> ## Use block 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=('block_jacobi', {'omega': 4.0/3.0, 'iterations' : 2, 'blocksize' : 4}),
... postsmoother=('block_jacobi', {'omega': 4.0/3.0, 'iterations' : 2, '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')
sweep = slice(None)
(row_start, row_stop, row_step) = sweep.indices(A.shape[0] / blocksize)
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])
for iter in xrange(iterations):
amg_core.block_jacobi(A.indptr, A.indices, numpy.ravel(A.data), x, b,
numpy.ravel(Dinv), temp, row_start, row_stop,
row_step, omega, blocksize)