def relaxation_vectors(A, R, k, alpha):
"""Generate test vectors by relaxing on Ax=0 for some random vectors x.
Parameters
----------
A : {csr_matrix}
Sparse NxN matrix
alpha : scalar
Weight for Jacobi
R : integer
Number of random vectors
k : integer
Number of relaxation passes
Returns
-------
x : {array}
Dense array N x k array of relaxation vectors
"""
# random n x R block in column ordering
n = A.shape[0]
x = np.random.rand(n * R) - 0.5
x = np.reshape(x, (n, R), order='F')
# for i in range(R):
# x[:,i] = x[:,i] - np.mean(x[:,i])
b = np.zeros((n, 1))
for r in range(0, R):
jacobi(A, x[:, r], b, iterations=k, omega=alpha)
# x[:,r] = x[:,r]/norm(x[:,r])
return x
def relaxation_vectors(A, R, k, alpha):
"""Generate test vectors by relaxing on Ax=0 for some random vectors x.
Parameters
----------
A : {csr_matrix}
Sparse NxN matrix
alpha : scalar
Weight for Jacobi
R : integer
Number of random vectors
k : integer
Number of relaxation passes
Returns
-------
x : {array}
Dense array N x k array of relaxation vectors
"""
# random n x R block in column ordering
n = A.shape[0]
x = np.random.rand(n * R) - 0.5
x = np.reshape(x, (n, R), order='F')
# for i in range(R):
# x[:,i] = x[:,i] - np.mean(x[:,i])
b = np.zeros((n, 1))
for r in range(0, R):
jacobi(A, x[:, r], b, iterations=k, omega=alpha)
# x[:,r] = x[:,r]/norm(x[:,r])
return x
def algebraic_distance(A, alpha=0.5, R=5, k=20, theta=0.1, p=2):
"""Construct an AMG strength of connection matrix using an algebraic
distance measure.
Parameters
----------
A : {csr_matrix}
Sparse NxN matrix
alpha : scalar
Weight for Jacobi
R : integer
Number of random vectors
k : integer
Number of relaxation passes
theta : scalar
Drop values larger than theta
p : scalar or inf
p-norm of the measure
Returns
-------
C : {csr_matrix}
Sparse matrix of strength values
References
----------
.. [1] "Advanced Coarsening Schemes for Graph Partitioning"
by Ilya Safro, Peter Sanders, and Christian Schulz
Notes
-----
No unit testing yet.
Does not handle BSR matrices yet.
"""
print(A.format)
if not sparse.isspmatrix_csr(A):
warn("Implicit conversion of A to csr", sparse.SparseEfficiencyWarning)
A = sparse.csr_matrix(A)
if alpha < 0:
raise ValueError('expected alpha>0')
if R <= 0 or not isinstance(R, int):
raise ValueError('expected integer R>0')
if k <= 0 or not isinstance(k, int):
raise ValueError('expected integer k>0')
if theta < 0:
raise ValueError('expected theta>0.0')
if p < 1:
raise ValueError('expected p>1 or equal to numpy.inf')
# random n x R block in column ordering
n = A.shape[0]
x = np.random.rand(n * R) - 0.5
x = np.reshape(x, (n, R), order='F')
b = np.zeros((n, 1))
# relax k times
for r in range(0, R):
jacobi(A, x[:, r], b, iterations=k, omega=alpha)
# get distance measure d
C = A.tocoo()
I = C.row
J = C.col
# TODO : this is not right vvvvvvv
if p != np.inf:
d = np.sum((x[I] - x[J])**p, axis=1)**(1.0/p)
else:
d = np.abs(x[I] - x[J]).max(axis=1)
del d
# drop weak connections larger than theta
weak = np.where(C.data > theta)[0]
C.data[weak] = 0
C = C.tocsr()
C.eliminate_zeros()
# Strength represents "distance", so take the magnitude
C.data = np.abs(C.data)
# Standardized strength values require small values be weak and large
# values be strong. So, we invert the distances.
C.data = 1.0/C.data
# Put an identity on the diagonal
C = C + sparse.eye(C.shape[0], C.shape[1], format='csr')
# Scale C by the largest magnitude entry in each row
C = scale_rows_by_largest_entry(C)
return C