def build_complex(self, simplex_array):
"""Compute faces and boundary operators for all dimensions"""
N,K = simplex_array.shape
s = simplex_array.copy()
parity = simplex_array_parity(s)
s.sort()
simplices = [s]
chain_complex = []
parities = [parity]
while s.shape[1] > 1:
s,boundary = simplex_array_boundary(s,parity)
parity = zeros(s.shape[0],dtype=s.dtype)
simplices.append( s )
chain_complex.append( boundary )
parities.append( parity )
B0 = sparse.csr_matrix( (1,len(s)), dtype='uint8')
chain_complex.append( B0 )
simplices = simplices[::-1]
chain_complex = chain_complex[::-1]
parities = parities[::-1]
Bn = chain_complex[-1]
cochain_complex = [ B.T for B in chain_complex[1:] ]
cochain_complex += [ sparse.csc_matrix( (1, Bn.shape[1]), dtype=Bn.dtype) ]
for n in range(K):
data = self.data_cache()
data.d = cochain_complex[n]
data.boundary = chain_complex[n]
data.complex = self
data.dim = n
data.simplices = simplices[n]
data.num_simplices = len(data.simplices)
data.simplex_parity = parities[n]
self.append(data)
def __init__(self, simplices):
"""Construct an abstract simplicial complex
Parameters
----------
simplices : list of arrays
Maximal simplices of each dimension
TODO
Examples
--------
>>> from pydec.dec import abstract_simplicial_complex
>>> from numpy import array
>>> simplices = [array([[4]]), array([[0,3]])]
>>> asc = abstract_simplicial_complex(simplices)
TODO
>>> print rc.simplices[0]
>>> print rc.simplices[1]
Notes
-----
TODO explain input handling
"""
# convert array-like objects to arrays
simplices = [atleast_2d(s) for s in simplices]
# find top simplex dimension
D = max([s.shape[1] for s in simplices]) - 1
# convert simplices list to canonical simplex_array list representation
old_simplices = simplices
simplices = [None] * (D + 1)
for s in old_simplices:
simplices[s.shape[1] - 1] = s
# top most simplex array
s = simplices[-1].copy()
parity = simplex_array_parity(s)
s.sort()
chain_complex = [None] * (D + 1)
for d in range(D - 1, -1, -1):
s,B = simplex_array_boundary(s,parity)
if simplices[d] is not None:
old_s = s
# sort columns to ensure user-defined faces are in canonical format
simplices[d].sort()
# merge user-defined faces with boundary faces
s = vstack((s,simplices[d]))
# sort rows to bring equivalent elements together
s = s[lexsort(s.T[::-1])]
# find unique simplices
mask = -hstack((array([False]),alltrue(s[1:] == s[:-1],axis=1)))
s = s[mask]
# indices of the boundary faces in the full face array
remap = simplex_array_searchsorted(s, old_s)
# remap indices of boundary operator
B = B.tocoo(copy=False)
B = sparse.coo_matrix((B.data, (remap[B.row], B.col)), (s.shape[0], B.shape[1]))
B = B.tocsr()
# faces are already in canonical format, so parity is even
parity = zeros(s.shape[0],dtype=s.dtype)
simplices[d] = s
chain_complex[d+1] = B
# compute 0-simplices and boundary operator
simplices[0] = arange(simplices[0].max() + 1).reshape(-1,1)
chain_complex[0] = sparse.csr_matrix( (1,len(s)), dtype='uint8')
# store the cochain complex
Bn = chain_complex[-1]
cochain_complex = [ B.T for B in chain_complex[1:] ]
cochain_complex += [ sparse.csc_matrix( (1, Bn.shape[1]), dtype=Bn.dtype) ]
# store the data members
self.simplices = simplices
self._chain_complex = chain_complex
self._cochain_complex = cochain_complex