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"""Functions to construct sparse matrices
"""
__docformat__ = "restructuredtext en"
__all__ = [ 'spdiags', 'eye', 'identity', 'kron', 'kronsum',
'hstack', 'vstack', 'bmat' ]
from warnings import warn
import numpy as np
from sputils import upcast
from csr import csr_matrix
from csc import csc_matrix
from bsr import bsr_matrix
from coo import coo_matrix
from lil import lil_matrix
from dia import dia_matrix
def spdiags(data, diags, m, n, format=None):
"""Return a sparse matrix from diagonals.
Parameters
----------
data : array_like
matrix diagonals stored row-wise
diags : diagonals to set
- k = 0 the main diagonal
- k > 0 the k-th upper diagonal
- k < 0 the k-th lower diagonal
m, n : int
shape of the result
format : format of the result (e.g. "csr")
By default (format=None) an appropriate sparse matrix
format is returned. This choice is subject to change.
See Also
--------
The dia_matrix class which implements the DIAgonal format.
Example
-------
>>> data = array([[1,2,3,4],[1,2,3,4],[1,2,3,4]])
>>> diags = array([0,-1,2])
>>> spdiags(data, diags, 4, 4).todense()
matrix([[1, 0, 3, 0],
[1, 2, 0, 4],
[0, 2, 3, 0],
[0, 0, 3, 4]])
"""
return dia_matrix((data, diags), shape=(m,n)).asformat(format)
def identity(n, dtype='d', format=None):
"""Identity matrix in sparse format
Returns an identity matrix with shape (n,n) using a given
sparse format and dtype.
Parameters
----------
n : integer
Shape of the identity matrix.
dtype :
Data type of the matrix
format : string
Sparse format of the result, e.g. format="csr", etc.
Examples
--------
>>> identity(3).todense()
matrix([[ 1., 0., 0.],
[ 0., 1., 0.],
[ 0., 0., 1.]])
>>> identity(3, dtype='int8', format='dia')
<3x3 sparse matrix of type '<type 'numpy.int8'>'
with 3 stored elements (1 diagonals) in DIAgonal format>
"""
if format in ['csr','csc']:
indptr = np.arange(n+1, dtype=np.intc)
indices = np.arange(n, dtype=np.intc)
data = np.ones(n, dtype=dtype)
cls = eval('%s_matrix' % format)
return cls((data,indices,indptr),(n,n))
elif format == 'coo':
row = np.arange(n, dtype=np.intc)
col = np.arange(n, dtype=np.intc)
data = np.ones(n, dtype=dtype)
return coo_matrix((data,(row,col)),(n,n))
elif format == 'dia':
data = np.ones(n, dtype=dtype)
diags = [0]
return dia_matrix((data,diags), shape=(n,n))
else:
return identity(n, dtype=dtype, format='csr').asformat(format)
def eye(m, n, k=0, dtype='d', format=None):
"""eye(m, n) returns a sparse (m x n) matrix where the k-th diagonal
is all ones and everything else is zeros.
"""
m,n = int(m),int(n)
diags = np.ones((1, max(0, min(m + k, n))), dtype=dtype)
return spdiags(diags, k, m, n).asformat(format)
def kron(A, B, format=None):
"""kronecker product of sparse matrices A and B
Parameters
----------
A : sparse or dense matrix
first matrix of the product
B : sparse or dense matrix
second matrix of the product
format : string
format of the result (e.g. "csr")
Returns
-------
kronecker product in a sparse matrix format
Examples
--------
>>> A = csr_matrix(array([[0,2],[5,0]]))
>>> B = csr_matrix(array([[1,2],[3,4]]))
>>> kron(A,B).todense()
matrix([[ 0, 0, 2, 4],
[ 0, 0, 6, 8],
[ 5, 10, 0, 0],
[15, 20, 0, 0]])
>>> kron(A,[[1,2],[3,4]]).todense()
matrix([[ 0, 0, 2, 4],
[ 0, 0, 6, 8],
[ 5, 10, 0, 0],
[15, 20, 0, 0]])
"""
B = coo_matrix(B)
if (format is None or format == "bsr") and 2*B.nnz >= B.shape[0] * B.shape[1]:
#B is fairly dense, use BSR
A = csr_matrix(A,copy=True)
output_shape = (A.shape[0]*B.shape[0], A.shape[1]*B.shape[1])
if A.nnz == 0 or B.nnz == 0:
# kronecker product is the zero matrix
return coo_matrix( output_shape )
B = B.toarray()
data = A.data.repeat(B.size).reshape(-1,B.shape[0],B.shape[1])
data = data * B
return bsr_matrix((data,A.indices,A.indptr), shape=output_shape)
else:
#use COO
A = coo_matrix(A)
output_shape = (A.shape[0]*B.shape[0], A.shape[1]*B.shape[1])
if A.nnz == 0 or B.nnz == 0:
# kronecker product is the zero matrix
return coo_matrix( output_shape )
# expand entries of a into blocks
row = A.row.repeat(B.nnz)
col = A.col.repeat(B.nnz)
data = A.data.repeat(B.nnz)
row *= B.shape[0]
col *= B.shape[1]
# increment block indices
row,col = row.reshape(-1,B.nnz),col.reshape(-1,B.nnz)
row += B.row
col += B.col
row,col = row.reshape(-1),col.reshape(-1)
# compute block entries
data = data.reshape(-1,B.nnz) * B.data
data = data.reshape(-1)
return coo_matrix((data,(row,col)), shape=output_shape).asformat(format)
def kronsum(A, B, format=None):
"""kronecker sum of sparse matrices A and B
Kronecker sum of two sparse matrices is a sum of two Kronecker
products kron(I_n,A) + kron(B,I_m) where A has shape (m,m)
and B has shape (n,n) and I_m and I_n are identity matrices
of shape (m,m) and (n,n) respectively.
Parameters
----------
A
square matrix
B
square matrix
format : string
format of the result (e.g. "csr")
Returns
-------
kronecker sum in a sparse matrix format
Examples
--------
"""
A = coo_matrix(A)
B = coo_matrix(B)
if A.shape[0] != A.shape[1]:
raise ValueError('A is not square')
if B.shape[0] != B.shape[1]:
raise ValueError('B is not square')
dtype = upcast(A.dtype, B.dtype)
L = kron(identity(B.shape[0],dtype=dtype), A, format=format)
R = kron(B, identity(A.shape[0],dtype=dtype), format=format)
return (L+R).asformat(format) #since L + R is not always same format
def hstack(blocks, format=None, dtype=None):
"""Stack sparse matrices horizontally (column wise)
Parameters
----------
blocks
sequence of sparse matrices with compatible shapes
format : string
sparse format of the result (e.g. "csr")
by default an appropriate sparse matrix format is returned.
This choice is subject to change.
Example
-------
>>> from scipy.sparse import coo_matrix, vstack
>>> A = coo_matrix([[1,2],[3,4]])
>>> B = coo_matrix([[5],[6]])
>>> hstack( [A,B] ).todense()
matrix([[1, 2, 5],
[3, 4, 6]])
"""
return bmat([blocks], format=format, dtype=dtype)
def vstack(blocks, format=None, dtype=None):
"""Stack sparse matrices vertically (row wise)
Parameters
----------
blocks
sequence of sparse matrices with compatible shapes
format : string
sparse format of the result (e.g. "csr")
by default an appropriate sparse matrix format is returned.
This choice is subject to change.
Example
-------
>>> from scipy.sparse import coo_matrix, vstack
>>> A = coo_matrix([[1,2],[3,4]])
>>> B = coo_matrix([[5,6]])
>>> vstack( [A,B] ).todense()
matrix([[1, 2],
[3, 4],
[5, 6]])
"""
return bmat([ [b] for b in blocks ], format=format, dtype=dtype)
def bmat(blocks, format=None, dtype=None):
"""Build a sparse matrix from sparse sub-blocks
Parameters
----------
blocks
grid of sparse matrices with compatible shapes
an entry of None implies an all-zero matrix
format : sparse format of the result (e.g. "csr")
by default an appropriate sparse matrix format is returned.
This choice is subject to change.
Example
-------
>>> from scipy.sparse import coo_matrix, bmat
>>> A = coo_matrix([[1,2],[3,4]])
>>> B = coo_matrix([[5],[6]])
>>> C = coo_matrix([[7]])
>>> bmat( [[A,B],[None,C]] ).todense()
matrix([[1, 2, 5],
[3, 4, 6],
[0, 0, 7]])
>>> bmat( [[A,None],[None,C]] ).todense()
matrix([[1, 2, 0],
[3, 4, 0],
[0, 0, 7]])
"""
blocks = np.asarray(blocks, dtype='object')
if np.rank(blocks) != 2:
raise ValueError('blocks must have rank 2')
M,N = blocks.shape
block_mask = np.zeros(blocks.shape, dtype=np.bool)
brow_lengths = np.zeros(blocks.shape[0], dtype=np.intc)
bcol_lengths = np.zeros(blocks.shape[1], dtype=np.intc)
# convert everything to COO format
for i in range(M):
for j in range(N):
if blocks[i,j] is not None:
A = coo_matrix(blocks[i,j])
blocks[i,j] = A
block_mask[i,j] = True
if brow_lengths[i] == 0:
brow_lengths[i] = A.shape[0]
else:
if brow_lengths[i] != A.shape[0]:
raise ValueError('blocks[%d,:] has incompatible row dimensions' % i)
if bcol_lengths[j] == 0:
bcol_lengths[j] = A.shape[1]
else:
if bcol_lengths[j] != A.shape[1]:
raise ValueError('blocks[:,%d] has incompatible column dimensions' % j)
# ensure that at least one value in each row and col is not None
if brow_lengths.min() == 0:
raise ValueError('blocks[%d,:] is all None' % brow_lengths.argmin() )
if bcol_lengths.min() == 0:
raise ValueError('blocks[:,%d] is all None' % bcol_lengths.argmin() )
nnz = sum([ A.nnz for A in blocks[block_mask] ])
if dtype is None:
dtype = upcast( *tuple([A.dtype for A in blocks[block_mask]]) )
row_offsets = np.concatenate(([0], np.cumsum(brow_lengths)))
col_offsets = np.concatenate(([0], np.cumsum(bcol_lengths)))
data = np.empty(nnz, dtype=dtype)
row = np.empty(nnz, dtype=np.intc)
col = np.empty(nnz, dtype=np.intc)
nnz = 0
for i in range(M):
for j in range(N):
if blocks[i,j] is not None:
A = blocks[i,j]
data[nnz:nnz + A.nnz] = A.data
row[nnz:nnz + A.nnz] = A.row
col[nnz:nnz + A.nnz] = A.col
row[nnz:nnz + A.nnz] += row_offsets[i]
col[nnz:nnz + A.nnz] += col_offsets[j]
nnz += A.nnz
shape = (np.sum(brow_lengths), np.sum(bcol_lengths))
return coo_matrix((data, (row, col)), shape=shape).asformat(format)
#################################
# Deprecated functions
################################
__all__ += [ 'speye','spidentity', 'spkron', 'lil_eye', 'lil_diags' ]
spkron = np.deprecate(kron, oldname='spkron', newname='scipy.sparse.kron')
speye = np.deprecate(eye, oldname='speye', newname='scipy.sparse.eye')
spidentity = np.deprecate(identity, oldname='spidentity', newname='scipy.sparse.identity')
def lil_eye((r,c), k=0, dtype='d'):
"""Generate a lil_matrix of dimensions (r,c) with the k-th
diagonal set to 1.
Parameters
----------
r,c : int
row and column-dimensions of the output.
k : int
- diagonal offset. In the output matrix,
- out[m,m+k] == 1 for all m.
dtype : dtype
data-type of the output array.
"""
warn("lil_eye is deprecated." \
"use scipy.sparse.eye(r, c, k, format='lil') instead", \
DeprecationWarning)
return eye(r, c, k, dtype=dtype, format='lil')
#TODO remove this function
def lil_diags(diags, offsets, (m,n), dtype='d'):
"""Generate a lil_matrix with the given diagonals.
Parameters
----------
diags : list of list of values e.g. [[1,2,3],[4,5]]
values to be placed on each indicated diagonal.
offsets : list of ints
diagonal offsets. This indicates the diagonal on which
the given values should be placed.
(r,c) : tuple of ints
row and column dimensions of the output.
dtype : dtype
output data-type.
Example
-------
>>> lil_diags([[1,2,3],[4,5],[6]],[0,1,2],(3,3)).todense()
matrix([[ 1., 4., 6.],
[ 0., 2., 5.],
[ 0., 0., 3.]])
"""
offsets_unsorted = list(offsets)
diags_unsorted = list(diags)
if len(diags) != len(offsets):
raise ValueError("Number of diagonals provided should "
"agree with offsets.")
sort_indices = np.argsort(offsets_unsorted)
diags = [diags_unsorted[k] for k in sort_indices]
offsets = [offsets_unsorted[k] for k in sort_indices]
for i,k in enumerate(offsets):
if len(diags[i]) < m-abs(k):
raise ValueError("Not enough values specified to fill "
"diagonal %s." % k)
out = lil_matrix((m,n),dtype=dtype)
from itertools import izip
for k,diag in izip(offsets,diags):
for ix,c in enumerate(xrange(np.clip(k,0,n),np.clip(m+k,0,n))):
out.rows[c-k].append(c)
out.data[c-k].append(diag[ix])
return out
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