1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
|
import numpy as np
from scipy.sparse.sputils import isshape
from scipy.sparse import isspmatrix
__all__ = ['LinearOperator', 'aslinearoperator']
class LinearOperator:
"""Common interface for performing matrix vector products
Many iterative methods (e.g. cg, gmres) do not need to know the
individual entries of a matrix to solve a linear system A*x=b.
Such solvers only require the computation of matrix vector
products, A*v where v is a dense vector. This class serves as
an abstract interface between iterative solvers and matrix-like
objects.
Parameters
----------
shape : tuple
Matrix dimensions (M,N)
matvec : callable f(v)
Returns returns A * v.
Optional Parameters
-------------------
rmatvec : callable f(v)
Returns A^H * v, where A^H is the conjugate transpose of A.
matmat : callable f(V)
Returns A * V, where V is a dense matrix with dimensions (N,K).
dtype : dtype
Data type of the matrix.
See Also
--------
aslinearoperator : Construct LinearOperators
Notes
-----
The user-defined matvec() function must properly handle the case
where v has shape (N,) as well as the (N,1) case. The shape of
the return type is handled internally by LinearOperator.
Examples
--------
>>> from scipy.sparse.linalg import LinearOperator
>>> from scipy import *
>>> def mv(v):
... return array([ 2*v[0], 3*v[1]])
...
>>> A = LinearOperator( (2,2), matvec=mv )
>>> A
<2x2 LinearOperator with unspecified dtype>
>>> A.matvec( ones(2) )
array([ 2., 3.])
>>> A * ones(2)
array([ 2., 3.])
"""
def __init__(self, shape, matvec, rmatvec=None, matmat=None, dtype=None):
shape = tuple(shape)
if not isshape(shape):
raise ValueError('invalid shape')
self.shape = shape
self._matvec = matvec
if rmatvec is None:
def rmatvec(v):
raise NotImplementedError('rmatvec is not defined')
self.rmatvec = rmatvec
else:
self.rmatvec = rmatvec
if matmat is not None:
# matvec each column of V
self._matmat = matmat
if dtype is not None:
self.dtype = np.dtype(dtype)
def _matmat(self, X):
"""Default matrix-matrix multiplication handler. Falls back on
the user-defined matvec() routine, which is always provided.
"""
return np.hstack( [ self.matvec(col.reshape(-1,1)) for col in X.T ] )
def matvec(self, x):
"""Matrix-vector multiplication
Performs the operation y=A*x where A is an MxN linear
operator and x is a column vector or rank-1 array.
Parameters
----------
x : {matrix, ndarray}
An array with shape (N,) or (N,1).
Returns
-------
y : {matrix, ndarray}
A matrix or ndarray with shape (M,) or (M,1) depending
on the type and shape of the x argument.
Notes
-----
This matvec wraps the user-specified matvec routine to ensure that
y has the correct shape and type.
"""
x = np.asanyarray(x)
M,N = self.shape
if x.shape != (N,) and x.shape != (N,1):
raise ValueError('dimension mismatch')
y = self._matvec(x)
if isinstance(x, np.matrix):
y = np.asmatrix(y)
else:
y = np.asarray(y)
if x.ndim == 1:
y = y.reshape(M)
elif x.ndim == 2:
y = y.reshape(M,1)
else:
raise ValueError('invalid shape returned by user-defined matvec()')
return y
def matmat(self, X):
"""Matrix-matrix multiplication
Performs the operation y=A*X where A is an MxN linear
operator and X dense N*K matrix or ndarray.
Parameters
----------
X : {matrix, ndarray}
An array with shape (N,K).
Returns
-------
Y : {matrix, ndarray}
A matrix or ndarray with shape (M,K) depending on
the type of the X argument.
Notes
-----
This matmat wraps any user-specified matmat routine to ensure that
y has the correct type.
"""
X = np.asanyarray(X)
if X.ndim != 2:
raise ValueError('expected rank-2 ndarray or matrix')
M,N = self.shape
if X.shape[0] != N:
raise ValueError('dimension mismatch')
Y = self._matmat(X)
if isinstance(Y, np.matrix):
Y = np.asmatrix(Y)
return Y
def __mul__(self,x):
x = np.asarray(x)
if x.ndim == 1 or x.ndim == 2 and x.shape[1] == 1:
return self.matvec(x)
elif x.ndim == 2:
return self.matmat(x)
else:
raise ValueError('expected rank-1 or rank-2 array or matrix')
def __repr__(self):
M,N = self.shape
if hasattr(self,'dtype'):
dt = 'dtype=' + str(self.dtype)
else:
dt = 'unspecified dtype'
return '<%dx%d LinearOperator with %s>' % (M,N,dt)
def aslinearoperator(A):
"""Return A as a LinearOperator.
'A' may be any of the following types:
- ndarray
- matrix
- sparse matrix (e.g. csr_matrix, lil_matrix, etc.)
- LinearOperator
- An object with .shape and .matvec attributes
See the LinearOperator documentation for additonal information.
Examples
--------
>>> from scipy import matrix
>>> M = matrix( [[1,2,3],[4,5,6]], dtype='int32' )
>>> aslinearoperator( M )
<2x3 LinearOperator with dtype=int32>
"""
if isinstance(A, LinearOperator):
return A
elif isinstance(A, np.ndarray) or isinstance(A, np.matrix):
if A.ndim > 2:
raise ValueError('array must have rank <= 2')
A = np.atleast_2d(np.asarray(A))
def matvec(v):
return np.dot(A, v)
def rmatvec(v):
return np.dot(A.conj().transpose(), v)
def matmat(V):
return np.dot(A, V)
return LinearOperator(A.shape, matvec, rmatvec=rmatvec,
matmat=matmat, dtype=A.dtype)
elif isspmatrix(A):
def matvec(v):
return A * v
def rmatvec(v):
return A.conj().transpose() * v
def matmat(V):
return A * V
return LinearOperator(A.shape, matvec, rmatvec=rmatvec,
matmat=matmat, dtype=A.dtype)
else:
if hasattr(A, 'shape') and hasattr(A, 'matvec'):
rmatvec = None
dtype = None
if hasattr(A, 'rmatvec'):
rmatvec = A.rmatvec
if hasattr(A, 'dtype'):
dtype = A.dtype
return LinearOperator(A.shape, A.matvec,
rmatvec=rmatvec, dtype=dtype)
else:
raise TypeError('type not understood')
|
