/
linsolve.py
567 lines (467 loc) · 18.7 KB
/
linsolve.py
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
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
from __future__ import division, print_function, absolute_import
from warnings import warn
import numpy as np
from numpy import asarray
from scipy.sparse import (isspmatrix_csc, isspmatrix_csr, isspmatrix,
SparseEfficiencyWarning, csc_matrix, csr_matrix)
from scipy.linalg import LinAlgError
from . import _superlu
noScikit = False
try:
import scikits.umfpack as umfpack
except ImportError:
noScikit = True
useUmfpack = not noScikit
__all__ = ['use_solver', 'spsolve', 'splu', 'spilu', 'factorized',
'MatrixRankWarning', 'spsolve_triangular']
class MatrixRankWarning(UserWarning):
pass
def use_solver(**kwargs):
"""
Select default sparse direct solver to be used.
Parameters
----------
useUmfpack : bool, optional
Use UMFPACK over SuperLU. Has effect only if scikits.umfpack is
installed. Default: True
assumeSortedIndices : bool, optional
Allow UMFPACK to skip the step of sorting indices for a CSR/CSC matrix.
Has effect only if useUmfpack is True and scikits.umfpack is installed.
Default: False
Notes
-----
The default sparse solver is umfpack when available
(scikits.umfpack is installed). This can be changed by passing
useUmfpack = False, which then causes the always present SuperLU
based solver to be used.
Umfpack requires a CSR/CSC matrix to have sorted column/row indices. If
sure that the matrix fulfills this, pass ``assumeSortedIndices=True``
to gain some speed.
"""
if 'useUmfpack' in kwargs:
globals()['useUmfpack'] = kwargs['useUmfpack']
if useUmfpack and 'assumeSortedIndices' in kwargs:
umfpack.configure(assumeSortedIndices=kwargs['assumeSortedIndices'])
def _get_umf_family(A):
"""Get umfpack family string given the sparse matrix dtype."""
_families = {
(np.float64, np.int32): 'di',
(np.complex128, np.int32): 'zi',
(np.float64, np.int64): 'dl',
(np.complex128, np.int64): 'zl'
}
f_type = np.sctypeDict[A.dtype.name]
i_type = np.sctypeDict[A.indices.dtype.name]
try:
family = _families[(f_type, i_type)]
except KeyError:
msg = 'only float64 or complex128 matrices with int32 or int64' \
' indices are supported! (got: matrix: %s, indices: %s)' \
% (f_type, i_type)
raise ValueError(msg)
return family
def spsolve(A, b, permc_spec=None, use_umfpack=True):
"""Solve the sparse linear system Ax=b, where b may be a vector or a matrix.
Parameters
----------
A : ndarray or sparse matrix
The square matrix A will be converted into CSC or CSR form
b : ndarray or sparse matrix
The matrix or vector representing the right hand side of the equation.
If a vector, b.shape must be (n,) or (n, 1).
permc_spec : str, optional
How to permute the columns of the matrix for sparsity preservation.
(default: 'COLAMD')
- ``NATURAL``: natural ordering.
- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
- ``COLAMD``: approximate minimum degree column ordering
use_umfpack : bool, optional
if True (default) then use umfpack for the solution. This is
only referenced if b is a vector and ``scikit-umfpack`` is installed.
Returns
-------
x : ndarray or sparse matrix
the solution of the sparse linear equation.
If b is a vector, then x is a vector of size A.shape[1]
If b is a matrix, then x is a matrix of size (A.shape[1], b.shape[1])
Notes
-----
For solving the matrix expression AX = B, this solver assumes the resulting
matrix X is sparse, as is often the case for very sparse inputs. If the
resulting X is dense, the construction of this sparse result will be
relatively expensive. In that case, consider converting A to a dense
matrix and using scipy.linalg.solve or its variants.
Examples
--------
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import spsolve
>>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float)
>>> B = csc_matrix([[2, 0], [-1, 0], [2, 0]], dtype=float)
>>> x = spsolve(A, B)
>>> np.allclose(A.dot(x).todense(), B.todense())
True
"""
if not (isspmatrix_csc(A) or isspmatrix_csr(A)):
A = csc_matrix(A)
warn('spsolve requires A be CSC or CSR matrix format',
SparseEfficiencyWarning)
# b is a vector only if b have shape (n,) or (n, 1)
b_is_sparse = isspmatrix(b)
if not b_is_sparse:
b = asarray(b)
b_is_vector = ((b.ndim == 1) or (b.ndim == 2 and b.shape[1] == 1))
A.sort_indices()
A = A.asfptype() # upcast to a floating point format
result_dtype = np.promote_types(A.dtype, b.dtype)
if A.dtype != result_dtype:
A = A.astype(result_dtype)
if b.dtype != result_dtype:
b = b.astype(result_dtype)
# validate input shapes
M, N = A.shape
if (M != N):
raise ValueError("matrix must be square (has shape %s)" % ((M, N),))
if M != b.shape[0]:
raise ValueError("matrix - rhs dimension mismatch (%s - %s)"
% (A.shape, b.shape[0]))
use_umfpack = use_umfpack and useUmfpack
if b_is_vector and use_umfpack:
if b_is_sparse:
b_vec = b.toarray()
else:
b_vec = b
b_vec = asarray(b_vec, dtype=A.dtype).ravel()
if noScikit:
raise RuntimeError('Scikits.umfpack not installed.')
if A.dtype.char not in 'dD':
raise ValueError("convert matrix data to double, please, using"
" .astype(), or set linsolve.useUmfpack = False")
umf = umfpack.UmfpackContext(_get_umf_family(A))
x = umf.linsolve(umfpack.UMFPACK_A, A, b_vec,
autoTranspose=True)
else:
if b_is_vector and b_is_sparse:
b = b.toarray()
b_is_sparse = False
if not b_is_sparse:
if isspmatrix_csc(A):
flag = 1 # CSC format
else:
flag = 0 # CSR format
options = dict(ColPerm=permc_spec)
x, info = _superlu.gssv(N, A.nnz, A.data, A.indices, A.indptr,
b, flag, options=options)
if info != 0:
warn("Matrix is exactly singular", MatrixRankWarning)
x.fill(np.nan)
if b_is_vector:
x = x.ravel()
else:
# b is sparse
Afactsolve = factorized(A)
if not isspmatrix_csc(b):
warn('spsolve is more efficient when sparse b '
'is in the CSC matrix format', SparseEfficiencyWarning)
b = csc_matrix(b)
# Create a sparse output matrix by repeatedly applying
# the sparse factorization to solve columns of b.
data_segs = []
row_segs = []
col_segs = []
for j in range(b.shape[1]):
bj = b[:, j].A.ravel()
xj = Afactsolve(bj)
w = np.flatnonzero(xj)
segment_length = w.shape[0]
row_segs.append(w)
col_segs.append(np.ones(segment_length, dtype=int)*j)
data_segs.append(np.asarray(xj[w], dtype=A.dtype))
sparse_data = np.concatenate(data_segs)
sparse_row = np.concatenate(row_segs)
sparse_col = np.concatenate(col_segs)
x = A.__class__((sparse_data, (sparse_row, sparse_col)),
shape=b.shape, dtype=A.dtype)
return x
def splu(A, permc_spec=None, diag_pivot_thresh=None,
relax=None, panel_size=None, options=dict()):
"""
Compute the LU decomposition of a sparse, square matrix.
Parameters
----------
A : sparse matrix
Sparse matrix to factorize. Should be in CSR or CSC format.
permc_spec : str, optional
How to permute the columns of the matrix for sparsity preservation.
(default: 'COLAMD')
- ``NATURAL``: natural ordering.
- ``MMD_ATA``: minimum degree ordering on the structure of A^T A.
- ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A.
- ``COLAMD``: approximate minimum degree column ordering
diag_pivot_thresh : float, optional
Threshold used for a diagonal entry to be an acceptable pivot.
See SuperLU user's guide for details [1]_
relax : int, optional
Expert option for customizing the degree of relaxing supernodes.
See SuperLU user's guide for details [1]_
panel_size : int, optional
Expert option for customizing the panel size.
See SuperLU user's guide for details [1]_
options : dict, optional
Dictionary containing additional expert options to SuperLU.
See SuperLU user guide [1]_ (section 2.4 on the 'Options' argument)
for more details. For example, you can specify
``options=dict(Equil=False, IterRefine='SINGLE'))``
to turn equilibration off and perform a single iterative refinement.
Returns
-------
invA : scipy.sparse.linalg.SuperLU
Object, which has a ``solve`` method.
See also
--------
spilu : incomplete LU decomposition
Notes
-----
This function uses the SuperLU library.
References
----------
.. [1] SuperLU http://crd.lbl.gov/~xiaoye/SuperLU/
Examples
--------
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import splu
>>> A = csc_matrix([[1., 0., 0.], [5., 0., 2.], [0., -1., 0.]], dtype=float)
>>> B = splu(A)
>>> x = np.array([1., 2., 3.], dtype=float)
>>> B.solve(x)
array([ 1. , -3. , -1.5])
>>> A.dot(B.solve(x))
array([ 1., 2., 3.])
>>> B.solve(A.dot(x))
array([ 1., 2., 3.])
"""
if not isspmatrix_csc(A):
A = csc_matrix(A)
warn('splu requires CSC matrix format', SparseEfficiencyWarning)
A.sort_indices()
A = A.asfptype() # upcast to a floating point format
M, N = A.shape
if (M != N):
raise ValueError("can only factor square matrices") # is this true?
_options = dict(DiagPivotThresh=diag_pivot_thresh, ColPerm=permc_spec,
PanelSize=panel_size, Relax=relax)
if options is not None:
_options.update(options)
return _superlu.gstrf(N, A.nnz, A.data, A.indices, A.indptr,
ilu=False, options=_options)
def spilu(A, drop_tol=None, fill_factor=None, drop_rule=None, permc_spec=None,
diag_pivot_thresh=None, relax=None, panel_size=None, options=None):
"""
Compute an incomplete LU decomposition for a sparse, square matrix.
The resulting object is an approximation to the inverse of `A`.
Parameters
----------
A : (N, N) array_like
Sparse matrix to factorize
drop_tol : float, optional
Drop tolerance (0 <= tol <= 1) for an incomplete LU decomposition.
(default: 1e-4)
fill_factor : float, optional
Specifies the fill ratio upper bound (>= 1.0) for ILU. (default: 10)
drop_rule : str, optional
Comma-separated string of drop rules to use.
Available rules: ``basic``, ``prows``, ``column``, ``area``,
``secondary``, ``dynamic``, ``interp``. (Default: ``basic,area``)
See SuperLU documentation for details.
Remaining other options
Same as for `splu`
Returns
-------
invA_approx : scipy.sparse.linalg.SuperLU
Object, which has a ``solve`` method.
See also
--------
splu : complete LU decomposition
Notes
-----
To improve the better approximation to the inverse, you may need to
increase `fill_factor` AND decrease `drop_tol`.
This function uses the SuperLU library.
Examples
--------
>>> from scipy.sparse import csc_matrix
>>> from scipy.sparse.linalg import spilu
>>> A = csc_matrix([[1., 0., 0.], [5., 0., 2.], [0., -1., 0.]], dtype=float)
>>> B = spilu(A)
>>> x = np.array([1., 2., 3.], dtype=float)
>>> B.solve(x)
array([ 1. , -3. , -1.5])
>>> A.dot(B.solve(x))
array([ 1., 2., 3.])
>>> B.solve(A.dot(x))
array([ 1., 2., 3.])
"""
if not isspmatrix_csc(A):
A = csc_matrix(A)
warn('splu requires CSC matrix format', SparseEfficiencyWarning)
A.sort_indices()
A = A.asfptype() # upcast to a floating point format
M, N = A.shape
if (M != N):
raise ValueError("can only factor square matrices") # is this true?
_options = dict(ILU_DropRule=drop_rule, ILU_DropTol=drop_tol,
ILU_FillFactor=fill_factor,
DiagPivotThresh=diag_pivot_thresh, ColPerm=permc_spec,
PanelSize=panel_size, Relax=relax)
if options is not None:
_options.update(options)
return _superlu.gstrf(N, A.nnz, A.data, A.indices, A.indptr,
ilu=True, options=_options)
def factorized(A):
"""
Return a function for solving a sparse linear system, with A pre-factorized.
Parameters
----------
A : (N, N) array_like
Input.
Returns
-------
solve : callable
To solve the linear system of equations given in `A`, the `solve`
callable should be passed an ndarray of shape (N,).
Examples
--------
>>> from scipy.sparse.linalg import factorized
>>> A = np.array([[ 3. , 2. , -1. ],
... [ 2. , -2. , 4. ],
... [-1. , 0.5, -1. ]])
>>> solve = factorized(A) # Makes LU decomposition.
>>> rhs1 = np.array([1, -2, 0])
>>> solve(rhs1) # Uses the LU factors.
array([ 1., -2., -2.])
"""
if useUmfpack:
if noScikit:
raise RuntimeError('Scikits.umfpack not installed.')
if not isspmatrix_csc(A):
A = csc_matrix(A)
warn('splu requires CSC matrix format', SparseEfficiencyWarning)
A = A.asfptype() # upcast to a floating point format
if A.dtype.char not in 'dD':
raise ValueError("convert matrix data to double, please, using"
" .astype(), or set linsolve.useUmfpack = False")
umf = umfpack.UmfpackContext(_get_umf_family(A))
# Make LU decomposition.
umf.numeric(A)
def solve(b):
return umf.solve(umfpack.UMFPACK_A, A, b, autoTranspose=True)
return solve
else:
return splu(A).solve
def spsolve_triangular(A, b, lower=True, overwrite_A=False, overwrite_b=False):
"""
Solve the equation `A x = b` for `x`, assuming A is a triangular matrix.
Parameters
----------
A : (M, M) sparse matrix
A sparse square triangular matrix. Should be in CSR format.
b : (M,) or (M, N) array_like
Right-hand side matrix in `A x = b`
lower : bool, optional
Whether `A` is a lower or upper triangular matrix.
Default is lower triangular matrix.
overwrite_A : bool, optional
Allow changing `A`. The indices of `A` are going to be sorted and zero
entries are going to be removed.
Enabling gives a performance gain. Default is False.
overwrite_b : bool, optional
Allow overwriting data in `b`.
Enabling gives a performance gain. Default is False.
If `overwrite_b` is True, it should be ensured that
`b` has an appropriate dtype to be able to store the result.
Returns
-------
x : (M,) or (M, N) ndarray
Solution to the system `A x = b`. Shape of return matches shape of `b`.
Raises
------
LinAlgError
If `A` is singular or not triangular.
ValueError
If shape of `A` or shape of `b` do not match the requirements.
Notes
-----
.. versionadded:: 0.19.0
Examples
--------
>>> from scipy.sparse import csr_matrix
>>> from scipy.sparse.linalg import spsolve_triangular
>>> A = csr_matrix([[3, 0, 0], [1, -1, 0], [2, 0, 1]], dtype=float)
>>> B = np.array([[2, 0], [-1, 0], [2, 0]], dtype=float)
>>> x = spsolve_triangular(A, B)
>>> np.allclose(A.dot(x), B)
True
"""
# Check the input for correct type and format.
if not isspmatrix_csr(A):
warn('CSR matrix format is required. Converting to CSR matrix.',
SparseEfficiencyWarning)
A = csr_matrix(A)
elif not overwrite_A:
A = A.copy()
if A.shape[0] != A.shape[1]:
raise ValueError(
'A must be a square matrix but its shape is {}.'.format(A.shape))
A.eliminate_zeros()
A.sort_indices()
b = np.asanyarray(b)
if b.ndim not in [1, 2]:
raise ValueError(
'b must have 1 or 2 dims but its shape is {}.'.format(b.shape))
if A.shape[0] != b.shape[0]:
raise ValueError(
'The size of the dimensions of A must be equal to '
'the size of the first dimension of b but the shape of A is '
'{} and the shape of b is {}.'.format(A.shape, b.shape))
# Init x as (a copy of) b.
x_dtype = np.result_type(A.data, b, np.float)
if overwrite_b:
if np.can_cast(b.dtype, x_dtype, casting='same_kind'):
x = b
else:
raise ValueError(
'Cannot overwrite b (dtype {}) with result '
'of type {}.'.format(b.dtype, x_dtype))
else:
x = b.astype(x_dtype, copy=True)
# Choose forward or backward order.
if lower:
row_indices = range(len(b))
else:
row_indices = range(len(b) - 1, -1, -1)
# Fill x iteratively.
for i in row_indices:
# Get indices for i-th row.
indptr_start = A.indptr[i]
indptr_stop = A.indptr[i + 1]
if lower:
A_diagonal_index_row_i = indptr_stop - 1
A_off_diagonal_indices_row_i = slice(indptr_start, indptr_stop - 1)
else:
A_diagonal_index_row_i = indptr_start
A_off_diagonal_indices_row_i = slice(indptr_start + 1, indptr_stop)
# Check regularity and triangularity of A.
if indptr_stop <= indptr_start or A.indices[A_diagonal_index_row_i] < i:
raise LinAlgError(
'A is singular: diagonal {} is zero.'.format(i))
if A.indices[A_diagonal_index_row_i] > i:
raise LinAlgError(
'A is not triangular: A[{}, {}] is nonzero.'
''.format(i, A.indices[A_diagonal_index_row_i]))
# Incorporate off-diagonal entries.
A_column_indices_in_row_i = A.indices[A_off_diagonal_indices_row_i]
A_values_in_row_i = A.data[A_off_diagonal_indices_row_i]
x[i] -= np.dot(x[A_column_indices_in_row_i].T, A_values_in_row_i)
# Compute i-th entry of x.
x[i] /= A.data[A_diagonal_index_row_i]
return x