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
|
from numpy.lib import add_newdoc
add_newdoc('scipy.sparse.linalg.dsolve._superlu', 'SuperLU',
"""
LU factorization of a sparse matrix.
Factorization is represented as::
Pr * A * Pc = L * U
To construct these `SuperLU` objects, call the `splu` and `spilu`
functions.
Attributes
----------
shape
nnz
perm_c
perm_r
L
U
Methods
-------
solve
Notes
-----
.. versionadded:: 0.14.0
Examples
--------
The LU decomposition can be used to solve matrix equations. Consider:
>>> import numpy as np
>>> from scipy.sparse import csc_matrix, linalg as sla
>>> A = csc_matrix([[1,2,0,4],[1,0,0,1],[1,0,2,1],[2,2,1,0.]])
This can be solved for a given right-hand side:
>>> lu = sla.splu(A)
>>> b = np.array([1, 2, 3, 4])
>>> x = lu.solve(b)
>>> A.dot(x)
array([ 1., 2., 3., 4.])
The ``lu`` object also contains an explicit representation of the
decomposition. The permutations are represented as mappings of
indices:
>>> lu.perm_r
array([0, 2, 1, 3], dtype=int32)
>>> lu.perm_c
array([2, 0, 1, 3], dtype=int32)
The L and U factors are sparse matrices in CSC format:
>>> lu.L.A
array([[ 1. , 0. , 0. , 0. ],
[ 0. , 1. , 0. , 0. ],
[ 0. , 0. , 1. , 0. ],
[ 1. , 0.5, 0.5, 1. ]])
>>> lu.U.A
array([[ 2., 0., 1., 4.],
[ 0., 2., 1., 1.],
[ 0., 0., 1., 1.],
[ 0., 0., 0., -5.]])
The permutation matrices can be constructed:
>>> Pr = csc_matrix((4, 4))
>>> Pr[lu.perm_r, np.arange(4)] = 1
>>> Pc = csc_matrix((4, 4))
>>> Pc[np.arange(4), lu.perm_c] = 1
We can reassemble the original matrix:
>>> (Pr.T * (lu.L * lu.U) * Pc.T).A
array([[ 1., 2., 0., 4.],
[ 1., 0., 0., 1.],
[ 1., 0., 2., 1.],
[ 2., 2., 1., 0.]])
""")
add_newdoc('scipy.sparse.linalg.dsolve._superlu', 'SuperLU', ('solve',
"""
solve(rhs[, trans])
Solves linear system of equations with one or several right-hand sides.
Parameters
----------
rhs : ndarray, shape (n,) or (n, k)
Right hand side(s) of equation
trans : {'N', 'T', 'H'}, optional
Type of system to solve::
'N': A * x == rhs (default)
'T': A^T * x == rhs
'H': A^H * x == rhs
i.e., normal, transposed, and hermitian conjugate.
Returns
-------
x : ndarray, shape ``rhs.shape``
Solution vector(s)
"""))
add_newdoc('scipy.sparse.linalg.dsolve._superlu', 'SuperLU', ('L',
"""
Lower triangular factor with unit diagonal as a
`scipy.sparse.csc_matrix`.
.. versionadded:: 0.14.0
"""))
add_newdoc('scipy.sparse.linalg.dsolve._superlu', 'SuperLU', ('U',
"""
Upper triangular factor as a `scipy.sparse.csc_matrix`.
.. versionadded:: 0.14.0
"""))
add_newdoc('scipy.sparse.linalg.dsolve._superlu', 'SuperLU', ('shape',
"""
Shape of the original matrix as a tuple of ints.
"""))
add_newdoc('scipy.sparse.linalg.dsolve._superlu', 'SuperLU', ('nnz',
"""
Number of nonzero elements in the matrix.
"""))
add_newdoc('scipy.sparse.linalg.dsolve._superlu', 'SuperLU', ('perm_c',
"""
Permutation Pc represented as an array of indices.
The column permutation matrix can be reconstructed via:
>>> Pc = np.zeros((n, n))
>>> Pc[np.arange(n), perm_c] = 1
"""))
add_newdoc('scipy.sparse.linalg.dsolve._superlu', 'SuperLU', ('perm_r',
"""
Permutation Pr represented as an array of indices.
The row permutation matrix can be reconstructed via:
>>> Pr = np.zeros((n, n))
>>> Pr[perm_r, np.arange(n)] = 1
"""))
|
