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
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<HTML>
<HEAD>
<TITLE>LU LU Decomposition for Matrices
</TITLE>
</HEAD>
<BODY>
<H2>LU LU Decomposition for Matrices
</H2>
<P>
Section: <A HREF=sec_transforms.html> Transforms/Decompositions </A>
<H3>Usage</H3>
Computes the LU decomposition for a matrix. The form of the
command depends on the type of the argument. For full (non-sparse)
matrices, the primary form for <code>lu</code> is
<PRE>
[L,U,P] = lu(A),
</PRE>
<P>
where <code>L</code> is lower triangular, <code>U</code> is upper triangular, and
<code>P</code> is a permutation matrix such that <code>L*U = P*A</code>. The second form is
<PRE>
[V,U] = lu(A),
</PRE>
<P>
where <code>V</code> is <code>P'*L</code> (a row-permuted lower triangular matrix),
and <code>U</code> is upper triangular. For sparse, square matrices,
the LU decomposition has the following form:
<PRE>
[L,U,P,Q,R] = lu(A),
</PRE>
<P>
where <code>A</code> is a sparse matrix of either <code>double</code> or <code>dcomplex</code> type.
The matrices are such that <code>L*U=P*R*A*Q</code>, where <code>L</code> is a lower triangular
matrix, <code>U</code> is upper triangular, <code>P</code> and <code>Q</code> are permutation vectors
and <code>R</code> is a diagonal matrix of row scaling factors. The decomposition
is computed using UMFPACK for sparse matrices, and LAPACK for dense
matrices.
<H3>Example</H3>
First, we compute the LU decomposition of a dense matrix.
<PRE>
--> a = float([1,2,3;4,5,8;10,12,3])
a =
1 2 3
4 5 8
10 12 3
--> [l,u,p] = lu(a)
l =
1.0000 0 0
0.1000 1.0000 0
0.4000 0.2500 1.0000
u =
10.0000 12.0000 3.0000
0 0.8000 2.7000
0 0 6.1250
p =
0 0 1
1 0 0
0 1 0
--> l*u
ans =
10 12 3
1 2 3
4 5 8
--> p*a
ans =
10 12 3
1 2 3
4 5 8
</PRE>
<P>
Now we repeat the exercise with a sparse matrix, and demonstrate
the use of the permutation vectors.
<PRE>
--> a = sparse([1,0,0,4;3,2,0,0;0,0,0,1;4,3,2,4])
a =
1 1 1
2 1 3
4 1 4
2 2 2
4 2 3
4 3 2
1 4 4
3 4 1
4 4 4
--> [l,u,p,q,r] = lu(a)
l =
1 1 1
2 2 1
3 3 1
4 4 1
u =
1 1 0.153846
1 2 0.230769
2 2 0.4
1 3 0.307692
2 3 0.6
3 3 0.2
1 4 0.307692
3 4 0.8
4 4 1
p =
4
2
1
3
q =
3
2
1
4
r =
1 1 0.2
2 2 0.2
3 3 1
4 4 0.0769231
--> full(l*a)
ans =
1 0 0 4
3 2 0 0
0 0 0 1
4 3 2 4
--> b = r*a
b =
1 1 0.2
2 1 0.6
3 1 0
4 1 0.307692
1 2 0
2 2 0.4
3 2 0
4 2 0.230769
1 3 0
2 3 0
3 3 0
4 3 0.153846
1 4 0.8
2 4 0
3 4 1
4 4 0.307692
--> full(b(p,q))
ans =
0.1538 0.2308 0.3077 0.3077
0 0.4000 0.6000 0
0 0 0.2000 0.8000
0 0 0 1.0000
</PRE>
<P>
</BODY>
</HTML>
|
