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
|
<HTML>
<HEAD><TITLE>MB04LD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04LD">MB04LD</A></H2>
<H3>
LQ factorization of a special structured block matrix
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To calculate an LQ factorization of the first block row and apply
the orthogonal transformations (from the right) also to the second
block row of a structured matrix, as follows
_
[ L A ] [ L 0 ]
[ ]*Q = [ ]
[ 0 B ] [ C D ]
_
where L and L are lower triangular. The matrix A can be full or
lower trapezoidal/triangular. The problem structure is exploited.
This computation is useful, for instance, in combined measurement
and time update of one iteration of the Kalman filter (square
root covariance filter).
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB04LD( UPLO, N, M, P, L, LDL, A, LDA, B, LDB, C, LDC,
$ TAU, DWORK )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDB, LDC, LDL, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ L(LDL,*), TAU(*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
UPLO CHARACTER*1
Indicates if the matrix A is or not triangular as follows:
= 'L': Matrix A is lower trapezoidal/triangular;
= 'F': Matrix A is full.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER _
The order of the matrices L and L. N >= 0.
M (input) INTEGER
The number of columns of the matrices A, B and D. M >= 0.
P (input) INTEGER
The number of rows of the matrices B, C and D. P >= 0.
L (input/output) DOUBLE PRECISION array, dimension (LDL,N)
On entry, the leading N-by-N lower triangular part of this
array must contain the lower triangular matrix L.
On exit, the leading N-by-N lower triangular part of this
_
array contains the lower triangular matrix L.
The strict upper triangular part of this array is not
referenced.
LDL INTEGER
The leading dimension of array L. LDL >= MAX(1,N).
A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
On entry, if UPLO = 'F', the leading N-by-M part of this
array must contain the matrix A. If UPLO = 'L', the
leading N-by-MIN(N,M) part of this array must contain the
lower trapezoidal (lower triangular if N <= M) matrix A,
and the elements above the diagonal are not referenced.
On exit, the leading N-by-M part (lower trapezoidal or
triangular, if UPLO = 'L') of this array contains the
trailing components (the vectors v, see Method) of the
elementary reflectors used in the factorization.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading P-by-M part of this array must
contain the matrix B.
On exit, the leading P-by-M part of this array contains
the computed matrix D.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,P).
C (output) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array contains the
computed matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
TAU (output) DOUBLE PRECISION array, dimension (N)
The scalar factors of the elementary reflectors used.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (N)
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The routine uses N Householder transformations exploiting the zero
pattern of the block matrix. A Householder matrix has the form
( 1 ),
H = I - tau *u *u', u = ( v )
i i i i i ( i)
where v is an M-vector, if UPLO = 'F', or an min(i,M)-vector, if
i
UPLO = 'L'. The components of v are stored in the i-th row of A,
i
and tau is stored in TAU(i).
i
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm is backward stable.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
<HR>
<p>
<A HREF=..\libindex.html><B>Return to index</B></A></BODY>
</HTML>
|
