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
|
<HTML>
<HEAD><TITLE>MB04OD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04OD">MB04OD</A></H2>
<H3>
QR factorization of a special structured block matrix (variant)
</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 a QR factorization of the first block column and
apply the orthogonal transformations (from the left) also to the
second block column of a structured matrix, as follows
_ _
[ R B ] [ R B ]
Q' * [ ] = [ _ ]
[ A C ] [ 0 C ]
_
where R and R are upper triangular. The matrix A can be full or
upper trapezoidal/triangular. The problem structure is exploited.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB04OD( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C, LDC,
$ TAU, DWORK )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDB, LDC, LDR, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ R(LDR,*), 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:
= 'U': Matrix A is upper trapezoidal/triangular;
= 'F': Matrix A is full.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER _
The order of the matrices R and R. N >= 0.
M (input) INTEGER
The number of columns of the matrices B and C. M >= 0.
P (input) INTEGER
The number of rows of the matrices A and C. P >= 0.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix R.
On exit, the leading N-by-N upper triangular part of this
_
array contains the upper triangular matrix R.
The strict lower triangular part of this array is not
referenced.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, if UPLO = 'F', the leading P-by-N part of this
array must contain the matrix A. If UPLO = 'U', the
leading MIN(P,N)-by-N part of this array must contain the
upper trapezoidal (upper triangular if P >= N) matrix A,
and the elements below the diagonal are not referenced.
On exit, the leading P-by-N part (upper trapezoidal or
triangular, if UPLO = 'U') 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,P).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
On entry, the leading N-by-M part of this array must
contain the matrix B.
On exit, the leading N-by-M part of this array contains
_
the computed matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,M)
On entry, the leading P-by-M part of this array must
contain the matrix C.
On exit, the leading P-by-M 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 (MAX(N-1,M))
</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 a P-vector, if UPLO = 'F', or a min(i,P)-vector, if
i
UPLO = 'U'. The components of v are stored in the i-th column
i
of A, and tau is stored in TAU(i).
i
In-line code for applying Householder transformations is used
whenever possible (see MB04OY routine).
</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>
* MB04OD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER (ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER MMAX, NMAX, PMAX
PARAMETER ( MMAX = 20, NMAX = 20, PMAX = 20 )
INTEGER LDA, LDB, LDC, LDR
PARAMETER ( LDA = PMAX, LDB = NMAX, LDC = PMAX,
$ LDR = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX-1,MMAX ) )
* .. Local Scalars ..
CHARACTER*1 UPLO
INTEGER I, J, M, N, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,MMAX),
$ DWORK(LDWORK), R(LDR,NMAX), TAU(NMAX)
* .. External Subroutines ..
EXTERNAL MB04OD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, M, P, UPLO
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) M
ELSE
IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) P
ELSE
READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,M ), I = 1,P )
* Compute and apply QR factorization.
CALL MB04OD( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C,
$ LDC, TAU, DWORK )
*
WRITE ( NOUT, FMT = 99997 )
DO 40 I = 1, N
DO 20 J = 1, I-1
R(I,J) = ZERO
20 CONTINUE
WRITE ( NOUT, FMT = 99996 ) ( R(I,J), J = 1,N )
40 CONTINUE
IF ( M.GT.0 ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 60 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,M )
60 CONTINUE
IF ( P.GT.0 ) THEN
WRITE ( NOUT, FMT = 99994 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99996 ) ( C(I,J), J = 1,M )
80 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' MB04OD EXAMPLE PROGRAM RESULTS',/1X)
99997 FORMAT (' The updated matrix R is ')
99996 FORMAT (20(1X,F10.4))
99995 FORMAT (' The updated matrix B is ')
99994 FORMAT (' The updated matrix C is ')
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' M is out of range.',/' M = ',I5)
99991 FORMAT (/' P is out of range.',/' P = ',I5)
END
</PRE>
<B>Program Data</B>
<PRE>
MB04OD EXAMPLE PROGRAM DATA
3 2 2 F
3. 2. 1.
0. 2. 1.
0. 0. 1.
2. 3. 1.
4. 6. 5.
3. 2.
1. 3.
3. 2.
1. 3.
3. 2.
</PRE>
<B>Program Results</B>
<PRE>
MB04OD EXAMPLE PROGRAM RESULTS
The updated matrix R is
-5.3852 -6.6850 -4.6424
0.0000 -2.8828 -2.0694
0.0000 0.0000 -1.7793
The updated matrix B is
-4.2710 -3.7139
-0.1555 -2.1411
-1.6021 0.9398
The updated matrix C is
0.5850 1.0141
-2.7974 -3.1162
</PRE>
<HR>
<p>
<A HREF=..\libindex.html><B>Return to index</B></A></BODY>
</HTML>
|
