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
|
<HTML>
<HEAD><TITLE>MB02SD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB02SD">MB02SD</A></H2>
<H3>
LU factorization of an upper Hessenberg 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 compute an LU factorization of an n-by-n upper Hessenberg
matrix H using partial pivoting with row interchanges.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB02SD( N, H, LDH, IPIV, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDH, N
C .. Array Arguments ..
INTEGER IPIV(*)
DOUBLE PRECISION H(LDH,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the matrix H. N >= 0.
H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the n-by-n upper Hessenberg matrix to be
factored.
On exit, the factors L and U from the factorization
H = P*L*U; the unit diagonal elements of L are not stored,
and L is lower bidiagonal.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
IPIV (output) INTEGER array, dimension (N)
The pivot indices; for 1 <= i <= N, row i of the matrix
was interchanged with row IPIV(i).
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, U(i,i) is exactly zero. The
factorization has been completed, but the factor U
is exactly singular, and division by zero will occur
if it is used to solve a system of equations.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The factorization has the form
H = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (and one nonzero subdiagonal), and U is upper
triangular.
This is the right-looking Level 1 BLAS version of the algorithm
(adapted after DGETF2).
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
-
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 2
The algorithm requires 0( N ) operations.
</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>
* MB02SD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, NRHMAX
PARAMETER ( NMAX = 20, NRHMAX = 20 )
INTEGER LDB, LDH
PARAMETER ( LDB = NMAX, LDH = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX )
* .. Local Scalars ..
DOUBLE PRECISION HNORM, RCOND
INTEGER I, INFO, INFO1, J, N, NRHS
CHARACTER*1 NORM, TRANS
* .. Local Arrays ..
DOUBLE PRECISION H(LDH,NMAX), B(LDB,NRHMAX), DWORK(LDWORK)
INTEGER IPIV(NMAX), IWORK(LIWORK)
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANHS
EXTERNAL DLAMCH, DLANHS
* .. External Subroutines ..
EXTERNAL DLASET, MB02RD, MB02SD, MB02TD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, NRHS, NORM, TRANS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( H(I,J), J = 1,N ), I = 1,N )
IF ( NRHS.LT.0 .OR. NRHS.GT.NRHMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) NRHS
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,NRHS ), I = 1,N )
IF ( N.GT.2 )
$ CALL DLASET( 'Lower', N-2, N-2, ZERO, ZERO, H(3,1), LDH )
* Compute the LU factorization of the upper Hessenberg matrix.
CALL MB02SD( N, H, LDH, IPIV, INFO )
* Estimate the reciprocal condition number of the matrix.
HNORM = DLANHS( NORM, N, H, LDH, DWORK )
CALL MB02TD( NORM, N, HNORM, H, LDH, IPIV, RCOND, IWORK,
$ DWORK, INFO1 )
IF ( INFO.EQ.0 .AND. RCOND.GT.DLAMCH( 'Epsilon' ) ) THEN
* Solve the linear system.
CALL MB02RD( TRANS, N, NRHS, H, LDH, IPIV, B, LDB, INFO )
*
WRITE ( NOUT, FMT = 99997 )
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,NRHS )
10 CONTINUE
WRITE ( NOUT, FMT = 99995 ) RCOND
END IF
END IF
STOP
*
99999 FORMAT (' MB02SD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02SD = ',I2)
99997 FORMAT (' The solution matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' Reciprocal condition number = ',D12.4)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' NRHS is out of range.',/' NRHS = ',I5)
END
</PRE>
<B>Program Data</B>
<PRE>
MB02SD EXAMPLE PROGRAM DATA
5 4 O N
1. 2. 6. 3. 5.
-2. -1. -1. 0. -2.
0. 3. 1. 5. 1.
0. 0. 2. 0. -4.
0. 0. 0. 1. 4.
5. 5. 1. 5.
-2. 1. 3. 1.
0. 0. 4. 5.
2. 1. 1. 3.
-1. 3. 3. 1.
</PRE>
<B>Program Results</B>
<PRE>
MB02SD EXAMPLE PROGRAM RESULTS
The solution matrix is
0.0435 1.2029 1.6377 1.1014
1.0870 -4.4275 -5.5580 -2.9638
0.9130 0.7609 -0.1087 0.6304
-0.8261 2.4783 4.2174 2.7391
-0.0435 0.1304 -0.3043 -0.4348
Reciprocal condition number = 0.1554D-01
</PRE>
<HR>
<p>
<A HREF=..\libindex.html><B>Return to index</B></A></BODY>
</HTML>
|
