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
|
<HTML>
<HEAD><TITLE>MB01OD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB01OD">MB01OD</A></H2>
<H3>
Computation of matrix expression alpha R + beta ( op(H) X op(E)' + op(E) X op(H)' ) with R, X symmetric, H upper Hessenberg, and E upper triangular
</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 the matrix formula
R := alpha*R + beta*( op( H )*X*op( E )' + op( E )*X*op( H )' ),
where alpha and beta are scalars, R and X are symmetric matrices,
H is an upper Hessenberg matrix, E is an upper triangular matrix,
and op( M ) is one of
op( M ) = M or op( M ) = M'.
The result is overwritten on R.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB01OD( UPLO, TRANS, N, ALPHA, BETA, R, LDR, H, LDH,
$ X, LDX, E, LDE, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER TRANS, UPLO
INTEGER INFO, LDE, LDH, LDR, LDWORK, LDX, N
DOUBLE PRECISION ALPHA, BETA
C .. Array Arguments ..
DOUBLE PRECISION DWORK(*), E(LDE,*), H(LDH,*), R(LDR,*), X(LDX,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
UPLO CHARACTER*1
Specifies which triangles of the symmetric matrices R
and X are given as follows:
= 'U': the upper triangular part is given;
= 'L': the lower triangular part is given.
TRANS CHARACTER*1
Specifies the form of op( M ) to be used in the matrix
multiplication as follows:
= 'N': op( M ) = M;
= 'T': op( M ) = M';
= 'C': op( M ) = M'.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the matrices R, H, E, and X. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then R need not be
set before entry, except when R is identified with X in
the call.
BETA (input) DOUBLE PRECISION
The scalar beta. When beta is zero then H and X are not
referenced.
R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
On entry with UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix R.
On entry with UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix R.
In both cases, the other strictly triangular part is not
referenced.
On exit, the leading N-by-N upper triangular part (if
UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
this array contains the corresponding triangular part of
the computed matrix R.
LDR INTEGER
The leading dimension of array R. LDR >= MAX(1,N).
H (input) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the leading N-by-N upper Hessenberg part of this
array must contain the upper Hessenberg matrix H.
If TRANS = 'N', the entries 3, 4,..., N of the first
column are modified internally, but are restored on exit.
The remaining part of this array is not referenced.
LDH INTEGER
The leading dimension of array H. LDH >= MAX(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
On entry, if UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the symmetric matrix X and the strictly
lower triangular part of the array is not referenced.
On entry, if UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the symmetric matrix X and the strictly
upper triangular part of the array is not referenced.
The diagonal elements of this array are modified
internally, but are restored on exit.
LDX INTEGER
The leading dimension of array X. LDX >= MAX(1,N).
E (input) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix E.
The remaining part of this array is not referenced.
LDE INTEGER
The leading dimension of array E. LDE >= MAX(1,N).
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (LDWORK)
This array is not referenced when beta = 0, or N = 0.
LDWORK The length of the array DWORK.
LDWORK >= N*N, if beta <> 0;
LDWORK >= 0, if beta = 0.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -k, the k-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The matrix expression is efficiently evaluated taking the symmetry
into account. Specifically, let X = U + L, with U and L upper and
lower triangular matrices, defined by
U = triu( X ) - (1/2)*diag( X ),
L = tril( X ) - (1/2)*diag( X ),
where triu, tril, and diag denote the upper triangular part, lower
triangular part, and diagonal part of X, respectively. Then,
if UPLO = 'U',
H*X*E' + E*X*H' = (H*U)*E' + E*(H*U)' + H*(E*U)' + (E*U)*H',
for TRANS = 'N',
H'*X*E + E'*X*H = H'*(U*E) + (U*E)'*H + (U*H)'*E + E'*(U*H),
for TRANS = 'T', or 'C',
and if UPLO = 'L',
H*X*E' + E*X*H' = (H*L')*E' + E*(H*L')' + H*(E*L')' + (E*L')*H',
for TRANS = 'N',
H'*X*E + E'*X*H = H'*(L'*E) + (L'*E)'*H + (L'*H)'*E + E'*(L'*H),
for TRANS = 'T', or 'C',
which involve operations like in BLAS 2 and 3 (DTRMV and DSYR2K).
This approach ensures that the matrices H*U, U*H, H*L', or L'*H
are upper Hessenberg, and E*U, U*E, E*L', or L'*E are upper
triangular.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm requires approximately N**3/2 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>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
<HR>
<A HREF=support.html><B>Return to Supporting Routines index</B></A></BODY>
</HTML>
|
