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
|
<HTML>
<HEAD><TITLE>MB01UX - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB01UX">MB01UX</A></H2>
<H3>
Computation of matrix expressions alpha T A or alpha A T, over A, T quasi-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 one of the matrix products
A : = alpha*op( T ) * A, or A : = alpha*A * op( T ),
where alpha is a scalar, A is an m-by-n matrix, T is a quasi-
triangular matrix, and op( T ) is one of
op( T ) = T or op( T ) = T', the transpose of T.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB01UX( SIDE, UPLO, TRANS, M, N, ALPHA, T, LDT, A, LDA,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDT, LDWORK, M, N
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), T(LDT,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
SIDE CHARACTER*1
Specifies whether the upper quasi-triangular matrix H
appears on the left or right in the matrix product as
follows:
= 'L': A := alpha*op( T ) * A;
= 'R': A := alpha*A * op( T ).
UPLO CHARACTER*1.
Specifies whether the matrix T is an upper or lower
quasi-triangular matrix as follows:
= 'U': T is an upper quasi-triangular matrix;
= 'L': T is a lower quasi-triangular matrix.
TRANS CHARACTER*1
Specifies the form of op( T ) to be used in the matrix
multiplication as follows:
= 'N': op( T ) = T;
= 'T': op( T ) = T';
= 'C': op( T ) = T'.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
ALPHA (input) DOUBLE PRECISION
The scalar alpha. When alpha is zero then T is not
referenced and A need not be set before entry.
T (input) DOUBLE PRECISION array, dimension (LDT,k)
where k is M when SIDE = 'L' and is N when SIDE = 'R'.
On entry with UPLO = 'U', the leading k-by-k upper
Hessenberg part of this array must contain the upper
quasi-triangular matrix T. The elements below the
subdiagonal are not referenced.
On entry with UPLO = 'L', the leading k-by-k lower
Hessenberg part of this array must contain the lower
quasi-triangular matrix T. The elements above the
supdiagonal are not referenced.
LDT INTEGER
The leading dimension of the array T. LDT >= max(1,k),
where k is M when SIDE = 'L' and is N when SIDE = 'R'.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-N part of this array must
contain the matrix A.
On exit, the leading M-by-N part of this array contains
the computed product.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 and ALPHA<>0, DWORK(1) returns the
optimal value of LDWORK.
On exit, if INFO = -12, DWORK(1) returns the minimum
value of LDWORK.
This array is not referenced when alpha = 0.
LDWORK The length of the array DWORK.
LDWORK >= 1, if alpha = 0 or MIN(M,N) = 0;
LDWORK >= 2*(M-1), if SIDE = 'L';
LDWORK >= 2*(N-1), if SIDE = 'R'.
For maximal efficiency LDWORK should be at least
NOFF*N + M - 1, if SIDE = 'L';
NOFF*M + N - 1, if SIDE = 'R';
where NOFF is the number of nonzero elements on the
subdiagonal (if UPLO = 'U') or supdiagonal (if UPLO = 'L')
of T.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The technique used in this routine is similiar to the technique
used in the SLICOT [1] subroutine MB01UW developed by Vasile Sima.
The required matrix product is computed in two steps. In the first
step, the triangle of T specified by UPLO is used; in the second
step, the contribution of the sub-/supdiagonal is added. If the
workspace can accommodate parts of A, a fast BLAS 3 DTRMM
operation is used in the first step.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Benner, P., Mehrmann, V., Sima, V., Van Huffel, S., and
Varga, A.
SLICOT - A subroutine library in systems and control theory.
In: Applied and computational control, signals, and circuits,
Vol. 1, pp. 499-539, Birkhauser, Boston, 1999.
</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>
|
