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
|
.TH DGGBAL l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
DGGBAL - balance a pair of general real matrices (A,B)
.SH SYNOPSIS
.TP 19
SUBROUTINE DGGBAL(
JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
RSCALE, WORK, INFO )
.TP 19
.ti +4
CHARACTER
JOB
.TP 19
.ti +4
INTEGER
IHI, ILO, INFO, LDA, LDB, N
.TP 19
.ti +4
DOUBLE
PRECISION A( LDA, * ), B( LDB, * ), LSCALE( * ),
RSCALE( * ), WORK( * )
.SH PURPOSE
DGGBAL balances a pair of general real matrices (A,B). This involves, first, permuting A and B by similarity transformations to
isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
elements on the diagonal; and second, applying a diagonal similarity
transformation to rows and columns ILO to IHI to make the rows
and columns as close in norm as possible. Both steps are optional.
Balancing may reduce the 1-norm of the matrices, and improve the
accuracy of the computed eigenvalues and/or eigenvectors in the
generalized eigenvalue problem A*x = lambda*B*x.
.br
.SH ARGUMENTS
.TP 8
JOB (input) CHARACTER*1
Specifies the operations to be performed on A and B:
.br
= 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
and RSCALE(I) = 1.0 for i = 1,...,N.
= 'P': permute only;
.br
= 'S': scale only;
.br
= 'B': both permute and scale.
.TP 8
N (input) INTEGER
The order of the matrices A and B. N >= 0.
.TP 8
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the input matrix A.
On exit, A is overwritten by the balanced matrix.
If JOB = 'N', A is not referenced.
.TP 8
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
.TP 8
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the input matrix B.
On exit, B is overwritten by the balanced matrix.
If JOB = 'N', B is not referenced.
.TP 8
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
.TP 8
ILO (output) INTEGER
IHI (output) INTEGER
ILO and IHI are set to integers such that on exit
A(i,j) = 0 and B(i,j) = 0 if i > j and
j = 1,...,ILO-1 or i = IHI+1,...,N.
If JOB = 'N' or 'S', ILO = 1 and IHI = N.
.TP 8
LSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied
to the left side of A and B. If P(j) is the index of the
row interchanged with row j, and D(j)
is the scaling factor applied to row j, then
LSCALE(j) = P(j) for J = 1,...,ILO-1
= D(j) for J = ILO,...,IHI
= P(j) for J = IHI+1,...,N.
The order in which the interchanges are made is N to IHI+1,
then 1 to ILO-1.
.TP 8
RSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied
to the right side of A and B. If P(j) is the index of the
column interchanged with column j, and D(j)
is the scaling factor applied to column j, then
LSCALE(j) = P(j) for J = 1,...,ILO-1
= D(j) for J = ILO,...,IHI
= P(j) for J = IHI+1,...,N.
The order in which the interchanges are made is N to IHI+1,
then 1 to ILO-1.
.TP 8
WORK (workspace) DOUBLE PRECISION array, dimension (6*N)
.TP 8
INFO (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value.
.SH FURTHER DETAILS
See R.C. WARD, Balancing the generalized eigenvalue problem,
SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
|
