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
|
.TH DLAED4 l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
DLAED4 - subroutine computes the I-th updated eigenvalue of a symmetric rank-one modification to a diagonal matrix whose elements are given in the array d, and that D(i) < D(j) for i < j and that RHO > 0
.SH SYNOPSIS
.TP 19
SUBROUTINE DLAED4(
N, I, D, Z, DELTA, RHO, DLAM, INFO )
.TP 19
.ti +4
INTEGER
I, INFO, N
.TP 19
.ti +4
DOUBLE
PRECISION DLAM, RHO
.TP 19
.ti +4
DOUBLE
PRECISION D( * ), DELTA( * ), Z( * )
.SH PURPOSE
This subroutine computes the I-th updated eigenvalue of a symmetric rank-one modification to a diagonal matrix whose elements are given in the array d, and that D(i) < D(j) for i < j and that RHO > 0. This is arranged by the calling routine, and is no loss in generality. The rank-one modified system is thus
diag( D ) + RHO * Z * Z_transpose.
.br
where we assume the Euclidean norm of Z is 1.
.br
The method consists of approximating the rational functions in the
secular equation by simpler interpolating rational functions.
.SH ARGUMENTS
.TP 7
N (input) INTEGER
The length of all arrays.
.TP 7
I (input) INTEGER
The index of the eigenvalue to be computed. 1 <= I <= N.
.TP 7
D (input) DOUBLE PRECISION array, dimension (N)
The original eigenvalues. It is assumed that they are in
order, D(I) < D(J) for I < J.
.TP 7
Z (input) DOUBLE PRECISION array, dimension (N)
The components of the updating vector.
.TP 7
DELTA (output) DOUBLE PRECISION array, dimension (N)
If N .ne. 1, DELTA contains (D(j) - lambda_I) in its j-th
component. If N = 1, then DELTA(1) = 1. The vector DELTA
contains the information necessary to construct the
eigenvectors.
.TP 7
RHO (input) DOUBLE PRECISION
The scalar in the symmetric updating formula.
.TP 7
DLAM (output) DOUBLE PRECISION
The computed lambda_I, the I-th updated eigenvalue.
.TP 7
INFO (output) INTEGER
= 0: successful exit
.br
> 0: if INFO = 1, the updating process failed.
.SH PARAMETERS
Logical variable ORGATI (origin-at-i?) is used for distinguishing
whether D(i) or D(i+1) is treated as the origin.
ORGATI = .true. origin at i
ORGATI = .false. origin at i+1
Logical variable SWTCH3 (switch-for-3-poles?) is for noting
if we are working with THREE poles!
MAXIT is the maximum number of iterations allowed for each
eigenvalue.
Further Details
===============
Based on contributions by
Ren-Cang Li, Computer Science Division, University of California
at Berkeley, USA
|
