DLASD4(l) ) DLASD4(l)NAME
DLASD4 - subroutine computes the square root of the I-th updated eigen‐
value of a positive symmetric rank-one modification to a positive diag‐
onal matrix whose entries are given as the squares of the corresponding
entries in the array d, and that 0 <= D(i) < D(j) for i < j and that
RHO > 0
SYNOPSIS
SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
INTEGER I, INFO, N
DOUBLE PRECISION RHO, SIGMA
DOUBLE PRECISION D( * ), DELTA( * ), WORK( * ), Z( * )
PURPOSE
This subroutine computes the square root of the I-th updated eigenvalue
of a positive symmetric rank-one modification to a positive diagonal
matrix whose entries are given as the squares of the corresponding
entries in the array d, and that 0 <= 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 ) * diag( D ) + RHO * Z * Z_transpose.
where we assume the Euclidean norm of Z is 1.
The method consists of approximating the rational functions in the sec‐
ular equation by simpler interpolating rational functions.
ARGUMENTS
N (input) INTEGER
The length of all arrays.
I (input) INTEGER
The index of the eigenvalue to be computed. 1 <= I <= N.
D (input) DOUBLE PRECISION array, dimension ( N )
The original eigenvalues. It is assumed that they are in order,
0 <= D(I) < D(J) for I < J.
Z (input) DOUBLE PRECISION array, dimension ( N )
The components of the updating vector.
DELTA (output) DOUBLE PRECISION array, dimension ( N )
If N .ne. 1, DELTA contains (D(j) - sigma_I) in its j-th compo‐
nent. If N = 1, then DELTA(1) = 1. The vector DELTA contains
the information necessary to construct the (singular) eigenvec‐
tors.
RHO (input) DOUBLE PRECISION
The scalar in the symmetric updating formula.
SIGMA (output) DOUBLE PRECISION
The computed lambda_I, the I-th updated eigenvalue.
WORK (workspace) DOUBLE PRECISION array, dimension ( N )
If N .ne. 1, WORK contains (D(j) + sigma_I) in its j-th compo‐
nent. If N = 1, then WORK( 1 ) = 1.
INFO (output) INTEGER
= 0: successful exit
> 0: if INFO = 1, the updating process failed.
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, Uni‐
versity of California at Berkeley, USA
LAPACK version 3.0 15 June 2000 DLASD4(l)
