DLASD4 - subroutine computes the square root of the I-th
updated eigenvalue of a positive symmetric rank-one modi
fication 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
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 modi
fication 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 func
tions in the secular 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
vector DELTA contains the information necessary to
construct the (singular) eigenvectors.
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 component. 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 distin
guishing 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 not
ing 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
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