man dlasd4 (Fonctions bibliothèques) - 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

NAME

DLASD4 - 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

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 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 j-th component. If N = 1, then DELTA(1) = 1. The 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 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