SLAED4(3) subroutine compute 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

## SYNOPSIS

SUBROUTINE SLAED4(
N, I, D, Z, DELTA, RHO, DLAM, INFO )

INTEGER I, INFO, N

REAL DLAM, RHO

REAL D( * ), DELTA( * ), Z( * )

## 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 no loss in generality. The rank-one modified system is thus
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) REAL array, dimension (N)
The original eigenvalues. It is assumed that they are in order, D(I) < D(J) for I < J.
Z (input) REAL array, dimension (N)
The components of the updating vector.
DELTA (output) REAL array, dimension (N)
If N .GT. 2, DELTA contains (D(j) - lambda_I) in its j-th component. If N = 1, then DELTA(1) = 1. If N = 2, see SLAED5 for detail. The vector DELTA contains the information necessary to construct the eigenvectors by SLAED3 and SLAED9.
RHO (input) REAL
The scalar in the symmetric updating formula.
DLAM (output) REAL
The computed lambda_I, the I-th updated eigenvalue.
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