DLAED4(l) — LAPACK routine (version 2.0)

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

SYNOPSIS

SUBROUTINE DLAED4(
N, I, D, Z, DELTA, RHO, DLAM, INFO )

INTEGER I, INFO, N

DOUBLE PRECISION DLAM, RHO

DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N)
The original eigenvalues. It is assumed that they are in order, 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) - 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.

RHO (input) DOUBLE PRECISION
The scalar in the symmetric updating formula.

DLAM (output) DOUBLE PRECISION
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.