SLASD4(3S)SLASD4(3S)NAME
SLASD4 - 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 SLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
INTEGER I, INFO, N
REAL RHO, SIGMA
REAL D( * ), DELTA( * ), WORK( * ), Z( * )
IMPLEMENTATION
These routines are part of the SCSL Scientific Library and can be loaded
using either the -lscs or the -lscs_mp option. The -lscs_mp option
directs the linker to use the multi-processor version of the library.
When linking to SCSL with -lscs or -lscs_mp, the default integer size is
4 bytes (32 bits). Another version of SCSL is available in which integers
are 8 bytes (64 bits). This version allows the user access to larger
memory sizes and helps when porting legacy Cray codes. It can be loaded
by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
only one of the two versions; 4-byte integer and 8-byte integer library
calls cannot be mixed.
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.
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SLASD4(3S)SLASD4(3S)
D (input) REAL array, dimension ( N )
The original eigenvalues. It is assumed that they are in order, 0
<= 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 .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) REAL
The scalar in the symmetric updating formula.
SIGMA (output) REAL
The computed lambda_I, the I-th updated eigenvalue.
WORK (workspace) REAL 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
SEE ALSOINTRO_LAPACK(3S), INTRO_SCSL(3S)
This man page is available only online.
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