DLASD4(1) LAPACK auxiliary routine (version 3.2) DLASD4(1)NAME
DLASD4 - subroutine compute the square root of the I-th updated eigen‐
value of a positive symmetric rank-one modification to a positive diag‐
onal 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
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( * )
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 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 sec‐
ular equation by simpler interpolating rational functions.
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 compo‐
nent. If N = 1, then DELTA(1) = 1. The vector DELTA contains
the information necessary to construct the (singular) eigenvec‐
RHO (input) DOUBLE PRECISION
The scalar in the symmetric updating formula.
SIGMA (output) DOUBLE PRECISION
The computed sigma_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 compo‐
nent. If N = 1, then WORK( 1 ) = 1.
INFO (output) INTEGER
= 0: successful exit
> 0: if INFO = 1, the updating process failed.
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
LAPACK auxiliary routine (versioNovember 2008 DLASD4(1)