DLASD4
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.
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 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
component. If N = 1, then WORK( 1 ) = 1. |
| INFO |
(output) INTEGER
= 0: successful exit
> 0: if INFO = 1, the updating process failed. |
Internal 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.
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
Ren-Cang Li, Computer Science Division, University of California
at Berkeley, USA