Provided by: liblapack-doc_3.3.1-1_all #### NAME

```       LAPACK-3 - subroutine compute 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( * )

```

#### 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
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) 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 sigma_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

LAPACK auxiliary routine (version 3.3.1)   April 2011                            SLASD4(3lapack)
```