# dlasd4.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
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dlasd4.f

## SYNOPSIS

### Functions/Subroutines

subroutine dlasd4 (N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO)
DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.

## Function/Subroutine Documentation

### subroutine dlasd4 (integer N, integer I, double precision, dimension( * ) D, double precision, dimension( * ) Z, double precision, dimension( * ) DELTA, double precision RHO, double precision SIGMA, double precision, dimension( * ) WORK, integer INFO)

DLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by dbdsdc.

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.
```

Parameters:

N

```          N is INTEGER
The length of all arrays.
```

I

```          I is INTEGER
The index of the eigenvalue to be computed.  1 <= I <= N.
```

D

```          D is 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

```          Z is DOUBLE PRECISION array, dimension ( N )
The components of the updating vector.
```

DELTA

```          DELTA is 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

```          RHO is DOUBLE PRECISION
The scalar in the symmetric updating formula.
```

SIGMA

```          SIGMA is DOUBLE PRECISION
The computed sigma_I, the I-th updated eigenvalue.
```

WORK

```          WORK is 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

```          INFO is 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.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

December 2016

Contributors:

Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Definition at line 155 of file dlasd4.f.

## Author

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