# laed4 - Man Page

laed4: D&C step: secular equation nonlinear solver

## Synopsis

### Functions

subroutine dlaed4 (n, i, d, z, delta, rho, dlam, info)
DLAED4 used by DSTEDC. Finds a single root of the secular equation.
subroutine slaed4 (n, i, d, z, delta, rho, dlam, info)
SLAED4 used by SSTEDC. Finds a single root of the secular equation.

## Function Documentation

### subroutine dlaed4 (integer n, integer i, double precision, dimension( * ) d, double precision, dimension( * ) z, double precision, dimension( * ) delta, double precision rho, double precision dlam, integer info)

DLAED4 used by DSTEDC. Finds a single root of the secular equation.

Purpose:

``` This subroutine computes the I-th updated eigenvalue of a symmetric
rank-one modification to a diagonal matrix whose elements are
given in the array d, and that

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 )  +  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, 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 > 2, DELTA contains (D(j) - lambda_I) in its  j-th
component.  If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5
for detail. The vector DELTA contains the information necessary
to construct the eigenvectors by DLAED3 and DLAED9.```

RHO

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

DLAM

```          DLAM is DOUBLE PRECISION
The computed lambda_I, the I-th updated eigenvalue.```

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

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Definition at line 144 of file dlaed4.f.

### subroutine slaed4 (integer n, integer i, real, dimension( * ) d, real, dimension( * ) z, real, dimension( * ) delta, real rho, real dlam, integer info)

SLAED4 used by SSTEDC. Finds a single root of the secular equation.

Purpose:

``` This subroutine computes the I-th updated eigenvalue of a symmetric
rank-one modification to a diagonal matrix whose elements are
given in the array d, and that

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 )  +  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 REAL array, dimension (N)
The original eigenvalues.  It is assumed that they are in
order, D(I) < D(J)  for I < J.```

Z

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

DELTA

```          DELTA is REAL array, dimension (N)
If N > 2, DELTA contains (D(j) - lambda_I) in its  j-th
component.  If N = 1, then DELTA(1) = 1. If N = 2, see SLAED5
for detail. The vector DELTA contains the information necessary
to construct the eigenvectors by SLAED3 and SLAED9.```

RHO

```          RHO is REAL
The scalar in the symmetric updating formula.```

DLAM

```          DLAM is REAL
The computed lambda_I, the I-th updated eigenvalue.```

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

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Definition at line 144 of file slaed4.f.

## Author

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## Info

Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK