Integer, Intent (In)	::	n, ilo, ihi, m, ldv
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	scale(*)
Complex (Kind=nag_wp), Intent (Inout)	::	v(ldv,*)
Character (1), Intent (In)	::	job, side

C Header Interface

#include <nag.h>

void	f08nwf_ (const char job, const char side, const Integer n, const Integer ilo, const Integer ihi, const double scal[], const Integer m, Complex v[], const Integer ldv, Integer info, const Charlen length_job, const Charlen length_side)

The routine may be called by the names f08nwf, nagf_lapackeig_zgebak or its LAPACK name zgebak.

3 Description

f08nwf is intended to be used after a complex general matrix

A

has been balanced by f08nvf, and eigenvectors of the balanced matrix

A_{22}^{''}

have subsequently been computed.

For a description of balancing, see the document for f08nvf. The balanced matrix

A^{''}

is obtained as

A^{''} = D P A P^{T} D^{- 1}

, where

P

is a permutation matrix and

D

is a diagonal scaling matrix. This routine transforms left or right eigenvectors as follows:

if $x$ is a right eigenvector of $A^{''}$ , $P^{T} D^{- 1} x$ is a right eigenvector of $A$ ;
if $y$ is a left eigenvector of $A^{''}$ , $P^{T} D y$ is a left eigenvector of $A$ .

4 References

None.

5 Arguments

1: $job$ – Character(1) Input

On entry: this must be the same argument job as supplied to f08nvf.

Constraint:

job ='N'

'P'

'S'

'B'

2: $side$ – Character(1) Input

On entry: indicates whether left or right eigenvectors are to be transformed.

$side ='L'$: The left eigenvectors are transformed.
$side ='R'$: The right eigenvectors are transformed.

Constraint:

side ='L'

'R'

3: $n$ – Integer Input

On entry:

n

, the number of rows of the matrix of eigenvectors.

Constraint:

n \geq 0

4: $ilo$ – Integer Input

5: $ihi$ – Integer Input

On entry: the values

i_{lo}

and

i_{hi}

, as returned by f08nvf.

Constraints:

if $n > 0$ , $1 \leq ilo \leq ihi \leq n$ ;
if $n = 0$ , $ilo = 1$ and $ihi = 0$ .

6: $scale (*)$ – Real (Kind=nag_wp) array Input

Note: the dimension of the array scale must be at least

\max (1, n)

On entry: details of the permutations and/or the scaling factors used to balance the original complex general matrix, as returned by f08nvf.

7: $m$ – Integer Input

On entry:

m

, the number of columns of the matrix of eigenvectors.

Constraint:

m \geq 0

8: $v (ldv, *)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array v must be at least

\max (1, m)

On entry: the matrix of left or right eigenvectors to be transformed.

On exit: the transformed eigenvectors.

9: $ldv$ – Integer Input

On entry: the first dimension of the array v as declared in the (sub)program from which f08nwf is called.

Constraint:

ldv \geq \max (1, n)

10: $info$ – Integer Output

On exit:

info = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

The errors are negligible.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08nwf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.