, by backward transformation on the computed eigenvectors given by f08ykf. It is necessary to call this routine only if the optional balancing routine f08whf was previously called to balance the matrix pair

(A, B)

2 Specification

Fortran Interface

Subroutine f08wjf (

job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info)

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

C Header Interface

#include <nag.h>

void	f08wjf_ (const char job, const char side, const Integer n, const Integer ilo, const Integer ihi, const double lscale[], const double rscale[], const Integer m, double v[], const Integer ldv, Integer info, const Charlen length_job, const Charlen length_side)

The routine may be called by the names f08wjf, nagf_lapackeig_dggbak or its LAPACK name dggbak.

3 Description

If the matrix pair has been previously balanced using the routine f08whf then f08wjf backtransforms the eigenvector solution given by f08ykf. This is usually the sixth and last step in the solution of the generalized eigenvalue problem.

For a description of balancing, see the document for f08whf.

4 References

Ward R C (1981) Balancing the generalized eigenvalue problem SIAM J. Sci. Stat. Comp. 2 141–152

5 Arguments

1: $job$ – Character(1) Input

On entry: specifies the backward transformation step required.

$job ='N'$: No transformations are done.
$job ='P'$: Only do backward transformations based on permutations.
$job ='S'$: Only do backward transformations based on scaling.
$job ='B'$: Do backward transformations for both permutations and scaling.

Note: this must be the same argument job as supplied to f08whf.

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 order of the matrices

A

and

B

of the generalized eigenvalue problem.

Constraint:

n \geq 0

4: $ilo$ – Integer Input

5: $ihi$ – Integer Input

On entry:

i_{lo}

and

i_{hi}

as determined by a previous call to f08whf.

Constraints:

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

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

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

\max (1, n)

On entry: details of the permutations and scaling factors applied to the left side of the matrices

A

and

B

, as returned by a previous call to f08whf.

7: $rscale (*)$ – Real (Kind=nag_wp) array Input

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

\max (1, n)

On entry: details of the permutations and scaling factors applied to the right side of the matrices

A

and

B

, as returned by a previous call to f08whf.

8: $m$ – Integer Input

On entry:

m

, the required number of left or right eigenvectors.

Constraint:

0 \leq m \leq n

9: $v (ldv, *)$ – Real (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 right or left eigenvectors, as returned by f08whf.

On exit: the transformed right or left eigenvectors.

10: $ldv$ – Integer Input

On entry: the first dimension of the array v as declared in the (sub)program from which f08wjf is called.

Constraint:

ldv \geq \max (1, n)

11: $info$ – 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, compared with the previous computations.

8 Parallelism and Performance

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

f08wjf 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.