# NAG FL Interfacef08nwf (zgebak)

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## 1Purpose

f08nwf transforms eigenvectors of a balanced matrix to those of the original complex general matrix.

## 2Specification

Fortran Interface
 Subroutine f08nwf ( job, side, n, ilo, ihi, m, v, ldv, info)
 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
#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.

## 3Description

f08nwf is intended to be used after a complex general matrix $A$ has been balanced by f08nvf, and eigenvectors of the balanced matrix ${A}_{22}^{\prime \prime }$ have subsequently been computed.
For a description of balancing, see the document for f08nvf. The balanced matrix ${A}^{\prime \prime }$ is obtained as ${A}^{\prime \prime }=DPA{P}^{\mathrm{T}}{D}^{-1}$, 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 ${A}^{\prime \prime }$, ${P}^{\mathrm{T}}{D}^{-1}x$ is a right eigenvector of $A$;
• if $y$ is a left eigenvector of ${A}^{\prime \prime }$, ${P}^{\mathrm{T}}Dy$ is a left eigenvector of $A$.

None.

## 5Arguments

1: $\mathbf{job}$Character(1) Input
On entry: this must be the same argument job as supplied to f08nvf.
Constraint: ${\mathbf{job}}=\text{'N'}$, $\text{'P'}$, $\text{'S'}$ or $\text{'B'}$.
2: $\mathbf{side}$Character(1) Input
On entry: indicates whether left or right eigenvectors are to be transformed.
${\mathbf{side}}=\text{'L'}$
The left eigenvectors are transformed.
${\mathbf{side}}=\text{'R'}$
The right eigenvectors are transformed.
Constraint: ${\mathbf{side}}=\text{'L'}$ or $\text{'R'}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of rows of the matrix of eigenvectors.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{ilo}$Integer Input
5: $\mathbf{ihi}$Integer Input
On entry: the values ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$, as returned by f08nvf.
Constraints:
• if ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
• if ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
6: $\mathbf{scale}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array scale must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: details of the permutations and/or the scaling factors used to balance the original complex general matrix, as returned by f08nvf.
7: $\mathbf{m}$Integer Input
On entry: $m$, the number of columns of the matrix of eigenvectors.
Constraint: ${\mathbf{m}}\ge 0$.
8: $\mathbf{v}\left({\mathbf{ldv}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array v must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry: the matrix of left or right eigenvectors to be transformed.
On exit: the transformed eigenvectors.
9: $\mathbf{ldv}$Integer Input
On entry: the first dimension of the array v as declared in the (sub)program from which f08nwf is called.
Constraint: ${\mathbf{ldv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

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

## 7Accuracy

The errors are negligible.

## 8Parallelism and Performance

The total number of real floating-point operations is approximately proportional to $nm$.