F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08NJF (DGEBAK)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08NJF (DGEBAK) transforms eigenvectors of a balanced matrix to those of the original real nonsymmetric matrix.

## 2  Specification

 SUBROUTINE F08NJF ( JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
 INTEGER N, ILO, IHI, M, LDV, INFO REAL (KIND=nag_wp) SCALE(*), V(LDV,*) CHARACTER(1) JOB, SIDE
The routine may be called by its LAPACK name dgebak.

## 3  Description

F08NJF (DGEBAK) is intended to be used after a real nonsymmetric matrix $A$ has been balanced by F08NHF (DGEBAL), and eigenvectors of the balanced matrix ${A}_{22}^{\prime \prime }$ have subsequently been computed.
For a description of balancing, see the document for F08NHF (DGEBAL). 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.

## 5  Parameters

1:     JOB – CHARACTER(1)Input
On entry: this must be the same parameter JOB as supplied to F08NHF (DGEBAL).
Constraint: ${\mathbf{JOB}}=\text{'N'}$, $\text{'P'}$, $\text{'S'}$ or $\text{'B'}$.
2:     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:     N – INTEGERInput
On entry: $n$, the number of rows of the matrix of eigenvectors.
Constraint: ${\mathbf{N}}\ge 0$.
4:     ILO – INTEGERInput
5:     IHI – INTEGERInput
On entry: the values ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$, as returned by F08NHF (DGEBAL).
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:     SCALE($*$) – REAL (KIND=nag_wp) arrayInput
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 real nonsymmetric matrix, as returned by F08NHF (DGEBAL).
7:     M – INTEGERInput
On entry: $m$, the number of columns of the matrix of eigenvectors.
Constraint: ${\mathbf{M}}\ge 0$.
8:     V(LDV,$*$) – REAL (KIND=nag_wp) arrayInput/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:     LDV – INTEGERInput
On entry: the first dimension of the array V as declared in the (sub)program from which F08NJF (DGEBAK) is called.
Constraint: ${\mathbf{LDV}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
10:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error 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.

## 7  Accuracy

The errors are negligible.

The total number of floating point operations is approximately proportional to $nm$.
The complex analogue of this routine is F08NWF (ZGEBAK).

## 9  Example

See Section 9 in F08NHF (DGEBAL).