f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dgebak (f08njc)

## 1  Purpose

nag_dgebak (f08njc) transforms eigenvectors of a balanced matrix to those of the original real nonsymmetric matrix.

## 2  Specification

 #include #include
 void nag_dgebak (Nag_OrderType order, Nag_JobType job, Nag_SideType side, Integer n, Integer ilo, Integer ihi, const double scale[], Integer m, double v[], Integer pdv, NagError *fail)

## 3  Description

nag_dgebak (f08njc) is intended to be used after a real nonsymmetric matrix $A$ has been balanced by nag_dgebal (f08nhc), and eigenvectors of the balanced matrix ${A}_{22}^{\prime \prime }$ have subsequently been computed.
For a description of balancing, see the document for nag_dgebal (f08nhc). 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 function 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  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     jobNag_JobTypeInput
On entry: this must be the same argument job as supplied to nag_dgebal (f08nhc).
Constraint: ${\mathbf{job}}=\mathrm{Nag_DoNothing}$, $\mathrm{Nag_Permute}$, $\mathrm{Nag_Scale}$ or $\mathrm{Nag_DoBoth}$.
3:     sideNag_SideTypeInput
On entry: indicates whether left or right eigenvectors are to be transformed.
${\mathbf{side}}=\mathrm{Nag_LeftSide}$
The left eigenvectors are transformed.
${\mathbf{side}}=\mathrm{Nag_RightSide}$
The right eigenvectors are transformed.
Constraint: ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_RightSide}$.
4:     nIntegerInput
On entry: $n$, the number of rows of the matrix of eigenvectors.
Constraint: ${\mathbf{n}}\ge 0$.
5:     iloIntegerInput
6:     ihiIntegerInput
On entry: the values ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$, as returned by nag_dgebal (f08nhc).
Constraints:
• if ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
• if ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
7:     scale[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, 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 nag_dgebal (f08nhc).
8:     mIntegerInput
On entry: $m$, the number of columns of the matrix of eigenvectors.
Constraint: ${\mathbf{m}}\ge 0$.
9:     v[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array v must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdv}}×{\mathbf{m}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdv}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $V$ is stored in
• ${\mathbf{v}}\left[\left(j-1\right)×{\mathbf{pdv}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{v}}\left[\left(i-1\right)×{\mathbf{pdv}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the matrix of left or right eigenvectors to be transformed.
On exit: the transformed eigenvectors.
10:   pdvIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array v.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
11:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdv}}>0$.
NE_INT_2
On entry, ${\mathbf{pdv}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pdv}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INT_3
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{ilo}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ihi}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
if ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

## 7  Accuracy

The errors are negligible.

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