f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dggbak (f08wjc)

## 1  Purpose

nag_dggbak (f08wjc) forms the right or left eigenvectors of the real generalized eigenvalue problem $Ax=\lambda Bx$, by backward transformation on the computed eigenvectors given by nag_dtgevc (f08ykc). It is necessary to call this function only if the optional balancing function nag_dggbal (f08whc) was previously called to balance the matrix pair $\left(A,B\right)$.

## 2  Specification

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

## 3  Description

If the matrix pair has been previously balanced using the function nag_dggbal (f08whc) then nag_dggbak (f08wjc) backtransforms the eigenvector solution given by nag_dtgevc (f08ykc). 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 nag_dggbal (f08whc).

## 4  References

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

## 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: specifies the backward transformation step required.
${\mathbf{job}}=\mathrm{Nag_DoNothing}$
No transformations are done.
${\mathbf{job}}=\mathrm{Nag_Permute}$
Only do backward transformations based on permutations.
${\mathbf{job}}=\mathrm{Nag_Scale}$
Only do backward transformations based on scaling.
${\mathbf{job}}=\mathrm{Nag_DoBoth}$
Do backward transformations for both permutations and scaling.
Note:  this must be the same argument job as supplied to nag_dggbal (f08whc).
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 order of the matrices $A$ and $B$ of the generalized eigenvalue problem.
Constraint: ${\mathbf{n}}\ge 0$.
5:     iloIntegerInput
6:     ihiIntegerInput
On entry: ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ as determined by a previous call to nag_dggbal (f08whc).
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:     lscale[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array lscale must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 nag_dggbal (f08whc).
8:     rscale[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array rscale must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 nag_dggbal (f08whc).
9:     mIntegerInput
On entry: $m$, the required number of left or right eigenvectors.
Constraint: $0\le {\mathbf{m}}\le {\mathbf{n}}$.
10:   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 right or left eigenvectors, as returned by nag_dggbal (f08whc).
On exit: the transformed right or left eigenvectors.
11:   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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
12:   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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdv}}>0$.
NE_INT_2
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $0\le {\mathbf{m}}\le {\mathbf{n}}$.
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, compared with the previous computations.

The number of operations is proportional to ${n}^{2}$.