f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zggbak (f08wwc)

## 1  Purpose

nag_zggbak (f08wwc) 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_ztgevc (f08yxc). It is necessary to call this function only if the optional balancing function nag_zggbal (f08wvc) was previously called to balance the matrix pair $\left(A,B\right)$.

## 2  Specification

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

## 3  Description

If the matrix pair has been previously balanced using the function nag_zggbal (f08wvc) then nag_zggbak (f08wwc) backtransforms the eigenvector solution given by nag_ztgevc (f08yxc). 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_zggbal (f08wvc).

## 4  References

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

## 5  Arguments

1:    $\mathbf{order}$Nag_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 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{job}$Nag_JobTypeInput
On entry: specifies the backtransformation 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 identical to the argument job as supplied to nag_zggbal (f08wvc).
Constraint: ${\mathbf{job}}=\mathrm{Nag_DoNothing}$, $\mathrm{Nag_Permute}$, $\mathrm{Nag_Scale}$ or $\mathrm{Nag_DoBoth}$.
3:    $\mathbf{side}$Nag_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:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$ of the generalized eigenvalue problem.
Constraint: ${\mathbf{n}}\ge 0$.
5:    $\mathbf{ilo}$IntegerInput
6:    $\mathbf{ihi}$IntegerInput
On entry: ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ as determined by a previous call to nag_zggbal (f08wvc).
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:    $\mathbf{lscale}\left[\mathit{dim}\right]$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_zggbal (f08wvc).
8:    $\mathbf{rscale}\left[\mathit{dim}\right]$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_zggbal (f08wvc).
9:    $\mathbf{m}$IntegerInput
On entry: $m$, the required number of left or right eigenvectors.
Constraint: $0\le {\mathbf{m}}\le {\mathbf{n}}$.
10:  $\mathbf{v}\left[\mathit{dim}\right]$ComplexInput/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_zggbal (f08wvc).
On exit: the transformed right or left eigenvectors.
11:  $\mathbf{pdv}$IntegerInput
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:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7  Accuracy

The errors are negligible.

## 8  Parallelism and Performance

nag_zggbak (f08wwc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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