f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zgebal (f08nvc)

## 1  Purpose

nag_zgebal (f08nvc) balances a complex general matrix in order to improve the accuracy of computed eigenvalues and/or eigenvectors.

## 2  Specification

 #include #include
 void nag_zgebal (Nag_OrderType order, Nag_JobType job, Integer n, Complex a[], Integer pda, Integer *ilo, Integer *ihi, double scale[], NagError *fail)

## 3  Description

nag_zgebal (f08nvc) balances a complex general matrix $A$. The term ‘balancing’ covers two steps, each of which involves a similarity transformation of $A$. The function can perform either or both of these steps.
1. The function first attempts to permute $A$ to block upper triangular form by a similarity transformation:
 $PAPT = A′ = A11′ A12′ A13′ 0 A22′ A23′ 0 0 A33′$
where $P$ is a permutation matrix, and ${A}_{11}^{\prime }$ and ${A}_{33}^{\prime }$ are upper triangular. Then the diagonal elements of ${A}_{11}^{\prime }$ and ${A}_{33}^{\prime }$ are eigenvalues of $A$. The rest of the eigenvalues of $A$ are the eigenvalues of the central diagonal block ${A}_{22}^{\prime }$, in rows and columns ${i}_{\mathrm{lo}}$ to ${i}_{\mathrm{hi}}$. Subsequent operations to compute the eigenvalues of $A$ (or its Schur factorization) need only be applied to these rows and columns; this can save a significant amount of work if ${i}_{\mathrm{lo}}>1$ and ${i}_{\mathrm{hi}}. If no suitable permutation exists (as is often the case), the function sets ${i}_{\mathrm{lo}}=1$ and ${i}_{\mathrm{hi}}=n$, and ${A}_{22}^{\prime }$ is the whole of $A$.
2. The function applies a diagonal similarity transformation to ${A}^{\prime }$, to make the rows and columns of ${A}_{22}^{\prime }$ as close in norm as possible:
 $A′′ = DA′D-1 = I 0 0 0 D22 0 0 0 I A11′ A12′ A13′ 0 A22′ A23′ 0 0 A33′ I 0 0 0 D22-1 0 0 0 I .$
This scaling can reduce the norm of the matrix (i.e., $‖{A}_{22}^{\prime \prime }‖<‖{A}_{22}^{\prime }‖$) and hence reduce the effect of rounding errors on the accuracy of computed eigenvalues and eigenvectors.

## 4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 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: indicates whether $A$ is to be permuted and/or scaled (or neither).
${\mathbf{job}}=\mathrm{Nag_DoNothing}$
$A$ is neither permuted nor scaled (but values are assigned to ilo, ihi and scale).
${\mathbf{job}}=\mathrm{Nag_Permute}$
$A$ is permuted but not scaled.
${\mathbf{job}}=\mathrm{Nag_Scale}$
$A$ is scaled but not permuted.
${\mathbf{job}}=\mathrm{Nag_DoBoth}$
$A$ is both permuted and scaled.
Constraint: ${\mathbf{job}}=\mathrm{Nag_DoNothing}$, $\mathrm{Nag_Permute}$, $\mathrm{Nag_Scale}$ or $\mathrm{Nag_DoBoth}$.
3:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     a[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
Where ${\mathbf{A}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: a is overwritten by the balanced matrix. If ${\mathbf{job}}=\mathrm{Nag_DoNothing}$, a is not referenced.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6:     iloInteger *Output
7:     ihiInteger *Output
On exit: the values ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ such that on exit ${\mathbf{A}}\left(i,j\right)$ is zero if $i>j$ and $1\le j<{i}_{\mathrm{lo}}$ or ${i}_{\mathrm{hi}}.
If ${\mathbf{job}}=\mathrm{Nag_DoNothing}$ or $\mathrm{Nag_Scale}$, ${i}_{\mathrm{lo}}=1$ and ${i}_{\mathrm{hi}}=n$.
8:     scale[n]doubleOutput
On exit: details of the permutations and scaling factors applied to $A$. More precisely, if ${p}_{j}$ is the index of the row and column interchanged with row and column $j$ and ${d}_{j}$ is the scaling factor used to balance row and column $j$ then
 $scale[j-1] = pj, j=1,2,…,ilo-1 dj, j=ilo,ilo+1,…,ihi and pj, j=ihi+1,ihi+2,…,n.$
The order in which the interchanges are made is $n$ to ${i}_{\mathrm{hi}}+1$ then $1$ to ${i}_{\mathrm{lo}}-1$.
9:     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, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 those in subsequent computations.

If the matrix $A$ is balanced by nag_zgebal (f08nvc), then any eigenvectors computed subsequently are eigenvectors of the matrix ${A}^{\prime \prime }$ (see Section 3) and hence nag_zgebak (f08nwc) must then be called to transform them back to eigenvectors of $A$.
If the Schur vectors of $A$ are required, then this function must not be called with ${\mathbf{job}}=\mathrm{Nag_Scale}$ or $\mathrm{Nag_DoBoth}$, because then the balancing transformation is not unitary. If this function is called with ${\mathbf{job}}=\mathrm{Nag_Permute}$, then any Schur vectors computed subsequently are Schur vectors of the matrix ${A}^{\prime \prime }$, and nag_zgebak (f08nwc) must be called (with ${\mathbf{side}}=\mathrm{Nag_RightSide}$) to transform them back to Schur vectors of $A$.
The total number of real floating point operations is approximately proportional to ${n}^{2}$.
The real analogue of this function is nag_dgebal (f08nhc).

## 9  Example

This example computes all the eigenvalues and right eigenvectors of the matrix $A$, where
 $A = 1.50-2.75i 0.00+0.00i 0.00+0.00i 0.00+0.00i -8.06-1.24i -2.50-0.50i 0.00+0.00i -0.75+0.50i -2.09+7.56i 1.39+3.97i -1.25+0.75i -4.82-5.67i 6.18+9.79i -0.92-0.62i 0.00+0.00i -2.50-0.50i .$
The program first calls nag_zgebal (f08nvc) to balance the matrix; it then computes the Schur factorization of the balanced matrix, by reduction to Hessenberg form and the $QR$ algorithm. Then it calls nag_ztrevc (f08qxc) to compute the right eigenvectors of the balanced matrix, and finally calls nag_zgebak (f08nwc) to transform the eigenvectors back to eigenvectors of the original matrix $A$.

### 9.1  Program Text

Program Text (f08nvce.c)

### 9.2  Program Data

Program Data (f08nvce.d)

### 9.3  Program Results

Program Results (f08nvce.r)