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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgebal (f08nh)

## Purpose

nag_lapack_dgebal (f08nh) balances a real general matrix in order to improve the accuracy of computed eigenvalues and/or eigenvectors.

## Syntax

[a, ilo, ihi, scale, info] = f08nh(job, a, 'n', n)
[a, ilo, ihi, scale, info] = nag_lapack_dgebal(job, a, 'n', n)

## Description

nag_lapack_dgebal (f08nh) balances a real 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.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{job}$ – string (length ≥ 1)
Indicates whether $A$ is to be permuted and/or scaled (or neither).
${\mathbf{job}}=\text{'N'}$
$A$ is neither permuted nor scaled (but values are assigned to ilo, ihi and scale).
${\mathbf{job}}=\text{'P'}$
$A$ is permuted but not scaled.
${\mathbf{job}}=\text{'S'}$
$A$ is scaled but not permuted.
${\mathbf{job}}=\text{'B'}$
$A$ is both permuted and scaled.
Constraint: ${\mathbf{job}}=\text{'N'}$, $\text{'P'}$, $\text{'S'}$ or $\text{'B'}$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ matrix $A$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a stores the balanced matrix. If ${\mathbf{job}}=\text{'N'}$, a is not referenced.
2:     $\mathrm{ilo}$int64int32nag_int scalar
3:     $\mathrm{ihi}$int64int32nag_int scalar
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 ${\mathbf{job}}=\text{'N'}$ or $\text{'S'}$, ${i}_{\mathrm{lo}}=1$ and ${i}_{\mathrm{hi}}=n$.
4:     $\mathrm{scale}\left({\mathbf{n}}\right)$ – double array
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
 $scalej = pj, j=1,2,…,ilo-1 dj, j=ilo,ilo+1,…,ihi and pj, j=ihi+1,ihi+2,…,n.$
The order in which the interchanges are made is $n$ to ${i}_{\mathrm{hi}}+1$ then $1$ to ${i}_{\mathrm{lo}}-1$.
5:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: job, 2: n, 3: a, 4: lda, 5: ilo, 6: ihi, 7: scale, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

## Accuracy

The errors are negligible.

If the matrix $A$ is balanced by nag_lapack_dgebal (f08nh), then any eigenvectors computed subsequently are eigenvectors of the matrix ${A}^{\prime \prime }$ (see Description) and hence nag_lapack_dgebak (f08nj) 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 ${\mathbf{job}}=\text{'S'}$ or $\text{'B'}$, because then the balancing transformation is not orthogonal. If this function is called with ${\mathbf{job}}=\text{'P'}$, then any Schur vectors computed subsequently are Schur vectors of the matrix ${A}^{\prime \prime }$, and nag_lapack_dgebak (f08nj) must be called (with ${\mathbf{side}}=\text{'R'}$) to transform them back to Schur vectors of $A$.
The total number of floating-point operations is approximately proportional to ${n}^{2}$.
The complex analogue of this function is nag_lapack_zgebal (f08nv).

## Example

This example computes all the eigenvalues and right eigenvectors of the matrix $A$, where
 $A = 5.14 0.91 0.00 -32.80 0.91 0.20 0.00 34.50 1.90 0.80 -0.40 -3.00 -0.33 0.35 0.00 0.66 .$
The program first calls nag_lapack_dgebal (f08nh) 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_lapack_dtrevc (f08qk) to compute the right eigenvectors of the balanced matrix, and finally calls nag_lapack_dgebak (f08nj) to transform the eigenvectors back to eigenvectors of the original matrix $A$.
```function f08nh_example

fprintf('f08nh example results\n\n');

n = int64(4);
a = [ 5.14, 0.91,  0.00, -32.80;
0.91, 0.20,  0.00,  34.50;
1.90, 0.80, -0.40,  -3.00;
-0.33, 0.35,  0.00,   0.66];

% Balance a
[a, ilo, ihi, scale, info] = f08nh( ...
'Both', a);

% Reduce a to upper Hessenberg form
[H, tau, info] = f08ne( ...
ilo, ihi, a);

% Form Q
[Q, info] = f08nf( ...
ilo, ihi, H, tau);

% Calculate the eigenvalues and Schur factorisation of A
[H, wr, wi, Z, info] = f08pe( ...
'Schur Form', 'Vectors', ilo, ihi, H, Q);

w = wr + i*wi;
disp('Eigenvalues:');
disp(w);

% Calculate the eigenvectors of A
[select, ~, VR, m, info] = ...
f08qk( ...
'Right', 'Backtransform', false, H, zeros(1), Z, n);

% Back scale to get eigenvectors of A
[VR, info] = f08nj( ...
'Both', 'Right', ilo, ihi, scale, VR);

% Normalize eigenvectors: largest element positive
for j = 1:n
[~,k] = max(abs(VR(:,j)));
VR(:,j) =VR(:,j)/norm(VR(:,j));
if VR(k,j) < 0;
VR(:,j) = -VR(:,j);
end
end

disp('Eigenvectors:');
disp(VR);

```
```f08nh example results

Eigenvalues:
-0.4000
-4.0208
3.0136
7.0072

Eigenvectors:
0   -0.4381    0.4654    0.9513
0    0.8923    0.7888   -0.1714
1.0000   -0.0481    0.3981    0.2494
0   -0.0976    0.0521   -0.0589

```