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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dggbal (f08wh)

## Purpose

nag_lapack_dggbal (f08wh) balances a pair of real square matrices $\left(A,B\right)$ of order $n$. Balancing usually improves the accuracy of computed generalized eigenvalues and eigenvectors.

## Syntax

[a, b, ilo, ihi, lscale, rscale, info] = f08wh(job, a, b, 'n', n)
[a, b, ilo, ihi, lscale, rscale, info] = nag_lapack_dggbal(job, a, b, 'n', n)

## Description

Balancing may reduce the $1$-norms of the matrices and improve the accuracy of the computed eigenvalues and eigenvectors in the real generalized eigenvalue problem
 $Ax=λBx.$
nag_lapack_dggbal (f08wh) is usually the first step in the solution of the above generalized eigenvalue problem. Balancing is optional but it is highly recommended.
The term ‘balancing’ covers two steps, each of which involves similarity transformations on $A$ and $B$. The function can perform either or both of these steps. Both steps are optional.
1. The function first attempts to permute $A$ and $B$ to block upper triangular form by a similarity transformation:
 $PAPT=F= F11 F12 F13 F22 F23 F33$
 $PBPT=G= G11 G12 G13 G22 G23 G33$
where $P$ is a permutation matrix, ${F}_{11}$, ${F}_{33}$, ${G}_{11}$ and ${G}_{33}$ are upper triangular. Then the diagonal elements of the matrix pairs $\left({F}_{11},{G}_{11}\right)$ and $\left({F}_{33},{G}_{33}\right)$ are generalized eigenvalues of $\left(A,B\right)$. The rest of the generalized eigenvalues are given by the matrix pair $\left({F}_{22},{G}_{22}\right)$ which are in rows and columns ${i}_{\mathrm{lo}}$ to ${i}_{\mathrm{hi}}$. Subsequent operations to compute the generalized eigenvalues of $\left(A,B\right)$ need only be applied to the matrix pair $\left({F}_{22},{G}_{22}\right)$; 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 ${i}_{\mathrm{hi}}=n$.
2. The function applies a diagonal similarity transformation to $\left(F,G\right)$, to make the rows and columns of $\left({F}_{22},{G}_{22}\right)$ as close in norm as possible:
 $DFD^= I 0 0 0 D22 0 0 0 I F11 F12 F13 F22 F23 F33 I 0 0 0 D^22 0 0 0 I$
 $DGD^= I 0 0 0 D22 0 0 0 I G11 G12 G13 G22 G23 G33 I 0 0 0 D^22 0 0 0 I$
This transformation usually improves the accuracy of computed generalized eigenvalues and eigenvectors.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{job}$ – string (length ≥ 1)
Specifies the operations to be performed on matrices $A$ and $B$.
${\mathbf{job}}=\text{'N'}$
No balancing is done. Initialize ${\mathbf{ilo}}=1$, ${\mathbf{ihi}}={\mathbf{n}}$, ${\mathbf{lscale}}\left(\mathit{i}\right)=1.0$ and ${\mathbf{rscale}}\left(\mathit{i}\right)=1.0$, for $\mathit{i}=1,2,\dots ,n$.
${\mathbf{job}}=\text{'P'}$
Only permutations are used in balancing.
${\mathbf{job}}=\text{'S'}$
Only scalings are are used in balancing.
${\mathbf{job}}=\text{'B'}$
Both permutations and scalings are used in balancing.
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 $A$.
3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ matrix $B$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the arrays a, b and the second dimension of the arrays a, b.
$n$, the order of the matrices $A$ and $B$.
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:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
b stores the balanced matrix. If ${\mathbf{job}}=\text{'N'}$, b is not referenced.
3:     $\mathrm{ilo}$int64int32nag_int scalar
4:     $\mathrm{ihi}$int64int32nag_int scalar
${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ are set such that ${\mathbf{a}}\left(i,j\right)=0$ and ${\mathbf{b}}\left(i,j\right)=0$ 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 ${i}_{\mathrm{hi}}=n$.
5:     $\mathrm{lscale}\left({\mathbf{n}}\right)$ – double array
Details of the permutations and scaling factors applied to the left side of the matrices $A$ and $B$. If ${P}_{i}$ is the index of the row interchanged with row $i$ and ${d}_{i}$ is the scaling factor applied to row $i$, then
• ${\mathbf{lscale}}\left(\mathit{i}\right)={P}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathit{i}}_{\mathrm{lo}}-1$;
• ${\mathbf{lscale}}\left(\mathit{i}\right)={\mathit{d}}_{\mathit{i}}$, for $\mathit{i}={\mathit{i}}_{\mathrm{lo}},\dots ,{\mathit{i}}_{\mathrm{hi}}$;
• ${\mathbf{lscale}}\left(\mathit{i}\right)={P}_{\mathit{i}}$, for $\mathit{i}={\mathit{i}}_{\mathrm{hi}}+1,\dots ,n$.
The order in which the interchanges are made is $n$ to ${i}_{\mathrm{hi}}+1$, then $1$ to ${i}_{\mathrm{lo}}-1$.
6:     $\mathrm{rscale}\left({\mathbf{n}}\right)$ – double array
Details of the permutations and scaling factors applied to the right side of the matrices $A$ and $B$.
If ${P}_{j}$ is the index of the column interchanged with column $j$ and ${\stackrel{^}{d}}_{j}$ is the scaling factor applied to column $j$, then
• ${\mathbf{rscale}}\left(\mathit{j}\right)={P}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathit{i}}_{\mathrm{lo}}-1$;
• ${\mathbf{rscale}}\left(\mathit{j}\right)={\stackrel{^}{d}}_{\mathit{j}}$, for $\mathit{j}={i}_{\mathrm{lo}},\dots ,{i}_{\mathrm{hi}}$;
• ${\mathbf{rscale}}\left(\mathit{j}\right)={P}_{\mathit{j}}$, for $\mathit{j}={i}_{\mathrm{hi}}+1,\dots ,n$.
The order in which the interchanges are made is $n$ to ${i}_{\mathrm{hi}}+1$, then $1$ to ${i}_{\mathrm{lo}}-1$.
7:     $\mathrm{info}$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: b, 6: ldb, 7: ilo, 8: ihi, 9: lscale, 10: rscale, 11: work, 12: 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, compared to those in subsequent computations.

nag_lapack_dggbal (f08wh) is usually the first step in computing the real generalized eigenvalue problem but it is an optional step. The matrix $B$ is reduced to the upper triangular form using the $QR$ factorization function nag_lapack_dgeqrf (f08ae) and this orthogonal transformation $Q$ is applied to the matrix $A$ by calling nag_lapack_dormqr (f08ag). This is followed by nag_lapack_dgghrd (f08we) which reduces the matrix pair into the generalized Hessenberg form.
If the matrix pair $\left(A,B\right)$ is balanced by this function, then any generalized eigenvectors computed subsequently are eigenvectors of the balanced matrix pair. In that case, to compute the generalized eigenvectors of the original matrix, nag_lapack_dggbak (f08wj) must be called.
The total number of floating-point operations is approximately proportional to ${n}^{2}$.
The complex analogue of this function is nag_lapack_zggbal (f08wv).

## Example

See Example in nag_lapack_dhgeqz (f08xe) and nag_lapack_dtgevc (f08yk).
```function f08wh_example

fprintf('f08wh example results\n\n');

a = [1,  1,   1,    1,    1;
2,  4,   8,   16,   32;
3,  9,  27,   81,  243;
4, 16,  64,  256, 1024;
5, 25, 125,  625, 3125];
b = [1,  2,   3,    4,    5;
1,  4,   9,   16,   25;
1,  8,  27,   64,  125;
1, 16,  81,  256,  625;
1, 32, 243, 1024, 3125];

% Balance matrix pair (A,B)
job = 'Balance';
[a, b, ilo, ihi, lscale, rscale, info] = ...
f08wh(job, a, b);

fprintf('Lower index for permuted central block matrix pair = %d\n', ilo);
fprintf('Upper index for permuted central block matrix pair = %d\n', ihi);
disp('Contents of left scaling vector:');
disp(lscale');
disp('Contents of right scaling vector:');
disp(rscale');
disp('Balanced matrix A');
disp(a);
disp('Balanced matrix B');
disp(b);

```
```f08wh example results

Lower index for permuted central block matrix pair = 1
Upper index for permuted central block matrix pair = 5
Contents of left scaling vector:
1.0000    1.0000    0.1000    0.1000    0.1000

Contents of right scaling vector:
1.0000    1.0000    0.1000    0.1000    0.1000

Balanced matrix A
1.0000    1.0000    0.1000    0.1000    0.1000
2.0000    4.0000    0.8000    1.6000    3.2000
0.3000    0.9000    0.2700    0.8100    2.4300
0.4000    1.6000    0.6400    2.5600   10.2400
0.5000    2.5000    1.2500    6.2500   31.2500

Balanced matrix B
1.0000    2.0000    0.3000    0.4000    0.5000
1.0000    4.0000    0.9000    1.6000    2.5000
0.1000    0.8000    0.2700    0.6400    1.2500
0.1000    1.6000    0.8100    2.5600    6.2500
0.1000    3.2000    2.4300   10.2400   31.2500

```