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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dpbequ (f07hf)

## Purpose

nag_lapack_dpbequ (f07hf) computes a diagonal scaling matrix $S$ intended to equilibrate a real $n$ by $n$ symmetric positive definite band matrix $A$, with bandwidth $\left(2{k}_{d}+1\right)$, and reduce its condition number.

## Syntax

[s, scond, amax, info] = f07hf(uplo, kd, ab, 'n', n)
[s, scond, amax, info] = nag_lapack_dpbequ(uplo, kd, ab, 'n', n)

## Description

nag_lapack_dpbequ (f07hf) computes a diagonal scaling matrix $S$ chosen so that
 $sj=1 / ajj .$
This means that the matrix $B$ given by
 $B=SAS ,$
has diagonal elements equal to unity. This in turn means that the condition number of $B$, ${\kappa }_{2}\left(B\right)$, is within a factor $n$ of the matrix of smallest possible condition number over all possible choices of diagonal scalings (see Corollary 7.6 of Higham (2002)).

## References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)
Indicates whether the upper or lower triangular part of $A$ is stored in the array ab, as follows:
${\mathbf{uplo}}=\text{'U'}$
The upper triangle of $A$ is stored.
${\mathbf{uplo}}=\text{'L'}$
The lower triangle of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{kd}$int64int32nag_int scalar
${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if ${\mathbf{uplo}}=\text{'L'}$.
Constraint: ${\mathbf{kd}}\ge 0$.
3:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array
The first dimension of the array ab must be at least ${\mathbf{kd}}+1$.
The second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The upper or lower triangle of the symmetric positive definite band matrix $A$ whose scaling factors are to be computed.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
Only the elements of the array ab corresponding to the diagonal elements of $A$ are referenced. (Row $\left({k}_{d}+1\right)$ of ab when ${\mathbf{uplo}}=\text{'U'}$, row $1$ of ab when ${\mathbf{uplo}}=\text{'L'}$.)

### Optional Input Parameters

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

### Output Parameters

1:     $\mathrm{s}\left({\mathbf{n}}\right)$ – double array
If ${\mathbf{info}}={\mathbf{0}}$, s contains the diagonal elements of the scaling matrix $S$.
2:     $\mathrm{scond}$ – double scalar
If ${\mathbf{info}}={\mathbf{0}}$, scond contains the ratio of the smallest value of s to the largest value of s. If ${\mathbf{scond}}\ge 0.1$ and amax is neither too large nor too small, it is not worth scaling by $S$.
3:     $\mathrm{amax}$ – double scalar
$\mathrm{max}\left|{a}_{ij}\right|$. If amax is very close to overflow or underflow, the matrix $A$ should be scaled.
4:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
The $_$th diagonal element of $A$ is not positive (and hence $A$ cannot be positive definite).

## Accuracy

The computed scale factors will be close to the exact scale factors.

The complex analogue of this function is nag_lapack_zpbequ (f07ht).

## Example

This example equilibrates the symmetric positive definite matrix $A$ given by
 $A = 5.49 -2.68×1010 -0 -0 2.68×1010 -5.63×1020 -2.39×1010 -0 0 -2.39×1010 -2.60 -2.22 0 -0 -2.22 -5.17 .$
Details of the scaling factors and the scaled matrix are output.
```function f07hf_example

fprintf('f07hf example results\n\n');

% Symmetric A (one lower/upper off-diagonal) in banded form
uplo = 'U';
kd = int64(1);
n  = int64(4);
ab = [0,    2.68e10, -2.39e10, -2.22;
5.49, 5.63e20,  2.60,     5.17];

% Scale A
[s, scond, amax, info] = f07hf( ...
uplo, kd, ab);

fprintf('scond = %8.1e, amax = %8.1e\n\n', scond, amax);
disp('Diagonal scaling factors');
fprintf('%10.1e',s);
fprintf('\n\n');

% Apply scalings
asp = ab*diag(s);
for i = 1:n
for j = 0:min(kd,n-i)
asp(kd+1-j,i+j) = s(i)*asp(kd+1-j,i+j);
end
end

kl = int64(0);
[ifail] = x04ce( ...
n, n, kl, kd, asp, 'Scaled matrix');

```
```f07hf example results

scond =  6.8e-11, amax =  5.6e+20

Diagonal scaling factors
4.3e-01   4.2e-11   6.2e-01   4.4e-01

Scaled matrix
1          2          3          4
1      1.0000     0.4821
2                 1.0000    -0.6247
3                            1.0000    -0.6055
4                                       1.0000
```