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# NAG Toolbox: nag_lapack_dpoequ (f07ff)

## Purpose

nag_lapack_dpoequ (f07ff) computes a diagonal scaling matrix $S$ intended to equilibrate a real $n$ by $n$ symmetric positive definite matrix $A$ and reduce its condition number.

## Syntax

[s, scond, amax, info] = f07ff(a, 'n', n)
[s, scond, amax, info] = nag_lapack_dpoequ(a, 'n', n)

## Description

nag_lapack_dpoequ (f07ff) 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 matrix $A$ whose scaling factors are to be computed. Only the diagonal elements of the array a are referenced.

### 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:     $\mathrm{s}\left({\mathbf{n}}\right)$ – 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.

## Further Comments

The complex analogue of this function is nag_lapack_zpoequ (f07ft).

## Example

This example equilibrates the symmetric positive definite matrix $A$ given by
 $A = -4.16 -3.12×105 -0.56 -0.10 -3.12×105 -5.03×1010 -0.83×105 -1.18×105 -0.56 -0.83×105 -0.76 -0.34 -0.10 -1.18×105 -0.34 -1.18 .$
Details of the scaling factors and the scaled matrix are output.
```function f07ff_example

fprintf('f07ff example results\n\n');

% Upper triangular part of symmetric matrix A
a = [ 4.16      -3.12e+05   0.56      -0.10    ;
0          5.03e+10  -8.30e+04   1.18e+05;
0          0          0.76       0.34    ;
0          0          0          1.18    ];

% Scale A
[s, scond, amax, info] = f07ff(a);

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

% Scaled matrix
as = diag(s)*a*diag(s);

[ifail] = x04ca( ...
'Upper', 'Non-unit', as, 'Scaled matrix');

```
```f07ff example results

scond =  3.9e-06, amax =  5.0e+10

Diagonal scaling factors
4.9e-01   4.5e-06   1.1e+00   9.2e-01

Scaled matrix
1          2          3          4
1      1.0000    -0.6821     0.3149    -0.0451
2                 1.0000    -0.4245     0.4843
3                            1.0000     0.3590
4                                       1.0000
```

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