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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dppequ (f07gf)

## Purpose

nag_lapack_dppequ (f07gf) computes a diagonal scaling matrix S $S$ intended to equilibrate a real n $n$ by n $n$ symmetric positive definite matrix A $A$, stored in packed format, and reduce its condition number.

## Syntax

[s, scond, amax, info] = f07gf(uplo, n, ap)
[s, scond, amax, info] = nag_lapack_dppequ(uplo, n, ap)

## Description

nag_lapack_dppequ (f07gf) computes a diagonal scaling matrix S $S$ chosen so that
 sj = 1 / sqrt(ajj) . $sj=1 / ajj .$
This means that the matrix B $B$ given by
 B = SAS , $B=SAS ,$
has diagonal elements equal to unity. This in turn means that the condition number of B $B$, κ2(B) ${\kappa }_{2}\left(B\right)$, is within a factor n $n$ 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:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of A$A$ is stored in the array ap, as follows:
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
The upper triangle of A$A$ is stored.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
The lower triangle of A$A$ is stored.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     n – int64int32nag_int scalar
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
3:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The n$n$ by n$n$ symmetric matrix A$A$, packed by columns.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + j(j1) / 2)${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for ij$i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + (2nj)(j1) / 2)${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for ij$i\ge j$.
Only the elements of ap corresponding to the diagonal elements A$A$ are referenced.

None.

None.

### Output Parameters

1:     s(n) – double array
If ${\mathbf{INFO}}={\mathbf{0}}$, s contains the diagonal elements of the scaling matrix S$S$.
2:     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 scond0.1${\mathbf{scond}}\ge 0.1$ and amax is neither too large nor too small, it is not worth scaling by S$S$.
3:     amax – double scalar
max|aij|$\mathrm{max}|{a}_{ij}|$. If amax is very close to overflow or underflow, the matrix A$A$ should be scaled.
4:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: ap, 4: s, 5: scond, 6: amax, 7: info.
INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, the i$i$th diagonal element of A$A$ is not positive (and hence A$A$ 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_zppequ (f07gt).

## Example

```function nag_lapack_dppequ_example
uplo = 'U';
n = int64(4);
ap = [4.16;
-312000;
50300000000;
0.56;
-83000;
0.76;
-0.1;
118000;
0.34;
1.18];
[s, scond, amax, info] = nag_lapack_dppequ(uplo, n, ap)
```
```

s =

0.4903
0.0000
1.1471
0.9206

scond =

3.8871e-06

amax =

5.0300e+10

info =

0

```
```function f07gf_example
uplo = 'U';
n = int64(4);
ap = [4.16;
-312000;
50300000000;
0.56;
-83000;
0.76;
-0.1;
118000;
0.34;
1.18];
[s, scond, amax, info] = f07gf(uplo, n, ap)
```
```

s =

0.4903
0.0000
1.1471
0.9206

scond =

3.8871e-06

amax =

5.0300e+10

info =

0

```