# NAG Library Routine Document

## 1Purpose

f07gtf (zppequ) computes a diagonal scaling matrix $S$ intended to equilibrate a complex $n$ by $n$ Hermitian positive definite matrix $A$, stored in packed format, and reduce its condition number.

## 2Specification

Fortran Interface
 Subroutine f07gtf ( uplo, n, ap, s, amax, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Out) :: s(n), scond, amax Complex (Kind=nag_wp), Intent (In) :: ap(*) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f07gtf_ (const char *uplo, const Integer *n, const Complex ap[], double s[], double *scond, double *amax, Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name zppequ.

## 3Description

f07gtf (zppequ) 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)).

## 4References

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

## 5Arguments

1:     $\mathbf{uplo}$ – Character(1)Input
On entry: indicates whether the upper or lower triangular part of $A$ is stored in the array ap, 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{ap}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the $n$ by $n$ Hermitian matrix $A$, packed by columns.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
Only the elements of ap corresponding to the diagonal elements $A$ are referenced.
4:     $\mathbf{s}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{info}}={\mathbf{0}}$, s contains the diagonal elements of the scaling matrix $S$.
5:     $\mathbf{scond}$ – Real (Kind=nag_wp)Output
On exit: 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 $S$.
6:     $\mathbf{amax}$ – Real (Kind=nag_wp)Output
On exit: $\mathrm{max}\left|{a}_{ij}\right|$. If amax is very close to overflow or underflow, the matrix $A$ should be scaled.
7:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error 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 $〈\mathit{\text{value}}〉$th diagonal element of $A$ is not positive (and hence $A$ cannot be positive definite).

## 7Accuracy

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

## 8Parallelism and Performance

f07gtf (zppequ) is not threaded in any implementation.

The real analogue of this routine is f07gff (dppequ).

## 10Example

This example equilibrates the Hermitian positive definite matrix $A$ given by
 $A = (3.23 ((1.51-1.92i 1.90+0.84i×105 ((0.42+2.50i (1.51+1.92i ((3.58 -0.23+1.11i×105 -1.18+1.37i 1.90-0.84i×105 -0.23-1.11i×105 4.09×1010 (2.33-0.14i×105 (0.42-2.50i (-1.18-1.37i 2.33+0.14i×105 ((4.29 .$
Details of the scaling factors and the scaled matrix are output.

### 10.1Program Text

Program Text (f07gtfe.f90)

### 10.2Program Data

Program Data (f07gtfe.d)

### 10.3Program Results

Program Results (f07gtfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017