F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07GFF (DPPEQU)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07GFF (DPPEQU) computes a diagonal scaling matrix $S$ intended to equilibrate a real $n$ by $n$ symmetric positive definite matrix $A$, stored in packed format, and reduce its condition number.

## 2  Specification

 SUBROUTINE F07GFF ( UPLO, N, AP, S, SCOND, AMAX, INFO)
 INTEGER N, INFO REAL (KIND=nag_wp) AP(*), S(N), SCOND, AMAX CHARACTER(1) UPLO
The routine may be called by its LAPACK name dppequ.

## 3  Description

F07GFF (DPPEQU) 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)).

## 4  References

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

## 5  Parameters

1:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     AP($*$) – REAL (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 $n$ symmetric 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:     S(N) – REAL (KIND=nag_wp) arrayOutput
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, S contains the diagonal elements of the scaling matrix $S$.
5:     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:     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:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, the $i$th diagonal element of $A$ is not positive (and hence $A$ cannot be positive definite).

## 7  Accuracy

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

The complex analogue of this routine is F07GTF (ZPPEQU).

## 9  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.

### 9.1  Program Text

Program Text (f07gffe.f90)

### 9.2  Program Data

Program Data (f07gffe.d)

### 9.3  Program Results

Program Results (f07gffe.r)