# NAG Library Routine Document

## 1Purpose

f07htf (zpbequ) computes a diagonal scaling matrix $S$ intended to equilibrate a complex $n$ by $n$ Hermitian positive definite band matrix $A$, with bandwidth $\left(2{k}_{d}+1\right)$, and reduce its condition number.

## 2Specification

Fortran Interface
 Subroutine f07htf ( uplo, n, kd, ab, ldab, s, amax, info)
 Integer, Intent (In) :: n, kd, ldab Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Out) :: s(n), scond, amax Complex (Kind=nag_wp), Intent (In) :: ab(ldab,*) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f07htf_ ( const char *uplo, const Integer *n, const Integer *kd, const Complex ab[], const Integer *ldab, double s[], double *scond, double *amax, Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name zpbequ.

## 3Description

f07htf (zpbequ) 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 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{kd}$ – IntegerInput
On entry: ${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if ${\mathbf{uplo}}=\text{'L'}$.
Constraint: ${\mathbf{kd}}\ge 0$.
4:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the upper or lower triangle of the Hermitian 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 ${\mathbf{uplo}}=\text{'L'}$.)
5:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07htf (zpbequ) is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
6:     $\mathbf{s}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{info}}={\mathbf{0}}$, s contains the diagonal elements of the scaling matrix $S$.
7:     $\mathbf{scond}$ – 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 $S$.
8:     $\mathbf{amax}$ – 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.
9:     $\mathbf{info}$ – 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

f07htf (zpbequ) is not threaded in any implementation.

The real analogue of this routine is f07hff (dpbequ).

## 10Example

This example equilibrates the Hermitian positive definite matrix $A$ given by
 $A = 9.39 -i1.08-1.73i -i0 -i0 1.08+1.73i -i1.69 -0.04+0.29i×1010 -i0 0 -0.04-0.29i×1010 2.65×1020 -0.33+2.24i×1010 0 -i0 -0.33-2.24i×1010 -i2.17 .$
Details of the scaling factors and the scaled matrix are output.

### 10.1Program Text

Program Text (f07htfe.f90)

### 10.2Program Data

Program Data (f07htfe.d)

### 10.3Program Results

Program Results (f07htfe.r)

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