F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07HFF (DPBEQU)

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

F07HFF (DPBEQU) computes a diagonal scaling matrix $S$ intended to equilibrate a real $n$ by $n$ symmetric positive definite band matrix $A$, with bandwidth $\left(2{k}_{d}+1\right)$, and reduce its condition number.

## 2  Specification

 SUBROUTINE F07HFF ( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO)
 INTEGER N, KD, LDAB, INFO REAL (KIND=nag_wp) AB(LDAB,*), S(N), SCOND, AMAX CHARACTER(1) UPLO
The routine may be called by its LAPACK name dpbequ.

## 3  Description

F07HFF (DPBEQU) 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 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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     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:     AB(LDAB,$*$) – REAL (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 symmetric 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:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07HFF (DPBEQU) is called.
Constraint: ${\mathbf{LDAB}}\ge {\mathbf{KD}}+1$.
6:     S(N) – REAL (KIND=nag_wp) arrayOutput
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, S contains the diagonal elements of the scaling matrix $S$.
7:     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:     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:     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 F07HTF (ZPBEQU).

## 9  Example

This example equilibrates the symmetric positive definite matrix $A$ given by
 $A = 5.49 -2.68×1010 -0 -0 2.68×1010 -5.63×1020 -2.39×1010 -0 0 -2.39×1010 -2.60 -2.22 0 -0 -2.22 -5.17 .$
Details of the scaling factors and the scaled matrix are output.

### 9.1  Program Text

Program Text (f07hffe.f90)

### 9.2  Program Data

Program Data (f07hffe.d)

### 9.3  Program Results

Program Results (f07hffe.r)