# NAG Library Function Document

## 1Purpose

nag_dpbequ (f07hfc) 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.

## 2Specification

 #include #include
 void nag_dpbequ (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, const double ab[], Integer pdab, double s[], double *scond, double *amax, NagError *fail)

## 3Description

nag_dpbequ (f07hfc) 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:    $\mathbf{order}$Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of $A$ is stored in the array ab, as follows:
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangle of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangle of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{kd}$IntegerInput
On entry: ${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, or the number of subdiagonals if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$.
Constraint: ${\mathbf{kd}}\ge 0$.
5:    $\mathbf{ab}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdab}}×{\mathbf{n}}\right)$.
On entry: the upper or lower triangle of the symmetric positive definite band matrix $A$ whose scaling factors are to be computed.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of ${A}_{ij}$, depends on the order and uplo arguments as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[{k}_{d}+i-j+\left(j-1\right)×{\mathbf{pdab}}\right]$, for $j=1,\dots ,n$ and $i=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right),\dots ,j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[i-j+\left(j-1\right)×{\mathbf{pdab}}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[j-i+\left(i-1\right)×{\mathbf{pdab}}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+{k}_{d}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[{k}_{d}+j-i+\left(i-1\right)×{\mathbf{pdab}}\right]$, for $i=1,\dots ,n$ and $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{k}_{d}\right),\dots ,i$.
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 ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, row $1$ of ab when ${\mathbf{uplo}}=\mathrm{Nag_Lower}$.)
6:    $\mathbf{pdab}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array ab.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{kd}}+1$.
7:    $\mathbf{s}\left[{\mathbf{n}}\right]$doubleOutput
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, s contains the diagonal elements of the scaling matrix $S$.
8:    $\mathbf{scond}$double *Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, 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 $S$.
9:    $\mathbf{amax}$double *Output
On exit: $\mathrm{max}\left|{a}_{ij}\right|$. If amax is very close to overflow or underflow, the matrix $A$ should be scaled.
10:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{kd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{kd}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}>0$.
NE_INT_2
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$ and ${\mathbf{kd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{kd}}+1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_MAT_NOT_POS_DEF
The $〈\mathit{\text{value}}〉$th diagonal element of $A$ is not positive (and hence $A$ cannot be positive definite).
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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

## 8Parallelism and Performance

nag_dpbequ (f07hfc) is not threaded in any implementation.

The complex analogue of this function is nag_zpbequ (f07htc).

## 10Example

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.

### 10.1Program Text

Program Text (f07hfce.c)

### 10.2Program Data

Program Data (f07hfce.d)

### 10.3Program Results

Program Results (f07hfce.r)

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