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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dpbequ (f07hf)

## Purpose

nag_lapack_dpbequ (f07hf) computes a diagonal scaling matrix S $S$ intended to equilibrate a real n $n$ by n $n$ symmetric positive definite band matrix A $A$, with bandwidth (2kd + 1) $\left(2{k}_{d}+1\right)$, and reduce its condition number.

## Syntax

[s, scond, amax, info] = f07hf(uplo, kd, ab, 'n', n)
[s, scond, amax, info] = nag_lapack_dpbequ(uplo, kd, ab, 'n', n)

## Description

nag_lapack_dpbequ (f07hf) computes a diagonal scaling matrix S $S$ chosen so that
 sj = 1 / sqrt(ajj) . $sj=1 / ajj .$
This means that the matrix B $B$ given by
 B = SAS , $B=SAS ,$
has diagonal elements equal to unity. This in turn means that the condition number of B $B$, κ2(B) ${\kappa }_{2}\left(B\right)$, is within a factor n $n$ of the matrix of smallest possible condition number over all possible choices of diagonal scalings (see Corollary 7.6 of Higham (2002)).

## References

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

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of A$A$ is stored in the array ab, as follows:
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
The upper triangle of A$A$ is stored.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
The lower triangle of A$A$ is stored.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     kd – int64int32nag_int scalar
kd${k}_{d}$, the number of superdiagonals of the matrix A$A$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$.
Constraint: kd0${\mathbf{kd}}\ge 0$.
3:     ab(ldab, : $:$) – double array
The first dimension of the array ab must be at least kd + 1${\mathbf{kd}}+1$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The upper or lower triangle of the symmetric positive definite band matrix A$A$ whose scaling factors are to be computed.
The matrix is stored in rows 1$1$ to kd + 1${k}_{d}+1$, more precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ij${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(1 + ij,j)​ for ​jimin (n,j + kd).${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
Only the elements of the array ab corresponding to the diagonal elements of A$A$ are referenced. (Row (kd + 1)$\left({k}_{d}+1\right)$ of ab when uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, row 1$1$ of ab when uplo = 'L'${\mathbf{uplo}}=\text{'L'}$.)

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The second dimension of the array ab.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

ldab

### Output Parameters

1:     s(n) – double array
If ${\mathbf{INFO}}={\mathbf{0}}$, s contains the diagonal elements of the scaling matrix S$S$.
2:     scond – double scalar
If ${\mathbf{INFO}}={\mathbf{0}}$, scond contains the ratio of the smallest value of s to the largest value of s. If scond0.1${\mathbf{scond}}\ge 0.1$ and amax is neither too large nor too small, it is not worth scaling by S$S$.
3:     amax – double scalar
max|aij|$\mathrm{max}|{a}_{ij}|$. If amax is very close to overflow or underflow, the matrix A$A$ should be scaled.
4:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: kd, 4: ab, 5: ldab, 6: s, 7: scond, 8: amax, 9: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, the i$i$th diagonal element of A$A$ is not positive (and hence A$A$ cannot be positive definite).

## Accuracy

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

The complex analogue of this function is nag_lapack_zpbequ (f07ht).

## Example

```function nag_lapack_dpbequ_example
uplo = 'U';
kd = int64(1);
ab = [0, 26800000000, -23900000000, -2.22;
5.49, 5.63e+20, 2.6, 5.17];
[s, scond, amax, info] = nag_lapack_dpbequ(uplo, kd, ab)
```
```

s =

0.4268
0.0000
0.6202
0.4398

scond =

6.7957e-11

amax =

5.6300e+20

info =

0

```
```function f07hf_example
uplo = 'U';
kd = int64(1);
ab = [0, 26800000000, -23900000000, -2.22;
5.49, 5.63e+20, 2.6, 5.17];
[s, scond, amax, info] = f07hf(uplo, kd, ab)
```
```

s =

0.4268
0.0000
0.6202
0.4398

scond =

6.7957e-11

amax =

5.6300e+20

info =

0

```