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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgbcon (f07bg)

## Purpose

nag_lapack_dgbcon (f07bg) estimates the condition number of a real band matrix $A$, where $A$ has been factorized by nag_lapack_dgbtrf (f07bd).

## Syntax

[rcond, info] = f07bg(norm_p, kl, ku, ab, ipiv, anorm, 'n', n)
[rcond, info] = nag_lapack_dgbcon(norm_p, kl, ku, ab, ipiv, anorm, 'n', n)

## Description

nag_lapack_dgbcon (f07bg) estimates the condition number of a real band matrix $A$, in either the $1$-norm or the $\infty$-norm:
 $κ1A=A1A-11 or κ∞A=A∞A-1∞ .$
Note that ${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{T}}\right)$.
Because the condition number is infinite if $A$ is singular, the function actually returns an estimate of the reciprocal of the condition number.
The function should be preceded by a computation of ${‖A‖}_{1}$ or ${‖A‖}_{\infty }$, and a call to nag_lapack_dgbtrf (f07bd) to compute the $LU$ factorization of $A$. The function then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate ${‖{A}^{-1}‖}_{1}$ or ${‖{A}^{-1}‖}_{\infty }$.

## References

Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{norm_p}$ – string (length ≥ 1)
Indicates whether ${\kappa }_{1}\left(A\right)$ or ${\kappa }_{\infty }\left(A\right)$ is estimated.
${\mathbf{norm_p}}=\text{'1'}$ or $\text{'O'}$
${\kappa }_{1}\left(A\right)$ is estimated.
${\mathbf{norm_p}}=\text{'I'}$
${\kappa }_{\infty }\left(A\right)$ is estimated.
Constraint: ${\mathbf{norm_p}}=\text{'1'}$, $\text{'O'}$ or $\text{'I'}$.
2:     $\mathrm{kl}$int64int32nag_int scalar
${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
3:     $\mathrm{ku}$int64int32nag_int scalar
${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
4:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array
The first dimension of the array ab must be at least $2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
The second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $LU$ factorization of $A$, as returned by nag_lapack_dgbtrf (f07bd).
5:     $\mathrm{ipiv}\left(:\right)$int64int32nag_int array
The dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The pivot indices, as returned by nag_lapack_dgbtrf (f07bd).
6:     $\mathrm{anorm}$ – double scalar
If ${\mathbf{norm_p}}=\text{'1'}$ or $\text{'O'}$, the $1$-norm of the original matrix $A$.
If ${\mathbf{norm_p}}=\text{'I'}$, the $\infty$-norm of the original matrix $A$.
anorm must be computed either before calling nag_lapack_dgbtrf (f07bd) or else from a copy of the original matrix $A$ (see Example).
Constraint: ${\mathbf{anorm}}\ge 0.0$.

### Optional Input Parameters

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

### Output Parameters

1:     $\mathrm{rcond}$ – double scalar
An estimate of the reciprocal of the condition number of $A$. rcond is set to zero if exact singularity is detected or the estimate underflows. If rcond is less than machine precision, $A$ is singular to working precision.
2:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error 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.

## Accuracy

The computed estimate rcond is never less than the true value $\rho$, and in practice is nearly always less than $10\rho$, although examples can be constructed where rcond is much larger.

A call to nag_lapack_dgbcon (f07bg) involves solving a number of systems of linear equations of the form $Ax=b$ or ${A}^{\mathrm{T}}x=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2n\left(2{k}_{l}+{k}_{u}\right)$ floating-point operations (assuming $n\gg {k}_{l}$ and $n\gg {k}_{u}$) but takes considerably longer than a call to nag_lapack_dgbtrs (f07be) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The complex analogue of this function is nag_lapack_zgbcon (f07bu).

## Example

This example estimates the condition number in the $1$-norm of the matrix $A$, where
 $A= -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 .$
Here $A$ is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_lapack_dgbtrf (f07bd). The true condition number in the $1$-norm is $56.40$.
```function f07bg_example

fprintf('f07bg example results\n\n');

m  = int64(4);
kl = int64(1);
ku = int64(2);
ab = [ 0,    0,    -3.66, -2.13;
0,    2.54, -2.73,  4.07;
-0.23, 2.46,  2.46, -3.82;
-6.98, 2.56, -4.78,  0];

norm_p = 'O';
anorm = f16rb(norm_p, m, kl, ku, ab);

ab = [zeros(kl,m); ab];

% Factorize A
[abf, ipiv, info] = f07bd( ...
m, kl, ku, ab);

% Estimate condition number
[rcond, info] = f07bg( ...
norm_p, kl, ku, abf, ipiv, anorm);

if rcond > x02aj
fprintf('Estimate of condition number = %10.2e\n', 1/rcond);
else
fprintf('A is singular to working precision\n');
end

```
```f07bg example results

Estimate of condition number =   5.64e+01
```