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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dpbcon (f07hg)

## Purpose

nag_lapack_dpbcon (f07hg) estimates the condition number of a real symmetric positive definite band matrix $A$, where $A$ has been factorized by nag_lapack_dpbtrf (f07hd).

## Syntax

[rcond, info] = f07hg(uplo, kd, ab, anorm, 'n', n)
[rcond, info] = nag_lapack_dpbcon(uplo, kd, ab, anorm, 'n', n)

## Description

nag_lapack_dpbcon (f07hg) estimates the condition number (in the $1$-norm) of a real symmetric positive definite band matrix $A$:
 $κ1A=A1A-11 .$
Since $A$ is symmetric, ${\kappa }_{1}\left(A\right)={\kappa }_{\infty }\left(A\right)={‖A‖}_{\infty }{‖{A}^{-1}‖}_{\infty }$.
Because ${\kappa }_{1}\left(A\right)$ is infinite if $A$ is singular, the function actually returns an estimate of the reciprocal of ${\kappa }_{1}\left(A\right)$.
The function should be preceded by a computation of ${‖A‖}_{1}$ and a call to nag_lapack_dpbtrf (f07hd) to compute the Cholesky factorization of $A$. The function then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate ${‖{A}^{-1}‖}_{1}$.

## 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:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{kd}$int64int32nag_int scalar
${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix $A$.
Constraint: ${\mathbf{kd}}\ge 0$.
3:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array
The first dimension of the array ab must be at least ${\mathbf{kd}}+1$.
The second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The Cholesky factor of $A$, as returned by nag_lapack_dpbtrf (f07hd).
4:     $\mathrm{anorm}$ – double scalar
The $1$-norm of the original matrix $A$. anorm must be computed either before calling nag_lapack_dpbtrf (f07hd) or else from a copy of the original matrix $A$.
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_dpbcon (f07hg) involves solving a number of systems of linear equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $4nk$ floating-point operations (assuming $n\gg k$) but takes considerably longer than a call to nag_lapack_dpbtrs (f07he) 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_zpbcon (f07hu).

## Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 .$
Here $A$ is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by nag_lapack_dpbtrf (f07hd). The true condition number in the $1$-norm is $74.15$.
```function f07hg_example

fprintf('f07hg example results\n\n');

% Symmetric banded matrix A in ab.
uplo = 'L';
kd  = int64(1);
ab  = [5.49,  5.63,  2.60, 5.17;
2.68, -2.39, -2.22, 0.00];

% To calculate 1-norm here, need to add superdiagonal
abn = [0.00,  2.68, -2.39, -2.22;
ab];
% 1-norm of A = 1-norm of abn
anorm = norm(abn,1);

% Factorize A
[abf, info] = f07hd( ...
uplo, kd, ab);

% Get reciprocal condition number
[rcond, info] = f07hg( ...
uplo, kd, abf, anorm);

fprintf('Condition number of A = %7.1f\n',1/rcond);

```
```f07hg example results

Condition number of A =    74.2
```