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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgecon (f07ag)

## Purpose

nag_lapack_dgecon (f07ag) estimates the condition number of a real matrix $A$, where $A$ has been factorized by nag_lapack_dgetrf (f07ad).

## Syntax

[rcond, info] = f07ag(norm_p, a, anorm, 'n', n)
[rcond, info] = nag_lapack_dgecon(norm_p, a, anorm, 'n', n)

## Description

nag_lapack_dgecon (f07ag) estimates the condition number of a real matrix $A$, in either the $1$-norm or the $\infty$-norm:
 $κ1 A = A1 A-11 or κ∞ A = A∞ A-1∞ .$
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_dgetrf (f07ad) 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a 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_dgetrf (f07ad).
3:     $\mathrm{anorm}$ – 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_dgetrf (f07ad) 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 first dimension of the array a and the second dimension of the array a.
$n$, 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_dgecon (f07ag) 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 $2{n}^{2}$ floating-point operations but takes considerably longer than a call to nag_lapack_dgetrs (f07ae) 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_zgecon (f07au).

## Example

This example estimates the condition number in the $1$-norm of the matrix $A$, where
 $A= 1.80 2.88 2.05 -0.89 5.25 -2.95 -0.95 -3.80 1.58 -2.69 -2.90 -1.04 -1.11 -0.66 -0.59 0.80 .$
Here $A$ is nonsymmetric and must first be factorized by nag_lapack_dgetrf (f07ad). The true condition number in the $1$-norm is $152.16$.
```function f07ag_example

fprintf('f07ag example results\n\n');

a = [ 1.80,  2.88,  2.05, -0.89;
5.25, -2.95, -0.95, -3.80;
1.58, -2.69, -2.90, -1.04;
-1.11, -0.66, -0.59,  0.80];

anorm = norm(a, 1);

% Factorize A

% Estimate condition number
norm_p = '1';
[rcond, info] = f07ag(norm_p, a, anorm);

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

```
```f07ag example results

Estimate of condition number =   1.52e+02
```