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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgecon (f07au)

## Purpose

nag_lapack_zgecon (f07au) estimates the condition number of a complex matrix $A$, where $A$ has been factorized by nag_lapack_zgetrf (f07ar).

## Syntax

[rcond, info] = f07au(norm_p, a, anorm, 'n', n)
[rcond, info] = nag_lapack_zgecon(norm_p, a, anorm, 'n', n)

## Description

nag_lapack_zgecon (f07au) estimates the condition number of a complex matrix $A$, in either the $1$-norm or the $\infty$-norm:
 $κ1 A = A1 A-11 or κ∞ A = A∞ A-1∞ .$
Note that ${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{H}}\right)$.
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_zgetrf (f07ar) 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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_zgetrf (f07ar).
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_zgetrf (f07ar) 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_zgecon (f07au) involves solving a number of systems of linear equations of the form $Ax=b$ or ${A}^{\mathrm{H}}x=b$; the number is usually $5$ and never more than $11$. Each solution involves approximately $8{n}^{2}$ real floating-point operations but takes considerably longer than a call to nag_lapack_zgetrs (f07as) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The real analogue of this function is nag_lapack_dgecon (f07ag).

## Example

This example estimates the condition number in the $1$-norm of the matrix $A$, where
 $A= -1.34+2.55i 0.28+3.17i -6.39-2.20i 0.72-0.92i -0.17-1.41i 3.31-0.15i -0.15+1.34i 1.29+1.38i -3.29-2.39i -1.91+4.42i -0.14-1.35i 1.72+1.35i 2.41+0.39i -0.56+1.47i -0.83-0.69i -1.96+0.67i .$
Here $A$ is nonsymmetric and must first be factorized by nag_lapack_zgetrf (f07ar). The true condition number in the $1$-norm is $231.86$.
```function f07au_example

fprintf('f07au example results\n\n');

a = [-1.34 + 2.55i,  0.28 + 3.17i, -6.39 - 2.20i,  0.72 - 0.92i;
-0.17 - 1.41i,  3.31 - 0.15i, -0.15 + 1.34i,  1.29 + 1.38i;
-3.29 - 2.39i, -1.91 + 4.42i, -0.14 - 1.35i,  1.72 + 1.35i;
2.41 + 0.39i, -0.56 + 1.47i, -0.83 - 0.69i, -1.96 + 0.67i];

norm_p = '1';
anorm = norm(a, 1);

% Factorise a
[LU, ipiv, info] = f07ar(a);

% Estimate condition number
[rcond, info] = f07au( ...
norm_p, LU, anorm);

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

```
```f07au example results

Estimate of condition number =   1.50e+02
```