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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$A$, where A$A$ 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$A$, in either the 1$1$-norm or the $\infty$-norm:
 κ1 (A) = ‖A‖1 ‖A − 1‖1   or   κ∞ (A) = ‖A‖∞ ‖A − 1‖∞ . $κ1 (A) = ‖A‖1 ‖A-1‖1 or κ∞ (A) = ‖A‖∞ ‖A-1‖∞ .$
Note that κ(A) = κ1(AH)${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{H}}\right)$.
Because the condition number is infinite if A$A$ is singular, the function actually returns an estimate of the reciprocal of the condition number.

## 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:     norm_p – string (length ≥ 1)
Indicates whether κ1(A)${\kappa }_{1}\left(A\right)$ or κ(A)${\kappa }_{\infty }\left(A\right)$ is estimated.
norm = '1'${\mathbf{norm}}=\text{'1'}$ or 'O'$\text{'O'}$
κ1(A)${\kappa }_{1}\left(A\right)$ is estimated.
norm = 'I'${\mathbf{norm}}=\text{'I'}$
κ(A)${\kappa }_{\infty }\left(A\right)$ is estimated.
Constraint: norm = '1'${\mathbf{norm}}=\text{'1'}$, 'O'$\text{'O'}$ or 'I'$\text{'I'}$.
2:     a(lda, : $:$) – complex array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
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 LU$LU$ factorization of A$A$, as returned by nag_lapack_zgetrf (f07ar).
3:     anorm – double scalar
If norm = '1'${\mathbf{norm}}=\text{'1'}$ or 'O'$\text{'O'}$, the 1$1$-norm of the original matrix A$A$.
If norm = 'I'${\mathbf{norm}}=\text{'I'}$, the $\infty$-norm of the original matrix A$A$.
anorm may be computed by calling nag_blas_zlange (f06ua) with the same value for the parameter norm_p.
anorm must be computed either before calling nag_lapack_zgetrf (f07ar) or else from a copy of the original matrix A$A$ (see Section [Example]).
Constraint: anorm0.0${\mathbf{anorm}}\ge 0.0$.

### Optional Input Parameters

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

lda work rwork

### Output Parameters

1:     rcond – double scalar
An estimate of the reciprocal of the condition number of A$A$. rcond is set to zero if exact singularity is detected or the estimate underflows. If rcond is less than machine precision, A$A$ is singular to working precision.
2:     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: norm_p, 2: n, 3: a, 4: lda, 5: anorm, 6: rcond, 7: work, 8: rwork, 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.

## Accuracy

The computed estimate rcond is never less than the true value ρ$\rho$, and in practice is nearly always less than 10ρ$10\rho$, 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$Ax=b$ or AHx = b${A}^{\mathrm{H}}x=b$; the number is usually 5$5$ and never more than 11$11$. Each solution involves approximately 8n2$8{n}^{2}$ 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$A$ is approximately singular.
The real analogue of this function is nag_lapack_dgecon (f07ag).

## Example

```function nag_lapack_zgecon_example
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,  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
[a, ipiv, info] = nag_lapack_zgetrf(a);

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

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

Estimate of condition number =   1.50e+02

```
```function f07au_example
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,  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
[a, ipiv, info] = f07ar(a);

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

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

Estimate of condition number =   1.50e+02

```