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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

```