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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zhecon (f07mu)

## Purpose

nag_lapack_zhecon (f07mu) estimates the condition number of a complex Hermitian indefinite matrix $A$, where $A$ has been factorized by nag_lapack_zhetrf (f07mr).

## Syntax

[rcond, info] = f07mu(uplo, a, ipiv, anorm, 'n', n)
[rcond, info] = nag_lapack_zhecon(uplo, a, ipiv, anorm, 'n', n)

## Description

nag_lapack_zhecon (f07mu) estimates the condition number (in the $1$-norm) of a complex Hermitian indefinite matrix $A$:
 $κ1A=A1A-11 .$
Since $A$ is Hermitian, ${\kappa }_{1}\left(A\right)={\kappa }_{\infty }\left(A\right)={‖A‖}_{\infty }{‖{A}^{-1}‖}_{\infty }$.
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_zhetrf (f07mr) to compute the Bunch–Kaufman 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.
${\mathbf{uplo}}=\text{'U'}$
$A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the factorization of $A$, as returned by nag_lapack_zhetrf (f07mr).
3:     $\mathrm{ipiv}\left(:\right)$int64int32nag_int array
The dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Details of the interchanges and the block structure of $D$, as returned by nag_lapack_zhetrf (f07mr).
4:     $\mathrm{anorm}$ – double scalar
The $1$-norm of the original matrix $A$. anorm must be computed either before calling nag_lapack_zhetrf (f07mr) 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 first dimension of the array a and the second dimension of the arrays a, ipiv.
$n$, 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_zhecon (f07mu) involves solving a number of systems of linear equations of the form $Ax=b$; 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_zhetrs (f07ms) 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_dsycon (f07mg).

## Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= -1.36+0.00i 1.58+0.90i 2.21-0.21i 3.91+1.50i 1.58-0.90i -8.87+0.00i -1.84-0.03i -1.78+1.18i 2.21+0.21i -1.84+0.03i -4.63+0.00i 0.11+0.11i 3.91-1.50i -1.78-1.18i 0.11-0.11i -1.84+0.00i .$
Here $A$ is Hermitian indefinite and must first be factorized by nag_lapack_zhetrf (f07mr). The true condition number in the $1$-norm is $9.10$.
```function f07mu_example

fprintf('f07mu example results\n\n');

% Hermitian indefinite matrix A (Lower triangular part stored)
uplo = 'L';
a = [-1.36 + 0i,     0    + 0i,     0    + 0i,      0    + 0i;
1.58 - 0.90i, -8.87 + 0i,     0    + 0i,      0    + 0i;
2.21 + 0.21i, -1.84 + 0.03i, -4.63 + 0i,      0    + 0i;
3.91 - 1.50i, -1.78 - 1.18i,  0.11 - 0.11i,  -1.84 + 0i];

% Factorize
[af, ipiv, info] = f07mr( ...
uplo, a);

% Norm of A
an = a + a' - diag(diag(a));
anorm = norm(an,1);

% Condition number estimator
[rcond, info] = f07mu( ...
uplo, af, ipiv, anorm);

fprintf('Estimate of condition number = %9.2e\n', 1/rcond);

```
```f07mu example results

Estimate of condition number =  6.68e+00
```