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# NAG Toolbox: nag_lapack_zppcon (f07gu)

## Purpose

nag_lapack_zppcon (f07gu) estimates the condition number of a complex Hermitian positive definite matrix $A$, where $A$ has been factorized by nag_lapack_zpptrf (f07gr), using packed storage.

## Syntax

[rcond, info] = f07gu(uplo, n, ap, anorm)
[rcond, info] = nag_lapack_zppcon(uplo, n, ap, anorm)

## Description

nag_lapack_zppcon (f07gu) estimates the condition number (in the $1$-norm) of a complex Hermitian positive definite 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_zpptrf (f07gr) to compute the Cholesky 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.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{n}$int64int32nag_int scalar
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathrm{ap}\left(:\right)$ – complex array
The dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
The Cholesky factor of $A$ stored in packed form, as returned by nag_lapack_zpptrf (f07gr).
4:     $\mathrm{anorm}$ – double scalar
The $1$-norm of the original matrix $A$. anorm must be computed either before calling nag_lapack_zpptrf (f07gr) or else from a copy of the original matrix $A$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.

None.

### 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_zppcon (f07gu) 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_zpptrs (f07gs) 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_dppcon (f07gg).

## Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .$
Here $A$ is Hermitian positive definite, stored in packed form, and must first be factorized by nag_lapack_zpptrf (f07gr). The true condition number in the $1$-norm is $201.92$.
```function f07gu_example

fprintf('f07gu example results\n\n');

uplo = 'L';
n = int64(4);
ap = [3.23 + 0i    1.51 + 1.92i    1.90 - 0.84i    0.42 - 2.50i ...
3.58 + 0i      -0.23 - 1.11i   -1.18 - 1.37i ...
4.09 + 0.00i    2.33 + 0.14i ...
4.29 + 0.00i];

[L, info] = f07gr( ...
uplo, n, ap);

anorm = norm([ap(4) ap(7) ap(9) ap(10)], 1);

[rcond, info] = f07gu( ...
uplo, n, L, anorm);

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

```
```f07gu example results

Estimate of condition number =  1.51e+02
```

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