NAG Toolbox: nag_lapack_zhpcon (f07pu)

Purpose

Syntax

Description

References

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:     $\mathrm{ap}\left(:\right)$ – complex array

The dimension of the array ap must be at least $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$

The factorization of $A$ stored in packed form, as returned by nag_lapack_zhptrf (f07pr).

3:     $\mathrm{ipiv}\left(:\right)$ – int64int32nag_int array

The dimension of the array ipiv must be at least $\max \left(1,\mathbf{n}\right)$

Details of the interchanges and the block structure of $D$, as returned by nag_lapack_zhptrf (f07pr).

4:     $\mathrm{anorm}$ – double scalar

The $1$-norm of the original matrix $A$. anorm must be computed either before calling nag_lapack_zhptrf (f07pr) 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 dimension of the array 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

Further Comments

Example

Open in the MATLAB editor: f07pu_example

function f07pu_example


fprintf('f07pu example results\n\n');

% Hermitian indefinite matrix A (Lower triangular part stored in packed form)

% A in full format first to get 1-norm of A
n = int64(4);
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];
anorm = norm(a,1);

% pack A in array ap
uplo = 'L';
ap = [];
for j = 1:n
  ap = [ap; a(j:n,j)];
end

% Factorize packed form
[apf, ipiv, info] = f07pr( ...
                           uplo, n, ap);

% Condition number estimator
[rcond, info] = f07pu( ...
                       uplo, apf, ipiv, anorm);

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

f07pu example results

Estimate of condition number =  6.68e+00