NAG Toolbox: nag_lapack_zppequ (f07gt)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)

Indicates whether the upper or lower triangular part of $A$ is stored in the array ap, as follows:

$\mathbf{uplo}=\text{'U'}$

The upper triangle of $A$ is stored.

$\mathbf{uplo}=\text{'L'}$

The lower triangle of $A$ is stored.

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 $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$

The $n$ by $n$ Hermitian matrix $A$, packed by columns.

More precisely,

• if $\mathbf{uplo}=\text{'U'}$, the upper triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the lower triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.

Only the elements of ap corresponding to the diagonal elements $A$ are referenced.

Optional Input Parameters

None.

Output Parameters

1:     $\mathrm{s}\left(\mathbf{n}\right)$ – double array

If $\mathbf{info}=\mathbf{0}$, s contains the diagonal elements of the scaling matrix $S$.

2:     $\mathrm{scond}$ – double scalar

If $\mathbf{info}=\mathbf{0}$, scond contains the ratio of the smallest value of s to the largest value of s. If $\mathbf{scond}\ge 0.1$ and amax is neither too large nor too small, it is not worth scaling by $S$.

3:     $\mathrm{amax}$ – double scalar

$\max \left|a_{ij}\right|$. If amax is very close to overflow or underflow, the matrix $A$ should be scaled.

4:     $\mathrm{info}$ – 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.

   $\mathbf{info}>0$

The $\_$th diagonal element of $A$ is not positive (and hence $A$ cannot be positive definite).

Accuracy

Further Comments

Example

Open in the MATLAB editor: f07gt_example

function f07gt_example


fprintf('f07gt example results\n\n');

% Upper triangular part of Hermitian matrix A
uplo = 'Upper';
n = int64(4);
ap = [ 3.23   + 0i,       ...
       1.51   - 1.92i,    3.58   + 0i,      ...
       1.90e5 + 0.84e5i, -0.23e5 + 1.11e5i, 4.09e10 + 0i,       ...
       0.42   + 2.50i,   -1.18   + 1.37i,   2.33e5  - 0.14e5i,  4.29 + 0i];

% Scale A
[s, scond, amax, info] = f07gt( ...
                                uplo, n, ap);

fprintf('scond = %8.1e, amax = %8.1e\n\n', scond, amax);
disp('Diagonal scaling factors');
fprintf('%10.1e',s);
fprintf('\n\n');

% Apply scalings
k = 0;
for i = 1:n
  for j = 1:i
    k = k + 1;
    asp(k) = s(i)*ap(k)*s(j);
  end
end

[ifail] = x04dc( ...
                 'Upper', 'Non-unit', n, asp, 'Scaled matrix');

f07gt example results

scond =  8.9e-06, amax =  4.1e+10

Diagonal scaling factors
   5.6e-01   5.3e-01   4.9e-06   4.8e-01

 Scaled matrix
          1       2       3       4
 1   1.0000  0.4441  0.5227  0.1128
     0.0000 -0.5646  0.2311  0.6716

 2           1.0000 -0.0601 -0.3011
             0.0000  0.2901  0.3496

 3                   1.0000  0.5562
                     0.0000 -0.0334

 4                           1.0000
                             0.0000