1: $a (lda :)$ – complex array: The first dimension of the array a must be at least $\max (1, n)$ .
The second dimension of the array a must be at least $\max (1, n)$ .

The matrix $A$ whose scaling factors are to be computed. Only the diagonal elements of the array a are referenced.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a and the second dimension of the array a.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $s (n)$ – double array: If $info = 0$ , s contains the diagonal elements of the scaling matrix $S$ .
2: $scond$ – double scalar: If $info = 0$ , scond contains the ratio of the smallest value of s to the largest value of s. If $scond \geq 0.1$ and amax is neither too large nor too small, it is not worth scaling by $S$ .
3: $amax$ – double scalar: $\max |a_{i j}|$ . If amax is very close to overflow or underflow, the matrix $A$ should be scaled.
4: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: The $_$ th diagonal element of $A$ is not positive (and hence $A$ cannot be positive definite).

Accuracy

The computed scale factors will be close to the exact scale factors.

Further Comments

The real analogue of this function is nag_lapack_dpoequ (f07ff).

Example

This example equilibrates the Hermitian positive definite matrix

A

given by

A = (\begin{array}{l} 3.23 & 1.51 - 1.92 i & (1.90 + 0.84 i) \times 10^{5} & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 & (- 0.23 + 1.11 i) \times 10^{5} & - 1.18 + 1.37 i \\ (1.90 - 0.84 i) \times 10^{5} & (- 0.23 - 1.11 i) \times 10^{5} & 4.09 \times 10^{10} & (2.33 - 0.14 i) \times 10^{5} \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & (2.33 + 0.14 i) \times 10^{5} & 4.29 \end{array}) .

Details of the scaling factors and the scaled matrix are output.

Open in the MATLAB editor: f07ft_example

function f07ft_example


fprintf('f07ft example results\n\n');

a = [3.23 + 0i    1.51 - 1.92i  1.90e+05 + 8.40e+04i   0.42     + 2.50i;
     0    + 0i    3.58 + 0i    -2.30e+04 + 1.11e+05i  -1.18     + 1.37i;
     0    + 0i    0    + 0i     4.09e+10 + 0i          2.33e+05 - 1.40e+04i;
     0    + 0i    0    + 0i     0        + 0i          4.29     + 0i];

% Scale A
[s, scond, amax, info] = f07ft(a);

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

% Scaled matrix
as = diag(s)*a*diag(s);

[ifail] = x04da( ...
                 'Upper', 'Non-unit', as, 'Scaled matrix');

f07ft 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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox