1: $a (lda :)$ – double 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 complex analogue of this function is nag_lapack_zpoequ (f07ft).

Example

This example equilibrates the symmetric positive definite matrix

A

given by

A = (\begin{array}{l} 4.16 & - 3.12 \times 10^{5} & 0.56 & - 0.10 \\ - 3.12 \times 10^{5} & 5.03 \times 10^{10} & - 0.83 \times 10^{5} & 1.18 \times 10^{5} \\ 0.56 & - 0.83 \times 10^{5} & 0.76 & 0.34 \\ - 0.10 & 1.18 \times 10^{5} & 0.34 & 1.18 \end{array}) .

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

Open in the MATLAB editor: f07ff_example

function f07ff_example


fprintf('f07ff example results\n\n');

% Upper triangular part of symmetric matrix A
a = [ 4.16      -3.12e+05   0.56      -0.10    ;
      0          5.03e+10  -8.30e+04   1.18e+05;
      0          0          0.76       0.34    ;
      0          0          0          1.18    ];

% Scale A
[s, scond, amax, info] = f07ff(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] = x04ca( ...
                 'Upper', 'Non-unit', as, 'Scaled matrix');

f07ff example results

scond =  3.9e-06, amax =  5.0e+10

Diagonal scaling factors
   4.9e-01   4.5e-06   1.1e+00   9.2e-01

 Scaled matrix
             1          2          3          4
 1      1.0000    -0.6821     0.3149    -0.0451
 2                 1.0000    -0.4245     0.4843
 3                            1.0000     0.3590
 4                                       1.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox