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

The $n$ by $n$ matrix $A$ in factorized form as returned by nag_lapack_zgetrf (f07ar).
2: $ipiv (n)$ – int64int32nag_int array: The row interchanges used to factorize matrix $A$ as returned by nag_lapack_zgetrf (f07ar).

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a and the dimension of the array ipiv. (An error is raised if these dimensions are not equal.)
$n$ , the order of the matrix $A$ .

Constraint: $n > 0$ .

Output Parameters

1: $d$ – complex scalar: The mantissa of the real and imaginary parts of the determinant.
2: $id (2)$ – int64int32nag_int array: The exponents for the real and imaginary parts of the determinant. The determinant, $d = (d_{r}, d_{i})$ , is returned as $d_{r} = D_{r} \times 2^{j}$ and $d_{i} = D_{i} \times 2^{k}$ , where $d = (D_{r}, D_{i})$ and $j$ and $k$ are stored in the first and second elements respectively of the array id on successful exit.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $n \geq 1$ .

$ifail = 3$: Constraint: $lda \geq n$ .

$ifail = 4$: The matrix $A$ is approximately singular.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The accuracy of the determinant depends on the conditioning of the original matrix. For a detailed error analysis, see page 107 of Wilkinson and Reinsch (1971).

Further Comments

The time taken by nag_det_complex_gen (f03bn) is approximately proportional to

n

Example

This example calculates the determinant of the complex matrix

(\begin{array}{lr} 1 & 1 + 2 i & 2 + 10 i \\ 1 + i & 3 i & - 5 + 14 i \\ 1 + i & 5 i & - 8 + 20 i \end{array}) .

Open in the MATLAB editor: f03bn_example

function f03bn_example


fprintf('f03bn example results\n\n');

a = [1,   1+2i, 2+10i;
     1+i, 3i,  -5+14i;
     1+i, 5i,  -8+20i];
% LU factorisation of a
[a, ipiv, info] = f07ar(a);

fprintf('\n');
[ifail] = x04da('g', 'n', a, 'Array a after factorization');

fprintf('\nPivots:\n');
fprintf(' %d', ipiv);
fprintf('\n\n');

[d, id, ifail] = f03bn(a, ipiv);

fprintf('\nd = %13.5f id = (%d, %d)\n', d, id);
fprintf('Value of determinant = (%13.5e, %13.5e)\n', ...
        real(d)*2^id(1),imag(d)*2^id(2));

f03bn example results


 Array a after factorization
             1          2          3
 1      1.0000     0.0000    -5.0000
        1.0000     3.0000    14.0000

 2      1.0000     0.0000    -3.0000
        0.0000     2.0000     6.0000

 3      0.5000     0.2500    -0.2500
       -0.5000     0.2500    -0.2500

Pivots:
 2 3 3


d =       0.06250 id = (4, 0)
Value of determinant = (  1.00000e+00,   0.00000e+00)

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

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