1: $a (lda :)$ – double 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_dgetrf (f07ad).
2: $ipiv (n)$ – int64int32nag_int array: The row interchanges used to factorize matrix $A$ as returned by nag_lapack_dgetrf (f07ad).

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: $d$ – double scalar
2: $id$ – int64int32nag_int scalar: The determinant of $A$ is given by $d \times {2.0}^{id}$ . It is given in this form to avoid overflow or underflow.
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_real_gen (f03ba) is approximately proportional to

n

Example

This example computes the

L U

factorization with partial pivoting, and calculates the determinant, of the real matrix

(\begin{array}{r} 33 & 16 & 72 \\ - 24 & - 10 & - 57 \\ - 8 & - 4 & - 17 \end{array}) .

Open in the MATLAB editor: f03ba_example

function f03ba_example


fprintf('f03ba example results\n\n');

a = [ 33,  16,  72;
     -24, -10, -57;
      -8,  -4, -17];
% Compute LU factorisation of a
[a, ipiv, info] = f07ad(a);

fprintf('\n');
[ifail] = x04ca('g', 'n', a, 'Array a after factorization');

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

[d, id, ifail] = f03ba(a, ipiv);

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

f03ba example results


 Array a after factorization
             1          2          3
 1     33.0000    16.0000    72.0000
 2     -0.7273     1.6364    -4.6364
 3     -0.2424    -0.0741     0.1111

Pivots:
 1 2 3

d =       0.37500 id = 4
Value of determinant =   6.00000e+00

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

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Chapter Introduction

NAG Toolbox