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NAG Toolbox

NAG Toolbox: nag_det_real_gen (f03ba)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_det_real_gen (f03ba) computes the determinant of a real n by n matrix A. nag_lapack_dgetrf (f07ad) must be called first to supply the matrix A in factorized form.

Syntax

[d, id, ifail] = f03ba(a, ipiv, 'n', n)
[d, id, ifail] = nag_det_real_gen(a, ipiv, 'n', n)

Description

nag_det_real_gen (f03ba) computes the determinant of a real n by n matrix A that has been factorized by a call to nag_lapack_dgetrf (f07ad). The determinant of A is the product of the diagonal elements of U with the correct sign determined by the row interchanges.

References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

Parameters

Compulsory Input Parameters

1:     alda: – 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:     ipivn 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×2.0id. 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: n1.
   ifail=3
Constraint: ldan.
   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 LU factorization with partial pivoting, and calculates the determinant, of the real matrix
33 16 72 -24 -10 -57 -8 -4 -17 .  
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

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Chapter Introduction
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