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NAG Toolbox: nag_det_complex_gen (f03bn)

Purpose

nag_det_complex_gen (f03bn) computes the determinant of a complex nn by nn matrix AA. nag_lapack_zgetrf (f07ar) must be called first to supply the matrix AA in factorized form.

Syntax

[d, id, ifail] = f03bn(a, ipiv, 'n', n)
[d, id, ifail] = nag_det_complex_gen(a, ipiv, 'n', n)

Description

nag_det_complex_gen (f03bn) computes the determinant of a complex nn by nn matrix AA that has been factorized by a call to nag_lapack_zgetrf (f07ar). The determinant of AA is the product of the diagonal elements of UU 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:     a(lda, : :) – complex array
The first dimension of the array a must be at least nn
The second dimension of the array must be at least nn
The nn by nn matrix AA in factorized form as returned by nag_lapack_zgetrf (f07ar).
2:     ipiv(n) – int64int32nag_int array
n, the dimension of the array, must satisfy the constraint n > 0n>0.
The row interchanges used to factorize matrix AA 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.)
nn, the order of the matrix AA.
Constraint: n > 0n>0.

Input Parameters Omitted from the MATLAB Interface

lda

Output Parameters

1:     d – complex scalar
The mantissa of the real and imaginary parts of the determinant.
2:     id(22) – int64int32nag_int array
The exponents for the real and imaginary parts of the determinant. The determinant, d = (dr,di)d=(dr,di), is returned as dr = Dr × 2jdr=Dr×2j and di = Di × 2kdi=Di×2k, where d = (Dr,Di)d=(Dr,Di) and jj and kk are stored in the first and second elements respectively of the array id on successful exit.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
Constraint: n1n1.
  ifail = 3ifail=3
Constraint: ldanldan.
  ifail = 4ifail=4
The matrix AA is approximately singular.

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 nn.

Example

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

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

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

[d, id, ifail] = nag_det_complex_gen(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));
 

 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)

function f03bn_example
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));
 

 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)
Chapter Contents
Chapter Introduction
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