1: $x (n)$ – double array: If the data values $z_{j}$ are written as $x_{j} + i y_{j}$ and if x is declared with bounds $(0 : n - 1)$ in the function from which nag_sum_conjugate_hermitian_rfmt (c06gb) is called, then for $0 \leq j \leq n / 2$ , $x (j)$ must contain $x_{j}$ ( $= x_{n - j}$ ), while for $n / 2 < j \leq n - 1$ , $x (j)$ must contain ${- y}_{j}$ ( $= y_{n - j}$ ). In other words, x must contain the Hermitian sequence in Hermitian form. (See also Real transforms in the C06 Chapter Introduction.)

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array x.
$n$ , the number of data values.

Constraint: $n \geq 1$ .

Output Parameters

1: $x (n)$ – double array: The imaginary parts $y_{j}$ are negated. The real parts $x_{j}$ are not referenced.
2: $ifail$ – 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:

$ifail = 1$

On entry,

n < 1

$ifail = - 99$: 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

Exact.

Further Comments

The time taken by nag_sum_conjugate_hermitian_rfmt (c06gb) is negligible.

Example

This example reads in a sequence of real data values, calls nag_sum_fft_real_1d_nowork (c06ea) followed by nag_sum_conjugate_hermitian_rfmt (c06gb) to compute their inverse discrete Fourier transform, and prints this after expanding it from Hermitian form into a full complex sequence.

Open in the MATLAB editor: c06gb_example

function c06gb_example


fprintf('c06gb example results\n\n');

% real data
n = 7;
x = [0.34907  0.54890  0.74776  0.94459  1.13850  1.32850  1.51370];

% transform
[xt, ifail] = c06ea(x);

% get result in form useful for printing.
zt = nag_herm2complex(xt);
disp('Discrete Fourier Transform of x:');
disp(transpose(zt));

% restore by conjugating and backtransforming
[xt, ifail] = c06gb(xt);
[xr, ifail] = c06eb(xt);

fprintf('Original sequence as restored by inverse transform\n\n');
fprintf('       Original   Restored\n');
for j = 1:n
  fprintf('%3d   %7.4f    %7.4f\n',j, x(j),xr(j));
end




function [z] = nag_herm2complex(x);
  n = size(x,2);
  z(1) = complex(x(1));
  for j = 2:floor((n-1)/2) + 1
    z(j) = x(j) + i*x(n-j+2);
    z(n-j+2) = x(j) - i*x(n-j+2);
  end
  if (mod(n,2)==0)
    z(n/2+1) = complex(x(n/2+1));
  end

c06gb example results

Discrete Fourier Transform of x:
   2.4836 + 0.0000i
  -0.2660 + 0.5309i
  -0.2577 + 0.2030i
  -0.2564 + 0.0581i
  -0.2564 - 0.0581i
  -0.2577 - 0.2030i
  -0.2660 - 0.5309i

Original sequence as restored by inverse transform

       Original   Restored
  1    0.3491     0.3491
  2    0.5489     0.5489
  3    0.7478     0.7478
  4    0.9446     0.9446
  5    1.1385     1.1385
  6    1.3285     1.3285
  7    1.5137     1.5137

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

Chapter Contents

Chapter Introduction

NAG Toolbox