NAG Toolbox: nag_sum_withdraw_fft_complex_1d_multi_rfmt (c06fr)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{m}$ – int64int32nag_int scalar

$m$, the number of sequences to be transformed.

Constraint: $\mathbf{m}\ge 1$.

2:     $\mathrm{n}$ – int64int32nag_int scalar

$n$, the number of complex values in each sequence.

Constraint: $\mathbf{n}\ge 1$.

3:     $\mathrm{x}\left(\mathbf{m}\times \mathbf{n}\right)$ – double array

4:     $\mathrm{y}\left(\mathbf{m}\times \mathbf{n}\right)$ – double array

The real and imaginary parts of the complex data must be stored in x and y respectively as if in a two-dimensional array of dimension $\left(1:\mathbf{m},0:\mathbf{n}-1\right)$; each of the $m$ sequences is stored in a row of each array. In other words, if the real parts of the $p$th sequence to be transformed are denoted by $x_{\mathit{j}}^p$, for $\mathit{j}=0,1,\dots ,n-1$, then the $mn$ elements of the array x must contain the values

$$x_0^1,x_0^2,\dots ,x_0^m,x_1^1,x_1^2,\dots ,x_1^m,\dots ,x_{n-1}^1,x_{n-1}^2,\dots ,x_{n-1}^m\text{.}$$

5:     $\mathrm{init}$ – string (length ≥ 1)

Indicates whether trigonometric coefficients are to be calculated.

$\mathbf{init}=\text{'I'}$

Calculate the required trigonometric coefficients for the given value of $n$, and store in the array trig.

$\mathbf{init}=\text{'S'}$ or $\text{'R'}$

The required trigonometric coefficients are assumed to have been calculated and stored in the array trig in a prior call to one of nag_sum_fft_real_1d_multi_rfmt (c06fp), nag_sum_fft_hermitian_1d_multi_rfmt (c06fq) or nag_sum_fft_complex_1d_multi_rfmt (c06fr). The function performs a simple check that the current value of $n$ is consistent with the values stored in trig.

Constraint: $\mathbf{init}=\text{'I'}$, $\text{'S'}$ or $\text{'R'}$.

6:     $\mathrm{trig}\left(2\times \mathbf{n}\right)$ – double array

If $\mathbf{init}=\text{'S'}$ or $\text{'R'}$, trig must contain the required trigonometric coefficients that have been previously calculated. Otherwise trig need not be set.

Optional Input Parameters

None.

Output Parameters

1:     $\mathrm{x}\left(\mathbf{m}\times \mathbf{n}\right)$ – double array

2:     $\mathrm{y}\left(\mathbf{m}\times \mathbf{n}\right)$ – double array

x and y store the real and imaginary parts of the complex transforms.

3:     $\mathrm{trig}\left(2\times \mathbf{n}\right)$ – double array

Contains the required coefficients (computed by the function if $\mathbf{init}=\text{'I'}$).

4:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry,

$\mathbf{m}<1$.

   $\mathbf{ifail}=2$

On entry,

$\mathbf{n}<1$.

   $\mathbf{ifail}=3$

On entry,

$\mathbf{init}\ne \text{'I'}$, $\text{'S'}$ or $\text{'R'}$.

   $\mathbf{ifail}=4$

Not used at this Mark.

   $\mathbf{ifail}=5$

On entry,

$\mathbf{init}=\text{'S'}$ or $\text{'R'}$, but the array trig and the current value of n are inconsistent.

   $\mathbf{ifail}=6$

An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: c06fr_example

function c06fr_example


fprintf('c06fr example results\n\n');

% 3 complex sequences, ral and imaginary parts stored as rows
% in separate arrays.
m = int64(3);
n = int64(6);
zr = [0.3854  0.6772  0.1138  0.6751  0.6362  0.1424;
      0.9172  0.0644  0.6037  0.6430  0.0428  0.4815;
      0.1156  0.0685  0.2060  0.8630  0.6967  0.2792];
zi = [0.5417  0.2983  0.1181  0.7255  0.8638  0.8723; 
      0.9089  0.3118  0.3465  0.6198  0.2668  0.1614; 
      0.6214  0.8681  0.7060  0.8652  0.9190  0.3355];

z = zr + i*zi;
title = 'Original sequences:';
[ifail] = x04da('General','Non-unit', z, title);

% Transform     
init = 'Initial';
trig = zeros(2*n,1);
[ztr, zti, trig, ifail] = c06fr(m, n, zr, zi, init, trig);

zt = ztr + i*zti;
disp(' ');
title = 'Discrete Fourier Transforms:';
[ifail] = x04da('General','Non-unit', zt, title);

% Restore by transform with pre- and post-conjugation
zti = -zti;
init = 'Subsequent';
[zrr, zri, trig, ifail] = c06fr(m, n, ztr, zti, init, trig);
zr = zrr - i*zri;

disp(' ');
title = 'Original data as restored by inverse transform';
[ifail] = x04da('General','Non-unit', zr, title);

c06fr example results

 Original sequences:
          1       2       3       4       5       6
 1   0.3854  0.6772  0.1138  0.6751  0.6362  0.1424
     0.5417  0.2983  0.1181  0.7255  0.8638  0.8723

 2   0.9172  0.0644  0.6037  0.6430  0.0428  0.4815
     0.9089  0.3118  0.3465  0.6198  0.2668  0.1614

 3   0.1156  0.0685  0.2060  0.8630  0.6967  0.2792
     0.6214  0.8681  0.7060  0.8652  0.9190  0.3355
 
 Discrete Fourier Transforms:
             1          2          3          4          5          6
 1      1.0737    -0.5706     0.1733    -0.1467     0.0518     0.3625
        1.3961    -0.0409    -0.2958    -0.1521     0.4517    -0.0321

 2      1.1237     0.1728     0.4185     0.1530     0.3686     0.0101
        1.0677     0.0386     0.7481     0.1752     0.0565     0.1403

 3      0.9100    -0.3054     0.4079    -0.0785    -0.1193    -0.5314
        1.7617     0.0624    -0.0695     0.0725     0.1285    -0.4335
 
 Original data as restored by inverse transform
          1       2       3       4       5       6
 1   0.3854  0.6772  0.1138  0.6751  0.6362  0.1424
     0.5417  0.2983  0.1181  0.7255  0.8638  0.8723

 2   0.9172  0.0644  0.6037  0.6430  0.0428  0.4815
     0.9089  0.3118  0.3465  0.6198  0.2668  0.1614

 3   0.1156  0.0685  0.2060  0.8630  0.6967  0.2792
     0.6214  0.8681  0.7060  0.8652  0.9190  0.3355