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NAG Toolbox: nag_sum_withdraw_fft_complex_1d_multi_rfmt (c06fr)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_sum_fft_complex_1d_multi_rfmt (c06fr) computes the discrete Fourier transforms of m sequences, each containing n complex data values. This function is designed to be particularly efficient on vector processors.
Note: this function is scheduled to be withdrawn, please see c06fr in Advice on Replacement Calls for Withdrawn/Superseded Routines..

Syntax

[x, y, trig, ifail] = c06fr(m, n, x, y, init, trig)
[x, y, trig, ifail] = nag_sum_withdraw_fft_complex_1d_multi_rfmt(m, n, x, y, init, trig)

Description

Given m sequences of n complex data values zjp , for j=0,1,,n-1 and p=1,2,,m, nag_sum_fft_complex_1d_multi_rfmt (c06fr) simultaneously calculates the Fourier transforms of all the sequences defined by
z^kp = 1n j=0 n-1 zjp × exp -i 2πjk n ,   k= 0, 1, , n-1 ​ and ​ p= 1,2,,m .  
(Note the scale factor 1n  in this definition.)
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term
z^kp = 1n j=0 n-1 zjp × exp +i 2πjk n .  
To compute this form, this function should be preceded and followed by a call of nag_sum_conjugate_complex_sep (c06gc) to form the complex conjugates of the zjp  and the z^kp .
The function uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special code is provided for the factors 2, 3, 4, 5 and 6. This function is designed to be particularly efficient on vector processors, and it becomes especially fast as m, the number of transforms to be computed in parallel, increases.

References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23

Parameters

Compulsory Input Parameters

1:     m int64int32nag_int scalar
m, the number of sequences to be transformed.
Constraint: m1.
2:     n int64int32nag_int scalar
n, the number of complex values in each sequence.
Constraint: n1.
3:     x m×n – double array
4:     y m×n – 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 1:m,0:n-1; each of the m sequences is stored in a row of each array. In other words, if the real parts of the pth sequence to be transformed are denoted by xjp, for j=0,1,,n-1, then the mn elements of the array x must contain the values
x01 , x02 ,, x0m , x11 , x12 ,, x1m ,, x n-1 1 , x n-1 2 ,, x n-1 m .  
5:     init – string (length ≥ 1)
Indicates whether trigonometric coefficients are to be calculated.
init='I'
Calculate the required trigonometric coefficients for the given value of n, and store in the array trig.
init='S' or '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: init='I', 'S' or 'R'.
6:     trig 2×n – double array
If init='S' or '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:     x m×n – double array
2:     y m×n – double array
x and y store the real and imaginary parts of the complex transforms.
3:     trig 2×n – double array
Contains the required coefficients (computed by the function if init='I').
4:     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,m<1.
   ifail=2
On entry,n<1.
   ifail=3
On entry,init'I', 'S' or 'R'.
   ifail=4
Not used at this Mark.
   ifail=5
On entry,init='S' or 'R', but the array trig and the current value of n are inconsistent.
   ifail=6
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.
   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

Some indication of accuracy can be obtained by performing a subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).

Further Comments

The time taken by nag_sum_fft_complex_1d_multi_rfmt (c06fr) is approximately proportional to nm logn, but also depends on the factors of n. nag_sum_fft_complex_1d_multi_rfmt (c06fr) is fastest if the only prime factors of n are 2, 3 and 5, and is particularly slow if n is a large prime, or has large prime factors.

Example

This example reads in sequences of complex data values and prints their discrete Fourier transforms (as computed by nag_sum_fft_complex_1d_multi_rfmt (c06fr)). Inverse transforms are then calculated using nag_sum_fft_complex_1d_multi_rfmt (c06fr) and nag_sum_conjugate_complex_sep (c06gc) and printed out, showing that the original sequences are restored.
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

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
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