hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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

NAG Toolbox: nag_sum_withdraw_fft_real_1d_multi_rfmt (c06fp)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_sum_fft_real_1d_multi_rfmt (c06fp) computes the discrete Fourier transforms of m sequences, each containing n real data values. This function is designed to be particularly efficient on vector processors.


[x, trig, ifail] = c06fp(m, n, x, init, trig)
[x, trig, ifail] = nag_sum_withdraw_fft_real_1d_multi_rfmt(m, n, x, init, trig)


Given m sequences of n real data values xjp , for j=0,1,,n-1 and p=1,2,,m, nag_sum_fft_real_1d_multi_rfmt (c06fp) simultaneously calculates the Fourier transforms of all the sequences defined by
z^ k p = 1n j=0 n-1 xjp × exp -i 2πjkn ,   k= 0, 1, , n-1 ​ and ​ p= 1,2,,m .  
(Note the scale factor 1n  in this definition.)
The transformed values z^kp  are complex, but for each value of p the z^kp  form a Hermitian sequence (i.e., z^n-kp  is the complex conjugate of z^kp ), so they are completely determined by mn  real numbers (see also the C06 Chapter Introduction).
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term:
z^kp = 1n j=0 n-1 xjp × exp +i 2πjkn .  
To compute this form, this function should be followed by forming the complex conjugates of the z^kp ; that is xk=-xk, for k=n/2+1×m+1,,m×n.
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 coding 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.


Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350


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 real values in each sequence.
Constraint: n1.
3:     x m×n – double array
The data must be stored in x 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 the array. In other words, if the data values 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 .  
4:     init – string (length ≥ 1)
Indicates whether trigonometric coefficients are to be calculated.
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'.
5:     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


Output Parameters

1:     x m×n – double array
The m discrete Fourier transforms stored as if in a two-dimensional array of dimension 1:m,0:n-1. Each of the m transforms is stored in a row of the array in Hermitian form, overwriting the corresponding original sequence. If the n components of the discrete Fourier transform z^ k p are written as akp + i bkp, then for 0 k n/2, akp is contained in xpk, and for 1 k n-1 / 2, bkp is contained in xpn-k. (See also Real transforms in the C06 Chapter Introduction.)
2:     trig 2×n – double array
Contains the required coefficients (computed by the function if init='I').
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:
On entry,m<1.
On entry,n<1.
On entry,init'I', 'S' or 'R'.
Not used at this Mark.
On entry,init='S' or 'R', but the array trig and the current value of n are inconsistent.
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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_real_1d_multi_rfmt (c06fp) is approximately proportional to nm logn, but also depends on the factors of n. nag_sum_fft_real_1d_multi_rfmt (c06fp) 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.


This example reads in sequences of real data values and prints their discrete Fourier transforms (as computed by nag_sum_fft_real_1d_multi_rfmt (c06fp)). The Fourier transforms are expanded into full complex form using and printed. Inverse transforms are then calculated by conjugating and calling nag_sum_fft_hermitian_1d_multi_rfmt (c06fq) showing that the original sequences are restored.
function c06fp_example

fprintf('c06fp example results\n\n');

% 3 real sequences stored as rows
m = int64(3);
n = int64(6);
x = [0.3854  0.6772  0.1138  0.6751  0.6362  0.1424;
     0.5417  0.2983  0.1181  0.7255  0.8638  0.8723;
     0.9172  0.0644  0.6037  0.6430  0.0428  0.4815];

% Transform to get Hermitian sequences
init = 'Initial';
trig = zeros(2*n,1);
[xt, trig, ifail] = c06fp(m, n, x, init, trig);
disp('Discrete Fourier transforms in Hermitian format:');

for j = 1:m
  zt(j,:) = nag_herm2complex(xt(j,:));
title = 'Discrete Fourier transforms in full complex format:';
[ifail] = x04da('General','Non-unit', zt, title);

% Restore data by conjugation and back transform
init = 'Subsequent';
nd = double(n);
xt(1:m,floor(nd/2)+2:n) = -xt(1:m,floor(nd/2)+2:n);
[xr, trig, ifail] = c06fq(m, n, xt, init, trig);

disp('Original data as restored by inverse transform:');

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);
  if (mod(n,2)==0)
    z(n/2+1) = complex(x(n/2+1));
c06fp example results

Discrete Fourier transforms in Hermitian format:
    1.0737   -0.1041    0.1126   -0.1467   -0.3738   -0.0044
    1.3961   -0.0365    0.0780   -0.1521   -0.0607    0.4666
    1.1237    0.0914    0.3936    0.1530    0.3458   -0.0508

 Discrete Fourier transforms in full complex format:
             1          2          3          4          5          6
 1      1.0737    -0.1041     0.1126    -0.1467     0.1126    -0.1041
        0.0000    -0.0044    -0.3738     0.0000     0.3738     0.0044

 2      1.3961    -0.0365     0.0780    -0.1521     0.0780    -0.0365
        0.0000     0.4666    -0.0607     0.0000     0.0607    -0.4666

 3      1.1237     0.0914     0.3936     0.1530     0.3936     0.0914
        0.0000    -0.0508     0.3458     0.0000    -0.3458     0.0508

Original data as restored by inverse transform:
    0.3854    0.6772    0.1138    0.6751    0.6362    0.1424
    0.5417    0.2983    0.1181    0.7255    0.8638    0.8723
    0.9172    0.0644    0.6037    0.6430    0.0428    0.4815

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015