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NAG Toolbox: nag_sum_fft_real_1d_multi_rfmt (c06fp)

Purpose

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

Syntax

[x, trig, ifail] = c06fp(m, n, x, init, trig)
[x, trig, ifail] = nag_sum_fft_real_1d_multi_rfmt(m, n, x, init, trig)

Description

Given mm sequences of nn real data values xjp xjp , for j = 0,1,,n1j=0,1,,n-1 and p = 1,2,,mp=1,2,,m, nag_sum_fft_real_1d_multi_rfmt (c06fp) simultaneously calculates the Fourier transforms of all the sequences defined by
n1
kp = 1/(sqrt(n))xjp × exp(i(2πjk)/n),  k = 0,1,,n1​ and ​p = 1,2,,m.
j = 0
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 1/(sqrt(n)) 1n  in this definition.)
The transformed values kp z^kp  are complex, but for each value of pp the kp z^kp  form a Hermitian sequence (i.e., nkp z^n-kp  is the complex conjugate of kp z^kp ), so they are completely determined by mn mn  real numbers (see also the C06 Chapter Introduction).
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term:
n1
kp = 1/(sqrt(n))xjp × exp( + i(2πjk)/n).
j = 0
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 kp z^kp ; that is x(k) = x(k)x(k)=-x(k), for k = (n / 2 + 1) × m + 1,,m × nk=(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 22, 33, 44, 55 and 66. This function is designed to be particularly efficient on vector processors, and it becomes especially fast as mm, the number of transforms to be computed in parallel, increases.

References

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

Parameters

Compulsory Input Parameters

1:     m – int64int32nag_int scalar
mm, the number of sequences to be transformed.
Constraint: m1m1.
2:     n – int64int32nag_int scalar
nn, the number of real values in each sequence.
Constraint: n1n1.
3:     x( m × n m×n ) – double array
The data must be stored in x as if in a two-dimensional array of dimension (1 : m,0 : n1)(1:m,0:n-1); each of the mm sequences is stored in a row of the array. In other words, if the data values of the ppth sequence to be transformed are denoted by xjpxjp, for j = 0,1,,n1j=0,1,,n-1, then the mnmn elements of the array x must contain the values
x01 , x02 ,, x0m , x11 , x12 ,, x1m ,, xn11 , xn12 ,, xn1m .
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.
init = 'I'init='I'
Calculate the required trigonometric coefficients for the given value of nn, and store in the array trig.
init = 'S'init='S' or 'R''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 nn is consistent with the values stored in trig.
Constraint: init = 'I'init='I', 'S''S' or 'R''R'.
5:     trig( 2 × n 2×n ) – double array
If init = 'S'init='S' or 'R''R', trig must contain the required trigonometric coefficients that have been previously calculated. Otherwise trig need not be set.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

work

Output Parameters

1:     x( m × n m×n ) – double array
The mm discrete Fourier transforms stored as if in a two-dimensional array of dimension (1 : m,0 : n1)(1:m,0:n-1). Each of the mm transforms is stored in a row of the array in Hermitian form, overwriting the corresponding original sequence. If the nn components of the discrete Fourier transform kp z^ k p are written as akp + i bkpakp + i bkp, then for 0 k n / 20 k n/2, akpakp is contained in x(p,k)xpk, and for 1 k (n1) / 21 k (n-1) / 2, bkpbkp is contained in x(p,nk)xpn-k. (See also Section [Real transforms] in the C06 Chapter Introduction.)
2:     trig( 2 × n 2×n ) – double array
Contains the required coefficients (computed by the function if init = 'I'init='I').
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
On entry,m < 1m<1.
  ifail = 2ifail=2
On entry,n < 1n<1.
  ifail = 3ifail=3
On entry,init'I'init'I', 'S''S' or 'R''R'.
  ifail = 4ifail=4
Not used at this Mark.
  ifail = 5ifail=5
On entry,init = 'S'init='S' or 'R''R', but the array trig and the current value of n are inconsistent.
  ifail = 6ifail=6
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.

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_real_1d_multi_rfmt (c06fp) is approximately proportional to nm log(n)nm log(n), but also depends on the factors of nn. nag_sum_fft_real_1d_multi_rfmt (c06fp) is fastest if the only prime factors of nn are 22, 33 and 55, and is particularly slow if nn is a large prime, or has large prime factors.

Example

function nag_sum_fft_real_1d_multi_rfmt_example
m = int64(3);
n = int64(6);
x = [0.3854;
     0.5417;
     0.9172;
     0.6772;
     0.2983;
     0.0644;
     0.1138;
     0.1181;
     0.6037;
     0.6751;
     0.7255;
     0.643;
     0.6362;
     0.8638;
     0.0428;
     0.1424;
     0.8723;
     0.4815];
init = 'Initial';
trig = [0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0];
[xOut, trigOut, ifail] = nag_sum_fft_real_1d_multi_rfmt(m, n, x, init, trig)
 

xOut =

    1.0737
    1.3961
    1.1237
   -0.1041
   -0.0365
    0.0914
    0.1126
    0.0780
    0.3936
   -0.1467
   -0.1521
    0.1530
   -0.3738
   -0.0607
    0.3458
   -0.0044
    0.4666
   -0.0508


trigOut =

     1
     1
     1
     1
     1
     6
     0
     0
     0
     0
     0
     6


ifail =

                    0


function c06fp_example
m = int64(3);
n = int64(6);
x = [0.3854;
     0.5417;
     0.9172;
     0.6772;
     0.2983;
     0.0644;
     0.1138;
     0.1181;
     0.6037;
     0.6751;
     0.7255;
     0.643;
     0.6362;
     0.8638;
     0.0428;
     0.1424;
     0.8723;
     0.4815];
init = 'Initial';
trig = [0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0];
[xOut, trigOut, ifail] = c06fp(m, n, x, init, trig)
 

xOut =

    1.0737
    1.3961
    1.1237
   -0.1041
   -0.0365
    0.0914
    0.1126
    0.0780
    0.3936
   -0.1467
   -0.1521
    0.1530
   -0.3738
   -0.0607
    0.3458
   -0.0044
    0.4666
   -0.0508


trigOut =

     1
     1
     1
     1
     1
     6
     0
     0
     0
     0
     0
     6


ifail =

                    0



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

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