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NAG Toolbox: nag_sum_fft_complex_1d_multi_rfmt (c06fr)

Purpose

nag_sum_fft_complex_1d_multi_rfmt (c06fr) computes the discrete Fourier transforms of mm sequences, each containing nn 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Note: this function is scheduled to be withdrawn, please see c06fr in Advice on Replacement Calls for Withdrawn/Superseded Routines..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_fft_complex_1d_multi_rfmt(m, n, x, y, init, trig)

Description

Given mm sequences of nn complex data values zjp zjp , for j = 0,1,,n1j=0,1,,n-1 and p = 1,2,,mp=1,2,,m, nag_sum_fft_complex_1d_multi_rfmt (c06fr) simultaneously calculates the Fourier transforms of all the sequences defined by
n1
kp = 1/(sqrt(n))zjp × exp(i(2πjk)/n),  k = 0,1,,n1​ and ​p = 1,2,,m.
j = 0
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 1/(sqrt(n)) 1n  in this definition.)
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term
n1
kp = 1/(sqrt(n))zjp × exp( + i(2πjk)/n).
j = 0
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 zjp  and the kp 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 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) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23

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 complex values in each sequence.
Constraint: n1n1.
3:     x( m × n m×n ) – double array
4:     y( m × n 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 : n1)(1:m,0:n-1); each of the mm sequences is stored in a row of each array. In other words, if the real parts 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 .
5:     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'.
6:     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
2:     y( m × n m×n ) – double array
x and y store the real and imaginary parts of the complex transforms.
3:     trig( 2 × n 2×n ) – double array
Contains the required coefficients (computed by the function if init = 'I'init='I').
4:     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_complex_1d_multi_rfmt (c06fr) is approximately proportional to nm log(n)nm log(n), but also depends on the factors of nn. nag_sum_fft_complex_1d_multi_rfmt (c06fr) 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_complex_1d_multi_rfmt_example
m = int64(3);
n = int64(6);
x = [0.3854;
     0.9172;
     0.1156;
     0.6772;
     0.0644;
     0.0685;
     0.1138;
     0.6037;
     0.206;
     0.6751;
     0.643;
     0.863;
     0.6362;
     0.0428;
     0.6967;
     0.1424;
     0.4815;
     0.2792];
y = [0.5417;
     0.9089;
     0.6214;
     0.2983;
     0.3118;
     0.8681;
     0.1181;
     0.3465;
     0.706;
     0.7255;
     0.6198;
     0.8652;
     0.8638;
     0.2668;
     0.919;
     0.8723;
     0.1614;
     0.3355];
init = 'Initial';
trig = zeros(12,1);
[xOut, yOut, trigOut, ifail] = nag_sum_fft_complex_1d_multi_rfmt(m, n, x, y, init, trig)
 

xOut =

    1.0737
    1.1237
    0.9100
   -0.5706
    0.1728
   -0.3054
    0.1733
    0.4185
    0.4079
   -0.1467
    0.1530
   -0.0785
    0.0518
    0.3686
   -0.1193
    0.3625
    0.0101
   -0.5314


yOut =

    1.3961
    1.0677
    1.7617
   -0.0409
    0.0386
    0.0624
   -0.2958
    0.7481
   -0.0695
   -0.1521
    0.1752
    0.0725
    0.4517
    0.0565
    0.1285
   -0.0321
    0.1403
   -0.4335


trigOut =

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


ifail =

                    0


function c06fr_example
m = int64(3);
n = int64(6);
x = [0.3854;
     0.9172;
     0.1156;
     0.6772;
     0.0644;
     0.0685;
     0.1138;
     0.6037;
     0.206;
     0.6751;
     0.643;
     0.863;
     0.6362;
     0.0428;
     0.6967;
     0.1424;
     0.4815;
     0.2792];
y = [0.5417;
     0.9089;
     0.6214;
     0.2983;
     0.3118;
     0.8681;
     0.1181;
     0.3465;
     0.706;
     0.7255;
     0.6198;
     0.8652;
     0.8638;
     0.2668;
     0.919;
     0.8723;
     0.1614;
     0.3355];
init = 'Initial';
trig = zeros(12,1);
[xOut, yOut, trigOut, ifail] = c06fr(m, n, x, y, init, trig)
 

xOut =

    1.0737
    1.1237
    0.9100
   -0.5706
    0.1728
   -0.3054
    0.1733
    0.4185
    0.4079
   -0.1467
    0.1530
   -0.0785
    0.0518
    0.3686
   -0.1193
    0.3625
    0.0101
   -0.5314


yOut =

    1.3961
    1.0677
    1.7617
   -0.0409
    0.0386
    0.0624
   -0.2958
    0.7481
   -0.0695
   -0.1521
    0.1752
    0.0725
    0.4517
    0.0565
    0.1285
   -0.0321
    0.1403
   -0.4335


trigOut =

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


ifail =

                    0



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