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NAG Toolbox: nag_sum_fft_complex_1d_multi_row (c06pr)

Purpose

nag_sum_fft_complex_1d_multi_row (c06pr) computes the discrete Fourier transforms of mm sequences, each containing nn complex data values.

Syntax

[x, ifail] = c06pr(direct, m, n, x)
[x, ifail] = nag_sum_fft_complex_1d_multi_row(direct, m, n, x)

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_row (c06pr) simultaneously calculates the (forward or backward) discrete 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 minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of nag_sum_fft_complex_1d_multi_row (c06pr) with direct = 'F'direct='F' followed by a call with direct = 'B'direct='B' will restore the original data.
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 and 55.

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:     direct – string (length ≥ 1)
If the forward transform as defined in Section [Description] is to be computed, then direct must be set equal to 'F'.
If the backward transform is to be computed then direct must be set equal to 'B'.
Constraint: direct = 'F'direct='F' or 'B''B'.
2:     m – int64int32nag_int scalar
mm, the number of sequences to be transformed.
Constraint: m1m1.
3:     n – int64int32nag_int scalar
nn, the number of complex values in each sequence.
Constraint: n1n1.
4:     x( m × n m×n ) – complex array
The complex 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 each array. In other words, if the elements of the ppth sequence to be transformed are denoted by zjpzjp, for j = 0,1,,n1j=0,1,,n-1, then x(j × m + p)xj×m+p must contain zjpzjp.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

work

Output Parameters

1:     x( m × n m×n ) – complex array
Stores the complex transforms.
2:     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,direct'F'direct'F' or 'B''B'.
  ifail = 4ifail=4
On entry,n has more than 3030 prime factors.
  ifail = 5ifail=5
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_row (c06pr) is approximately proportional to nm log(n)nm log(n), but also depends on the factors of nn. nag_sum_fft_complex_1d_multi_row (c06pr) 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_row_example
direct = 'F';
m = int64(3);
n = int64(6);
x = [ 0.3854 + 0.5417i;
      0.9172 + 0.9089i;
      0.1156 + 0.6214i;
      0.6772 + 0.2983i;
      0.0644 + 0.3118i;
      0.0685 + 0.8681i;
      0.1138 + 0.1181i;
      0.6037 + 0.3465i;
      0.206 + 0.706i;
      0.6751 + 0.7255i;
      0.643 + 0.6198i;
      0.863 + 0.8652i;
      0.6362 + 0.8638i;
      0.0428 + 0.2668i;
      0.6967 + 0.919i;
      0.1424 + 0.8723i;
      0.4815 + 0.1614i;
      0.2792 + 0.3355i];
[xOut, ifail] = nag_sum_fft_complex_1d_multi_row(direct, m, n, x)
 

xOut =

   1.0737 + 1.3961i
   1.1237 + 1.0677i
   0.9100 + 1.7617i
  -0.5706 - 0.0409i
   0.1728 + 0.0386i
  -0.3054 + 0.0624i
   0.1733 - 0.2958i
   0.4185 + 0.7481i
   0.4079 - 0.0695i
  -0.1467 - 0.1521i
   0.1530 + 0.1752i
  -0.0785 + 0.0725i
   0.0518 + 0.4517i
   0.3686 + 0.0565i
  -0.1193 + 0.1285i
   0.3625 - 0.0321i
   0.0101 + 0.1403i
  -0.5314 - 0.4335i


ifail =

                    0


function c06pr_example
direct = 'F';
m = int64(3);
n = int64(6);
x = [ 0.3854 + 0.5417i;
      0.9172 + 0.9089i;
      0.1156 + 0.6214i;
      0.6772 + 0.2983i;
      0.0644 + 0.3118i;
      0.0685 + 0.8681i;
      0.1138 + 0.1181i;
      0.6037 + 0.3465i;
      0.206 + 0.706i;
      0.6751 + 0.7255i;
      0.643 + 0.6198i;
      0.863 + 0.8652i;
      0.6362 + 0.8638i;
      0.0428 + 0.2668i;
      0.6967 + 0.919i;
      0.1424 + 0.8723i;
      0.4815 + 0.1614i;
      0.2792 + 0.3355i];
[xOut, ifail] = c06pr(direct, m, n, x)
 

xOut =

   1.0737 + 1.3961i
   1.1237 + 1.0677i
   0.9100 + 1.7617i
  -0.5706 - 0.0409i
   0.1728 + 0.0386i
  -0.3054 + 0.0624i
   0.1733 - 0.2958i
   0.4185 + 0.7481i
   0.4079 - 0.0695i
  -0.1467 - 0.1521i
   0.1530 + 0.1752i
  -0.0785 + 0.0725i
   0.0518 + 0.4517i
   0.3686 + 0.0565i
  -0.1193 + 0.1285i
   0.3625 - 0.0321i
   0.0101 + 0.1403i
  -0.5314 - 0.4335i


ifail =

                    0



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

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