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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_sum_fft_cosine (c06rf)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_sum_fft_cosine (c06rf) computes the discrete Fourier cosine transforms of m sequences of real data values. The elements of each sequence and its transform are stored contiguously.


[x, ifail] = c06rf(n, x, 'm', m)
[x, ifail] = nag_sum_fft_cosine(n, x, 'm', m)


Given m sequences of n+1 real data values xjp , for j=0,1,,n and p=1,2,,m, nag_sum_fft_cosine (c06rf) simultaneously calculates the Fourier cosine transforms of all the sequences defined by
x^ k p = 2n 12 x0p + j=1 n-1 xjp × cos jk πn + 12 -1k xnp ,   k= 0, 1, , n ​ and ​ p= 1, 2, , m .  
(Note the scale factor 2n  in this definition.)
This transform is also known as type-I DCT.
Since the Fourier cosine transform defined above is its own inverse, two consecutive calls of nag_sum_fft_cosine (c06rf) will restore the original data.
The transform calculated by this function can be used to solve Poisson's equation when the derivative of the solution is specified at both left and right boundaries (see Swarztrauber (1977)).
The function uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, described in Temperton (1983), together with pre- and post-processing stages described in Swarztrauber (1982). Special coding is provided for the factors 2, 3, 4 and 5.


Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19(3) 490–501
Swarztrauber P N (1982) Vectorizing the FFT's Parallel Computation (ed G Rodrique) 51–83 Academic Press
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350


Compulsory Input Parameters

1:     n int64int32nag_int scalar
One less than the number of real values in each sequence, i.e., the number of values in each sequence is n+1.
Constraint: n1.
2:     x0:nm – double array
The data values of the pth sequence to be transformed, denoted by xjp, for j=0,1,,n and p=1,2,,m, must be stored in xjp.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: 1
m, the number of sequences to be transformed.
Constraint: m1.

Output Parameters

1:     x0:nm – double array
The n+1 components of the pth Fourier cosine transform, denoted by x^kp, for k=0,1,,n and p=1,2,,m, are stored in xkp, overwriting the corresponding original values.
2:     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:
Constraint: m1.
Constraint: n1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
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_cosine (c06rf) is approximately proportional to nm logn, but also depends on the factors of n. nag_sum_fft_cosine (c06rf) 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. Workspace of order On is internally allocated by this function.


This example reads in sequences of real data values and prints their Fourier cosine transforms (as computed by nag_sum_fft_cosine (c06rf)). It then calls nag_sum_fft_cosine (c06rf) again and prints the results which may be compared with the original sequence.
function c06rf_example

fprintf('c06rf example results\n\n');

% Discrete cosine transform of 3 sequences of length 7
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   0.6430;
      0.6362   0.8638   0.0428;
      0.1424   0.8723   0.4815;
      0.9562   0.4936   0.2057];

[x, ifail] = c06rf(n,x);
disp('X under discrete cosine transform:');

% Reconstruct using same transform
[x, ifail] = c06rf(n,x);
disp('X reconstructed under second cosine transform:');

c06rf example results

X under discrete cosine transform:
    1.6833    1.9605    1.3838
   -0.0482   -0.4884    0.1588
    0.0176   -0.0655   -0.0761
    0.1368    0.4444   -0.1184
    0.3240    0.0964    0.3512
   -0.5830    0.0856    0.5759
   -0.0427   -0.2289    0.0110

X reconstructed under second cosine transform:
    0.3854    0.5417    0.9172
    0.6772    0.2983    0.0644
    0.1138    0.1181    0.6037
    0.6751    0.7255    0.6430
    0.6362    0.8638    0.0428
    0.1424    0.8723    0.4815
    0.9562    0.4936    0.2057

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

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