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

References

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

Parameters

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: $n \geq 1$ .
2: $x (0 : n, m)$ – double array: The data values of the $p$ th sequence to be transformed, denoted by $x_{j}^{p}$ , for $j = 0, 1, \dots, n$ and $p = 1, 2, \dots, m$ , must be stored in $x (j, p)$ .

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: $1$
$m$ , the number of sequences to be transformed.

Constraint: $m \geq 1$ .

Output Parameters

1: $x (0 : n, m)$ – double array: The $(n + 1)$ components of the $p$ th Fourier cosine transform, denoted by ${\hat{x}}_{k}^{p}$ , for $k = 0, 1, \dots, n$ and $p = 1, 2, \dots, m$ , are stored in $x (k, p)$ , 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:

$ifail = 1$: Constraint: $m \geq 1$ .

$ifail = 2$: Constraint: $n \geq 1$ .

$ifail = 3$: 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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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_cosine (c06rf) is approximately proportional to

n m \log (n)

, 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

O (n)

is internally allocated by this function.

Example

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.

Open in the MATLAB editor: c06rf_example

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:');
disp(x);

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

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