The transform calculated by this function can be used to solve Poisson's equation when 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 more 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: $x (n - 1, m)$ – double array: The data values of the $p$ th sequence to be transformed, denoted by $x_{j}^{p}$ , for $j = 1, 2, \dots, n - 1$ 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: $x (n - 1, m)$ – double array: The $(n - 1)$ components of the $p$ th Fourier sine transform, denoted by ${\hat{x}}_{k}^{p}$ , for $k = 1, 2, \dots, n - 1$ 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_sine (c06re) is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. nag_sum_fft_sine (c06re) 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 sine transforms (as computed by nag_sum_fft_sine (c06re)). It then calls nag_sum_fft_sine (c06re) again and prints the results which may be compared with the original sequence.

Open in the MATLAB editor: c06re_example

function c06re_example


fprintf('c06re example results\n\n');

% Discrete sine transform of 3 sequences of length 5
m = int64(3);
n = int64(6);
x = [0.6772  0.6037 0.8638;
     0.2983  0.6751 0.0428;
     0.0644  0.7255 0.1424;
     0.1138  0.6430 0.8723;
     0.1181  0.6362 0.4815];

[x, ifail] = c06re(n,x);
disp('X under discrete sine transform:');
disp(x);

% Reconstruct using same transform
[x, ifail] = c06re(n,x);
disp('X reconstructed under second sine transform:');
disp(x);

c06re example results

X under discrete sine transform:
    0.4728    1.4358    0.9281
    0.3718   -0.0002   -0.2236
    0.4220    0.2970    0.6945
    0.1873   -0.0323    0.6059
    0.0607    0.1177    0.0130

X reconstructed under second sine transform:
    0.6772    0.6037    0.8638
    0.2983    0.6751    0.0428
    0.0644    0.7255    0.1424
    0.1138    0.6430    0.8723
    0.1181    0.6362    0.4815

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox