The Fourier sine transform defined above is its own inverse, and two consecutive calls of this function with the same data will restore the original data.

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)). (See the C06 Chapter Introduction.)

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

5

and

6

. This function is designed to be particularly efficient on vector processors, and it becomes especially fast as

m

, the number of transforms to be computed in parallel, increases.

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: $m$ – int64int32nag_int scalar

m

, the number of sequences to be transformed.

Constraint:

m \geq 1

2: $n$ – 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

3: $x (m \times n)$ – double array

The data must be stored in x as if in a two-dimensional array of dimension

(1 : m, 1 : n)

; each of the

m

sequences is stored in a row of the array. In other words, if the

n - 1

data values of the

p

th sequence to be transformed are denoted by

x_{j}^{p}

, for

j = 1, 2, \dots, n - 1

and

p = 1, 2, \dots, m

, then the first

m (n - 1)

elements of the array x must contain the values

x_{1}^{1}, x_{1}^{2}, \dots, x_{1}^{m}, x_{2}^{1}, x_{2}^{2}, \dots, x_{2}^{m}, \dots, x_{n - 1}^{1}, x_{n - 1}^{2}, \dots, x_{n - 1}^{m} .

The

n

th element of each row

x_{n}^{p}

, for

p = 1, 2, \dots, m

, is required as workspace. These

m

elements may contain arbitrary values on entry, and are set to zero by the function.

4: $init$ – string (length ≥ 1)

Indicates whether trigonometric coefficients are to be calculated.

$init ='I'$: Calculate the required trigonometric coefficients for the given value of $n$ , and store in the array trig.
$init ='S'$ or $'R'$: 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_sine (c06ha), nag_sum_fft_real_cosine (c06hb), nag_sum_fft_real_qtrsine (c06hc) or nag_sum_fft_real_qtrcosine (c06hd). The function performs a simple check that the current value of $n$ is consistent with the values stored in trig.

Constraint:

init ='I'

'S'

'R'

5: $trig (2 \times n)$ – double array

init ='S'

'R'

, trig must contain the required trigonometric coefficients calculated in a previous call of the function. Otherwise trig need not be set.

Optional Input Parameters

None.

Output Parameters

1: $x (m \times n)$ – double array

The

m

Fourier transforms stored as if in a two-dimensional array of dimension

(1 : m, 1 : n)

. Each of the

m

transforms is stored in a row of the array, overwriting the corresponding original sequence. If the

n - 1

components of the

p

th Fourier sine transform are denoted by

{\hat{x}}_{k}^{p}

, for

k = 1, 2, \dots, n - 1

and

p = 1, 2, \dots, m

, then the

m n

elements of the array x contain the values

{\hat{x}}_{1}^{1}, {\hat{x}}_{1}^{2}, \dots, {\hat{x}}_{1}^{m}, {\hat{x}}_{2}^{1}, {\hat{x}}_{2}^{2}, \dots, {\hat{x}}_{2}^{m}, \dots, {\hat{x}}_{n - 1}^{1}, {\hat{x}}_{n - 1}^{2}, \dots, {\hat{x}}_{n - 1}^{m}, 0, 0, \dots, 0 (m times) .

n = 1

, the

m

elements of x are set to zero.

2: $trig (2 \times n)$ – double array

Contains the required coefficients (computed by the function if

init ='I'

3: $ifail$ – 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:

$ifail = 1$

On entry,

m < 1

$ifail = 2$

On entry,

n < 1

$ifail = 3$

On entry,

init \neq'I'

'S'

'R'

$ifail = 4$: Not used at this Mark.

$ifail = 5$

On entry,

init ='S'

'R'

, but the array trig and the current value of n are inconsistent.

$ifail = 6$: An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.

$ifail = - 99$: 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_real_sine (c06ha) is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. nag_sum_fft_real_sine (c06ha) 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.

Example

This example reads in sequences of real data values and prints their Fourier sine transforms (as computed by nag_sum_fft_real_sine (c06ha)). It then calls nag_sum_fft_real_sine (c06ha) again and prints the results which may be compared with the original sequence.

Open in the MATLAB editor: c06ha_example

function c06ha_example


fprintf('c06ha example results\n\n');

% 3 sequences of real data
m = int64(3);
n = int64(6);
x = [ 0.6772  0.1138  0.6751  0.6362  0.1424  0;
      0.2983  0.1181  0.7255  0.8638  0.8723  0;
      0.0644  0.6037  0.6430  0.0428  0.4815  0];

disp('Original data values:');
disp(x(:,1:n-1));

% Discrete Fourier Sine transform
init = 'Initial';
trig = zeros(2*n,1);
[xt, trig, ifail] = c06ha(m, n, x, init, trig);

disp('Discrete Fourier sine transforms:');
disp(xt(:,1:n-1));

% Restore data by another Sine transform
init = 'Subsequent';
[xr, trig, ifail] = c06ha(m, n, xt, init, trig);

disp('Original data as restored by inverse transform:');
disp(xr(:,1:n-1));

c06ha example results

Original data values:
    0.6772    0.1138    0.6751    0.6362    0.1424
    0.2983    0.1181    0.7255    0.8638    0.8723
    0.0644    0.6037    0.6430    0.0428    0.4815

Discrete Fourier sine transforms:
    1.0014    0.0062    0.0834    0.5286    0.2514
    1.2477   -0.6599    0.2570    0.0858    0.2658
    0.8521    0.0719   -0.0561   -0.4890    0.2056

Original data as restored by inverse transform:
    0.6772    0.1138    0.6751    0.6362    0.1424
    0.2983    0.1181    0.7255    0.8638    0.8723
    0.0644    0.6037    0.6430    0.0428    0.4815

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

Chapter Contents

Chapter Introduction

NAG Toolbox