The transform calculated by this function can be used to solve Poisson's equation when the solution is specified at the left boundary, and the derivative of the solution is specified at the right boundary (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: $direct$ – string (length ≥ 1)

If the forward transform as defined in 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'

'B'

2: $m$ – int64int32nag_int scalar

m

, the number of sequences to be transformed.

Constraint:

m \geq 1

3: $n$ – int64int32nag_int scalar

n

, the number of real values in each sequence.

Constraint:

n \geq 1

4: $x (m \times n)$ – 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 data values of the

p

th sequence to be transformed are denoted by

x_{j}^{p}

, for

j = 1, 2, \dots, n

and

p = 1, 2, \dots, m

, then the

m n

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

5: $init$ – 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'

6: $trig (2 \times n)$ – 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

quarter-wave sine 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

components of the

p

th quarter-wave sine transform are denoted by

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

, for

k = 1, 2, \dots, n

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}, {\hat{x}}_{n}^{2}, \dots, {\hat{x}}_{n}^{m} .

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.

$ifail = 6$

On entry,

direct \neq'F'

'B'

$ifail = 7$: 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_qtrsine (c06hc) is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. nag_sum_fft_real_qtrsine (c06hc) 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 quarter-wave sine transforms as computed by nag_sum_fft_real_qtrsine (c06hc) with

direct ='F'

. It then calls the function again with

direct ='B'

and prints the results which may be compared with the original data.

Open in the MATLAB editor: c06hc_example

function c06hc_example


fprintf('c06hc example results\n\n');

m = int64(3);
n = int64(6);
x = [0.3854  0.6772  0.1138  0.6751  0.6362  0.1424;
     0.5417  0.2983  0.1181  0.7255  0.8638  0.8723;
     0.9172  0.0644  0.6037  0.6430  0.0428  0.4815];

direct = 'Forward';
init = 'Initial';
trig = zeros(2*n,1);
[xt, trig, ifail] = c06hc(direct, m, n, x, init, trig);

disp('Original data values:');
disp(x);

disp('Discrete quarter-wave Fourier sine transforms:');
disp(xt);

% Restore data by inverse transform
direct = 'Backward';
init = 'Subsequent';
[xr, trig, ifail] = c06hc(direct, m, n, xt, init, trig);

disp('Original data as restored by inverse transform:');
disp(xr);

c06hc example results

Original data values:
    0.3854    0.6772    0.1138    0.6751    0.6362    0.1424
    0.5417    0.2983    0.1181    0.7255    0.8638    0.8723
    0.9172    0.0644    0.6037    0.6430    0.0428    0.4815

Discrete quarter-wave Fourier sine transforms:
    0.7304    0.2078    0.1150    0.2577   -0.2869   -0.0815
    0.9274   -0.1152    0.2532    0.2883   -0.0026   -0.0635
    0.6268    0.3547    0.0760    0.3078    0.4987   -0.0507

Original data as restored by inverse transform:
    0.3854    0.6772    0.1138    0.6751    0.6362    0.1424
    0.5417    0.2983    0.1181    0.7255    0.8638    0.8723
    0.9172    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