The transform calculated by this function can be used to solve Poisson's equation when the derivative of the solution is specified at the left boundary, and the solution is specified at the right boundary (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: $direct$ – 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: $x (m \times (n + 2))$ – double array

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

(1 : m, 0 : n + 1)

; 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 = 0, 1, \dots, n - 1

and

p = 1, 2, \dots, m

, then the first

m n

elements of the array x must contain the values

x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{m}, x_{1}^{1}, x_{1}^{2}, \dots, x_{1}^{m}, \dots, x_{n - 1}^{1}, x_{n - 1}^{2}, \dots, x_{n - 1}^{m} .

The

(n + 1)

th and

(n + 2)

th elements of each row

x_{n}^{p}, x_{n + 1}^{p}

, for

p = 1, 2, \dots, m

, are required as workspace. These

2 m

elements may contain arbitrary values as they are set to zero by the function.

Optional Input Parameters

None.

Output Parameters

1: $x (m \times (n + 2))$ – double array

the

m

quarter-wave cosine transforms stored as if in a two-dimensional array of dimension

(1 : m, 0 : n + 1)

. 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 cosine transform are denoted by

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

, for

k = 0, 1, \dots, n - 1

and

p = 1, 2, \dots, m

, then the

m (n + 2)

elements of the array x contain the values

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

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:

$ifail = 1$

On entry,

m < 1

$ifail = 2$

On entry,

n < 1

$ifail = 3$

On entry,

direct \neq'F'

'B'

$ifail = 4$: 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_qtrcosine_simple (c06rd) is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. nag_sum_fft_real_qtrcosine_simple (c06rd) 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 cosine transforms as computed by nag_sum_fft_real_qtrcosine_simple (c06rd) 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: c06rd_example

function c06rd_example


fprintf('c06rd example results\n\n');

% Discrete quarter-wave cosine transform of 3 sequences of length 6
m = int64(3);
n = int64(6);
x = zeros(m,(n+2));
x(1:m,1:n) = [ 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';
[xt, ifail] = c06rd(direct, m, n, x);
disp('X under discrete quarter-wave cosine transform');
disp(reshape(xt(1:m*n),m,n));

direct = 'Backward';
[xr, ifail] = c06rd(direct, m, n, xt);
disp('X reconstructed by inverse quarter-wave cosine transform');
y = reshape(xr(1:m*n),m,n);
disp(y);

c06rd example results

X under discrete quarter-wave cosine transform
    0.7257   -0.2216    0.1011    0.2355   -0.1406   -0.2282
    0.7479   -0.6172    0.4112    0.0791    0.1331   -0.0906
    0.6713   -0.1363   -0.0064   -0.0285    0.4758    0.1475

X reconstructed by inverse quarter-wave cosine 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