1: $m$ – int64int32nag_int scalar: $m$ , the first dimension of the transform.

Constraint: $m \geq 1$ .
2: $n$ – int64int32nag_int scalar: $n$ , the second dimension of the transform.

Constraint: $n \geq 1$ .
3: $x (m \times n)$ – double array: The real input dataset $x$ , where $x_{j1 j2}$ is stored in $x (j2 \times m + j1)$ , for $j1 = 0, 1, \dots, m - 1$ and $j2 = 0, 1, \dots, n - 1$ . That is, if x is regarded as a two-dimensional array of dimension $(0 : m - 1, 0 : n - 1)$ , then $x (j_{1}, j_{2})$ must contain $x_{j_{1} j_{2}}$ .

Optional Input Parameters

None.

Output Parameters

1: $y ((m / 2 + 1) \times n)$ – complex array: The complex output dataset $\hat{z}$ , where ${\hat{z}}_{k1 k2}$ is stored in $y (k2 \times (m / 2 + 1) + k_{1})$ , for $k1 = 0, 1, \dots, m / 2$ and $k2 = 0, 1, \dots, n - 1$ . That is, if y is regarded as a two-dimensional array of dimension $(0 : m / 2, 0 : n - 1)$ , then $y (k_{1}, k_{2})$ contains ${\hat{z}}_{k_{1} k_{2}}$ . Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.
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 forward transform using nag_sum_fft_real_2d (c06pv) and a backward transform using nag_sum_fft_hermitian_2d (c06pw), 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_2d (c06pv) is approximately proportional to

m n \log (m n)

, but also depends on the factors of

m

and

n

. nag_sum_fft_real_2d (c06pv) is fastest if the only prime factors of

m

and

n

are

2

3

and

5

, and is particularly slow if

m

n

is a large prime, or has large prime factors.

Workspace is internally allocated by nag_sum_fft_real_2d (c06pv). The total size of these arrays is approximately proportional to

m n

Example

This example reads in a bivariate sequence of real data values and prints their discrete Fourier transforms as computed by nag_sum_fft_real_2d (c06pv). Inverse transforms are then calculated by calling nag_sum_fft_hermitian_2d (c06pw) showing that the original sequences are restored.

Open in the MATLAB editor: c06pv_example

function c06pv_example


fprintf('c06pv example results\n\n');

m = int64(5);
n = int64(2);
x = [0.010  0.346;
     1.284  1.960;
     1.754  0.855;
     0.089  0.161;
     1.004  1.844];

% Compute Transform
[y, ifail] = c06pv(m, n, x);
fprintf('\nComponents of discrete Fourier transform\n');
% Display as 2-d array
disp(reshape(y, m/2, n));

% Compute Inverse Transform
[x, ifail] = c06pw(m, n, y);
fprintf('Original sequence as restored by inverse transform\n');
% Display as 2-d array
disp(reshape(x, m, n));

c06pv example results


Components of discrete Fourier transform
   2.9431 + 0.0000i  -0.3241 + 0.0000i
  -0.0235 - 0.5576i  -0.4660 - 0.2298i
  -1.1666 + 0.6359i   0.3624 + 0.2615i

Original sequence as restored by inverse transform
    0.0100    0.3460
    1.2840    1.9600
    1.7540    0.8550
    0.0890    0.1610
    1.0040    1.8440

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

Chapter Contents

Chapter Introduction

NAG Toolbox