NAG CL Interface
c06pvc (fft_real_2d)
1
Purpose
c06pvc computes the twodimensional discrete Fourier transform of a bivariate sequence of real data values.
2
Specification
void 
c06pvc (Integer m,
Integer n,
const double x[],
Complex y[],
NagError *fail) 

The function may be called by the names: c06pvc or nag_sum_fft_real_2d.
3
Description
c06pvc computes the twodimensional discrete Fourier transform of a bivariate sequence of real data values ${x}_{{j}_{1}{j}_{2}}$, for ${j}_{1}=0,1,\dots ,m1$ and ${j}_{2}=0,1,\dots ,n1$.
The discrete Fourier transform is here defined by
where
${k}_{1}=0,1,\dots ,m1$ and
${k}_{2}=0,1,\dots ,n1$. (Note the scale factor of
$\frac{1}{\sqrt{mn}}$ in this definition.)
The transformed values ${\hat{z}}_{{k}_{1}{k}_{2}}$ are complex. Because of conjugate symmetry (i.e., ${\hat{z}}_{{k}_{1}{k}_{2}}$ is the complex conjugate of ${\hat{z}}_{\left(m{k}_{1}\right)\left(n{k}_{2}\right)}$), only slightly more than half of the Fourier coefficients need to be stored in the output.
A call of
c06pvc followed by a call of
c06pwc will restore the original data.
This function performs multiple onedimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in
Brigham (1974) and
Temperton (1983).
4
References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixedradix real Fourier transforms J. Comput. Phys. 52 340–350
5
Arguments

1:
$\mathbf{m}$ – Integer
Input

On entry: $m$, the first dimension of the transform.
Constraint:
${\mathbf{m}}\ge 1$.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the second dimension of the transform.
Constraint:
${\mathbf{n}}\ge 1$.

3:
$\mathbf{x}\left[{\mathbf{m}}\times {\mathbf{n}}\right]$ – const double
Input

On entry: the real input dataset $x$, where
${x}_{{j}_{1}{j}_{2}}$ is stored in ${\mathbf{x}}\left[{j}_{2}\times m+{j}_{1}\right]$, for ${j}_{1}=0,1,\dots ,m1$ and ${j}_{2}=0,1,\dots ,n1$.

4:
$\mathbf{y}\left[\left({\mathbf{m}}/2+1\right)\times {\mathbf{n}}\right]$ – Complex
Output

On exit: the complex output dataset $\hat{z}$, where
${\hat{z}}_{{k}_{1}{k}_{2}}$ is stored in ${\mathbf{y}}\left[{k}_{2}\times \left(m/2+1\right)+{k}_{1}\right]$, for ${k}_{1}=0,1,\dots ,m/2$ and ${k}_{2}=0,1,\dots ,n1$. Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.

5:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
 NE_INTERNAL_ERROR

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.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
Some indication of accuracy can be obtained by performing a forward transform using
c06pvc and a backward transform using
c06pwc, and comparing the results with the original sequence (in exact arithmetic they would be identical).
8
Parallelism and Performance
c06pvc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pvc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time taken by c06pvc is approximately proportional to $mn\mathrm{log}\left(mn\right)$, but also depends on the factors of $m$ and $n$. c06pvc is fastest if the only prime factors of $m$ and $n$ are $2$, $3$ and $5$, and is particularly slow if $m$ or $n$ is a large prime, or has large prime factors.
Workspace is internally allocated by c06pvc. The total size of these arrays is approximately proportional to $mn$.
10
Example
This example reads in a bivariate sequence of real data values and prints their discrete Fourier transforms as computed by
c06pvc. Inverse transforms are then calculated by calling
c06pwc showing that the original sequences are restored.
10.1
Program Text
10.2
Program Data
10.3
Program Results