The function may be called by the names: c06pvc or nag_sum_fft_real_2d.
3Description
c06pvc computes the two-dimensional discrete Fourier transform of a bivariate sequence of real data values ${x}_{{j}_{1}{j}_{2}}$, for ${j}_{1}=0,1,\dots ,m-1$ and ${j}_{2}=0,1,\dots ,n-1$.
where ${k}_{1}=0,1,\dots ,m-1$ and ${k}_{2}=0,1,\dots ,n-1$. (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 ${\hat{z}}_{(m-{k}_{1})(n-{k}_{2})}$), 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 one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974) and Temperton (1983).
4References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys.52 340–350
5Arguments
1: $\mathbf{m}$ – IntegerInput
On entry: $m$, the first dimension of the transform.
Constraint:
${\mathbf{m}}\ge 1$.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the second dimension of the transform.
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 ${j}_{1}=0,1,\dots ,m-1$ and ${j}_{2}=0,1,\dots ,n-1$.
On exit: the complex output dataset $\hat{z}$, where
${\hat{z}}_{{k}_{1}{k}_{2}}$ is stored in ${\mathbf{y}}\left[{k}_{2}\times (m/2+1)+{k}_{1}\right]$, for ${k}_{1}=0,1,\dots ,m/2$ and ${k}_{2}=0,1,\dots ,n-1$. 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).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
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.
7Accuracy
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).
8Parallelism 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 implementation-specific information.
9Further Comments
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 $mn$.
10Example
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.