The function may be called by the names: c06psc, nag_sum_fft_complex_1d_multi_col or nag_sum_fft_complex_1d_multi.
3Description
Given $m$ sequences of $n$ complex data values ${z}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, c06psc simultaneously calculates the (forward or backward) discrete Fourier transforms of all the sequences defined by
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of c06psc with ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$ followed by a call with ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$ will restore the original data.
The function uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special code is provided for the factors $2$, $3$ and $5$.
4References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys.52 1–23
On entry: the complex data values
${z}_{\mathit{j}}^{p}$ stored in ${\mathbf{x}}\left[(\mathit{p}-1)\times {\mathbf{n}}+\mathit{j}\right]$, for $\mathit{j}=0,1,\dots ,{\mathbf{n}}-1$ and $\mathit{p}=1,2,\dots ,{\mathbf{m}}$.
On exit: is overwritten by the complex transforms.
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 subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).
8Parallelism and Performance
c06psc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06psc 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 c06psc is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06psc 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.
This function internally allocates a workspace of $nm+n+15$ Complex values.
10Example
This example reads in sequences of complex data values and prints their discrete Fourier transforms (as computed by c06psc with ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$). Inverse transforms are then calculated using c06psc with ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$ and printed out, showing that the original sequences are restored.