c06ppc computes the discrete Fourier transforms of $m$ sequences, each containing $n$ real data values or a Hermitian complex sequence stored in a complex storage format.
The function may be called by the names: c06ppc or nag_sum_fft_realherm_1d_multi_row.
3Description
Given $m$ sequences of $n$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, c06ppc simultaneously calculates the Fourier transforms of all the sequences defined by
The transformed values ${\hat{z}}_{k}^{p}$ are complex, but for each value of $p$ the ${\hat{z}}_{k}^{p}$ form a Hermitian sequence (i.e., ${\hat{z}}_{n-k}^{p}$ is the complex conjugate of ${\hat{z}}_{k}^{p}$), so they are completely determined by $mn$ real numbers (since ${\hat{z}}_{0}^{p}$ is real, as is ${\hat{z}}_{n/2}^{p}$ for $n$ even).
Alternatively, given $m$ Hermitian sequences of $n$ complex data values ${z}_{j}^{p}$, this function simultaneously calculates their inverse (backward) discrete Fourier transforms defined by
The transformed values ${\hat{x}}_{k}^{p}$ are real.
(Note the scale factor $\frac{1}{\sqrt{n}}$ in the above definition.)
A call of c06ppc 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 coding is provided for the factors $2$, $3$, $4$ and $5$.
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
On entry: the data must be stored such that consecutive elements of the same sequence are stored with a stride of m and corresponding elements of different sequences are stored consecutively. An additional two spaces are reserved for each sequence to allow for the pairwise storage of real and imaginary parts in the transformed domain. In other words, if the data values of the
$p$th sequence to be transformed are denoted by ${x}_{\mathit{j}}^{p}$, for $\mathit{j}=0,1,\dots ,n-1$:
if ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$,
${\mathbf{x}}\left[\mathit{j}\times {\mathbf{m}}+\mathit{p}-1\right]$ must contain ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$;
if ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$, ${\mathbf{x}}\left[2\times \mathit{k}\times {\mathbf{m}}+\mathit{p}-1\right]$ and ${\mathbf{x}}\left[(2\times \mathit{k}+1)\times {\mathbf{m}}+\mathit{p}-1\right]$ must contain the real and imaginary parts respectively of ${\hat{z}}_{k}^{p}$, for $\mathit{k}=0,1,\dots ,n/2$ and $\mathit{p}=1,2,\dots ,m$. (Note that for the sequence ${\hat{z}}_{k}^{p}$ to be Hermitian, the imaginary part of ${\hat{z}}_{0}^{p}$, and of ${\hat{z}}_{n/2}^{p}$ for $n$ even, must be zero.)
On exit:
if ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$ then
${\mathbf{x}}\left[2\times \mathit{k}\times {\mathbf{m}}+\mathit{p}-1\right]$ and ${\mathbf{x}}\left[(2\times \mathit{k}+1)\times {\mathbf{m}}+\mathit{p}-1\right]$ will contain the real and imaginary parts respectively of ${\hat{z}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n/2$ and $\mathit{p}=1,2,\dots ,m$;
if ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$ then
${\mathbf{x}}\left[\mathit{j}\times {\mathbf{m}}+\mathit{p}-1\right]$ will contain ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$;
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
c06ppc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06ppc 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 c06ppc is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06ppc 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.
10Example
This example reads in sequences of real data values and prints their discrete Fourier transforms (as computed by c06ppc with ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$), after expanding them from complex Hermitian form into a full complex sequences. Inverse transforms are then calculated by calling c06ppc with ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$ showing that the original sequences are restored.