NAG CL Interface
c06pac (fft_realherm_1d)
1
Purpose
c06pac calculates the discrete Fourier transform of a sequence of $n$ real data values or of a Hermitian sequence of $n$ complex data values stored in compact form in a double array.
2
Specification
void 
c06pac (Nag_TransformDirection direct,
double x[],
Integer n,
NagError *fail) 

The function may be called by the names: c06pac or nag_sum_fft_realherm_1d.
3
Description
Given a sequence of
$n$ real data values
${x}_{\mathit{j}}$, for
$\mathit{j}=0,1,\dots ,n1$,
c06pac calculates their discrete Fourier transform (in the
forward direction) defined by
The transformed values
${\hat{z}}_{k}$ are complex, but they form a Hermitian sequence (i.e.,
${\hat{z}}_{nk}$ is the complex conjugate of
${\hat{z}}_{k}$), so they are completely determined by
$n$ real numbers (since
${\hat{z}}_{0}$ is real, as is
${\hat{z}}_{n/2}$ for
$n$ even).
Alternatively, given a Hermitian sequence of
$n$ complex data values
${z}_{j}$, this function calculates their inverse (
backward) discrete Fourier transform defined by
The transformed values
${\hat{x}}_{k}$ are real.
(Note the scale factor of $\frac{1}{\sqrt{n}}$ in the above definitions.)
A call of c06pac with ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$ followed by a call with ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$ will restore the original data.
c06pac uses a variant of the fast Fourier transform (FFT) algorithm (see
Brigham (1974)) known as the Stockham selfsorting algorithm, which is described in
Temperton (1983).
The same functionality is available using the forward and backward transform function pair:
c06pvc and
c06pwc on setting
${\mathbf{n}}=1$. This pair use a different storage solution; real data is stored in a double array, while Hermitian data (the first unconjugated half) is stored in a Complex array.
4
References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Selfsorting mixedradix fast Fourier transforms J. Comput. Phys. 52 1–23
5
Arguments

1:
$\mathbf{direct}$ – Nag_TransformDirection
Input

On entry: if the forward transform as defined in
Section 3 is to be computed,
direct must be set equal to
$\mathrm{Nag\_ForwardTransform}$.
If the backward transform is to be computed,
direct must be set equal to
$\mathrm{Nag\_BackwardTransform}$.
Constraint:
${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$ or $\mathrm{Nag\_BackwardTransform}$.

2:
$\mathbf{x}\left[{\mathbf{n}}+2\right]$ – double
Input/Output

On entry:
 if ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$,
${\mathbf{x}}\left[\mathit{j}\right]$ must contain ${x}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n1$;

if ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$, ${\mathbf{x}}\left[2\times \mathit{k}\right]$ and ${\mathbf{x}}\left[2\times \mathit{k}+1\right]$ must contain the real and imaginary parts respectively of ${z}_{\mathit{k}}$, for $\mathit{k}=0,1,\dots ,n/2$. (Note that for the sequence ${z}_{k}$ to be Hermitian, the imaginary part of ${z}_{0}$, and of ${z}_{n/2}$ for $n$ even, must be zero.)
On exit:
 if ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$,
${\mathbf{x}}\left[2\times \mathit{k}\right]$ and ${\mathbf{x}}\left[2\times \mathit{k}+1\right]$ will contain the real and imaginary parts respectively of ${\hat{z}}_{\mathit{k}}$, for $\mathit{k}=0,1,\dots ,n/2$;
 if ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$,
${\mathbf{x}}\left[\mathit{j}\right]$ will contain ${\hat{x}}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n1$.

3:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of data values.
Constraint:
${\mathbf{n}}\ge 1$.

4:
$\mathbf{fail}$ – 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, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
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 subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).
8
Parallelism and Performance
c06pac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pac 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 is approximately proportional to $n\times \mathrm{log}\left(n\right)$, but also depends on the factorization of $n$. c06pac is faster if the only prime factors of $n$ are $2$, $3$ or $5$; and fastest of all if $n$ is a power of $2$. This function internally allocates a workspace of $3n+100$ double values.
10
Example
This example reads in a sequence of real data values and prints their discrete Fourier transform (as computed by c06pac with ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$), after expanding it from complex Hermitian form into a full complex sequence. It then performs an inverse transform using c06pac with ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$, and prints the sequence so obtained alongside the original data values.
10.1
Program Text
10.2
Program Data
10.3
Program Results