NAG Library Function Document
nag_sum_fft_complex_1d (c06pcc)
1 Purpose
nag_sum_fft_complex_1d (c06pcc) calculates the discrete Fourier transform of a sequence of $n$ complex data values (using complex data type).
2 Specification
#include <nag.h> 
#include <nagc06.h> 
void 
nag_sum_fft_complex_1d (Nag_TransformDirection direct,
Complex x[],
Integer n,
NagError *fail) 

3 Description
Given a sequence of
$n$ complex data values
${z}_{\mathit{j}}$, for
$\mathit{j}=0,1,\dots ,n1$, nag_sum_fft_complex_1d (c06pcc) calculates their (
forward or
backward) discrete Fourier transform (DFT) defined by
(Note the scale factor of
$\frac{1}{\sqrt{n}}$ 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 nag_sum_fft_complex_1d (c06pcc) with ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$ followed by a call with ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$ will restore the original data.
nag_sum_fft_complex_1d (c06pcc) 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). If
$n$ is a large prime number or if
$n$ contains large prime factors, then the Fourier transform is performed using Bluestein's algorithm (see
Bluestein (1968)), which expresses the DFT as a convolution that in turn can be efficiently computed using FFTs of highly composite sizes.
4 References
Bluestein L I (1968) A linear filtering approach to the computation of the discrete Fourier transform Northeast Electronics Research and Engineering Meeting Record 10 218–219
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_TransformDirectionInput

On entry: if the forward transform as defined in
Section 3 is to be computed, then
direct must be set equal to
$\mathrm{Nag\_ForwardTransform}$.
If the backward transform is to be computed then
direct must be set equal to
$\mathrm{Nag\_BackwardTransform}$.
Constraint:
${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$ or $\mathrm{Nag\_BackwardTransform}$.
 2:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – ComplexInput/Output

On entry:
${\mathbf{x}}\left[\mathit{j}\right]$ must contain ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n1$.
On exit: the components of the discrete Fourier transform.
${\hat{z}}_{k}$ is contained in ${\mathbf{x}}\left[k\right]$, for $0\le k\le n1$.
 3:
$\mathbf{n}$ – IntegerInput

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 2.7 in How to Use the NAG Library and its Documentation).
6 Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
$\u2329\mathit{\text{value}}\u232a$ is an invalid value of
direct.
 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.
An unexpected error has been triggered by this function. Please contact
NAG.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation 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
nag_sum_fft_complex_1d (c06pcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sum_fft_complex_1d (c06pcc) 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 $n$. nag_sum_fft_complex_1d (c06pcc) 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 $2n+15$ Complex values.
When the Bluestein's FFT algorithm is in use, an additional Complex workspace of size approximately $8n$ is allocated.
10 Example
This example reads in a sequence of complex data values and prints their discrete Fourier transform (as computed by nag_sum_fft_complex_1d (c06pcc) with ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$). It then performs an inverse transform using nag_sum_fft_complex_1d (c06pcc) with ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$, and prints the sequence so obtained alongside the original data values.
10.1 Program Text
Program Text (c06pcce.c)
10.2 Program Data
Program Data (c06pcce.d)
10.3 Program Results
Program Results (c06pcce.r)