NAG Library Function Document
nag_sum_fft_complex_1d_multi (c06psc)
1
Purpose
nag_sum_fft_complex_1d_multi (c06psc) computes the discrete Fourier transforms of $m$ sequences each containing $n$ complex data values.
2
Specification
#include <nag.h> 
#include <nagc06.h> 
void 
nag_sum_fft_complex_1d_multi (Nag_TransformDirection direct,
Integer n,
Integer m,
Complex x[],
NagError *fail) 

3
Description
Given
$m$ sequences of
$n$ complex data values
${z}_{\mathit{j}}^{\mathit{p}}$, for
$\mathit{j}=0,1,\dots ,n1$ and
$\mathit{p}=1,2,\dots ,m$,
nag_sum_fft_complex_1d_multi (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 nag_sum_fft_complex_1d_multi (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 selfsorting algorithm, which is described in
Temperton (1983). Special code is provided for the factors
$2$,
$3$ and
$5$.
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_TransformDirectionInput

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:
$\mathbf{n}$ – IntegerInput

On entry: $n$, the number of complex values in each sequence.
Constraint:
${\mathbf{n}}\ge 1$.
 3:
$\mathbf{m}$ – IntegerInput

On entry: $m$, the number of sequences to be transformed.
Constraint:
${\mathbf{m}}\ge 1$.
 4:
$\mathbf{x}\left[{\mathbf{n}}\times {\mathbf{m}}\right]$ – ComplexInput/Output

On entry: the complex data values
${z}_{\mathit{j}}^{p}$ stored in ${\mathbf{x}}\left[\left(\mathit{p}1\right)\times {\mathbf{n}}+\mathit{j}\right]$, 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 3.7 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, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\ge 1$.
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 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 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_multi (c06psc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sum_fft_complex_1d_multi (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 implementationspecific information.
The time taken by nag_sum_fft_complex_1d_multi (c06psc) is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. nag_sum_fft_complex_1d_multi (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.
10
Example
This example reads in sequences of complex data values and prints their discrete Fourier transforms (as computed by nag_sum_fft_complex_1d_multi (c06psc) with ${\mathbf{direct}}=\mathrm{Nag\_ForwardTransform}$). Inverse transforms are then calculated using nag_sum_fft_complex_1d_multi (c06psc) with ${\mathbf{direct}}=\mathrm{Nag\_BackwardTransform}$ and printed out, showing that the original sequences are restored.
10.1
Program Text
Program Text (c06psce.c)
10.2
Program Data
Program Data (c06psce.d)
10.3
Program Results
Program Results (c06psce.r)