NAG Library Routine Document
c06faf (fft_real_1d_rfmt)
1
Purpose
c06faf calculates the discrete Fourier transform of a sequence of $n$ real data values (using a work array for extra speed).
2
Specification
Fortran Interface
Integer, Intent (In)  ::  n  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (Inout)  ::  x(n)  Real (Kind=nag_wp), Intent (Out)  ::  work(n) 

C Header Interface
#include nagmk26.h
void 
c06faf_ (double x[], const Integer *n, double work[], Integer *ifail) 

3
Description
Given a sequence of
$n$ real data values
${x}_{\mathit{j}}$, for
$\mathit{j}=0,1,\dots ,n1$,
c06faf calculates their discrete Fourier transform defined by
(Note the scale factor of
$\frac{1}{\sqrt{n}}$ in this definition.) 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 (see also the
C06 Chapter Introduction).
To compute the inverse discrete Fourier transform defined by
this routine should be followed by forming the complex conjugates of the
${\hat{z}}_{k}$; that is,
$x\left(\mathit{k}\right)=x\left(\mathit{k}\right)$, for
$\mathit{k}=n/2+2,\dots ,n$.
c06faf uses the fast Fourier transform (FFT) algorithm (see
Brigham (1974)). There are some restrictions on the value of
$n$ (see
Section 5).
4
References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
5
Arguments
 1: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: if
x is declared with bounds
$\left(0:{\mathbf{n}}1\right)$ in the subroutine from which
c06faf is called,
${\mathbf{x}}\left(\mathit{j}\right)$ must contain
${x}_{\mathit{j}}$, for
$\mathit{j}=0,1,\dots ,n1$.
On exit: the discrete Fourier transform stored in Hermitian form. If the components of the transform
${\hat{z}}_{k}$ are written as
${a}_{k}+i{b}_{k}$, and if
x is declared with bounds
$\left(0:{\mathbf{n}}1\right)$ in the subroutine from which
c06faf is called, then for
$0\le k\le n/2$,
${a}_{k}$ is contained in
${\mathbf{x}}\left(k\right)$, and for
$1\le k\le \left(n1\right)/2$,
${b}_{k}$ is contained in
${\mathbf{x}}\left(nk\right)$. (See also
Section 2.1.2 in the C06 Chapter Introduction and
Section 10.)
 2: $\mathbf{n}$ – IntegerInput

On entry:
$n$, the number of data values. The largest prime factor of
n must not exceed
$19$, and the total number of prime factors of
n, counting repetitions, must not exceed
$20$.
Constraint:
${\mathbf{n}}>1$.
 3: $\mathbf{work}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace

 4: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

At least one of the prime factors of
n is greater than
$19$.
 ${\mathbf{ifail}}=2$

n has more than
$20$ prime factors.
 ${\mathbf{ifail}}=3$

On entry,  ${\mathbf{n}}\le 1$. 
 ${\mathbf{ifail}}=4$

An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 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
c06faf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06faf 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 routine. 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 $n$. c06faf is faster if the only prime factors of $n$ are $2$, $3$ or $5$; and fastest of all if $n$ is a power of $2$.
10
Example
This example reads in a sequence of real data values and prints their discrete Fourier transform (as computed by
c06faf), after expanding it from Hermitian form into a full complex sequence. It then performs an inverse transform using
c06fbf and conjugation, and prints the sequence so obtained alongside the original data values.
10.1
Program Text
Program Text (c06fafe.f90)
10.2
Program Data
Program Data (c06fafe.d)
10.3
Program Results
Program Results (c06fafe.r)