# NAG Library Routine Document

## 1Purpose

c06faf calculates the discrete Fourier transform of a sequence of $n$ real data values (using a work array for extra speed).

## 2Specification

Fortran Interface
 Subroutine c06faf ( x, n, work,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: x(n) Real (Kind=nag_wp), Intent (Out) :: work(n)
#include <nagmk26.h>
 void c06faf_ (double x[], const Integer *n, double work[], Integer *ifail)

## 3Description

Given a sequence of $n$ real data values ${x}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$, c06faf calculates their discrete Fourier transform defined by
 $z^k = 1n ∑ j=0 n-1 xj × exp -i 2πjk n , k= 0, 1, …, n-1 .$
(Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.) The transformed values ${\stackrel{^}{z}}_{k}$ are complex, but they form a Hermitian sequence (i.e., ${\stackrel{^}{z}}_{n-k}$ is the complex conjugate of ${\stackrel{^}{z}}_{k}$), so they are completely determined by $n$ real numbers (see also the C06 Chapter Introduction).
To compute the inverse discrete Fourier transform defined by
 $w^k = 1n ∑ j=0 n-1 xj × exp +i 2πjk n ,$
this routine should be followed by forming the complex conjugates of the ${\stackrel{^}{z}}_{k}$; 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)).

## 4References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall

## 5Arguments

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 $\mathit{j}=0,1,\dots ,n-1$.
On exit: the discrete Fourier transform stored in Hermitian form. If the components of the transform ${\stackrel{^}{z}}_{k}$ 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 $1\le k\le \left(n-1\right)/2$, ${b}_{k}$ is contained in ${\mathbf{x}}\left(n-k\right)$. (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.
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 $0$, . 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  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  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).

## 6Error 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:
${\mathbf{ifail}}=3$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.
${\mathbf{ifail}}=-99$
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.

## 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

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 implementation-specific information.

The time taken is approximately proportional to $n×\mathrm{log}\left(n\right)$, 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 $2$.

## 10Example

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.

### 10.1Program Text

Program Text (c06fafe.f90)

### 10.2Program Data

Program Data (c06fafe.d)

### 10.3Program Results

Program Results (c06fafe.r)