C06 Chapter Contents
C06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC06FAF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 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

 SUBROUTINE C06FAF ( X, N, WORK, IFAIL)
 INTEGER N, IFAIL REAL (KIND=nag_wp) X(N), WORK(N)

## 3  Description

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)). There are some restrictions on the value of $n$ (see Section 5).
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall

## 5  Parameters

1:     X(N) – 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, then ${\mathbf{X}}\left(\mathit{j}\right)$ 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 9.)
2:     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:     WORK(N) – REAL (KIND=nag_wp) arrayWorkspace
4:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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 ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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.

## 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).

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

## 9  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.

### 9.1  Program Text

Program Text (c06fafe.f90)

### 9.2  Program Data

Program Data (c06fafe.d)

### 9.3  Program Results

Program Results (c06fafe.r)