# NAG Library Routine Document

## 1Purpose

c06paf calculates the discrete Fourier transform of a sequence of $n$ real data values or of a Hermitian sequence of $n$ complex data values stored in compact form in a real array.

## 2Specification

Fortran Interface
 Subroutine c06paf ( x, n, work,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: x(n+2), work(*) Character (1), Intent (In) :: direct
C Header Interface
#include nagmk26.h
 void c06paf_ (const char *direct, double x[], const Integer *n, double work[], Integer *ifail, const Charlen length_direct)

## 3Description

Given a sequence of $n$ real data values ${x}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$, c06paf calculates their discrete Fourier transform (in the forward direction) defined by
 $z^k = 1n ∑ j=0 n-1 xj × exp -i 2πjk n , k= 0, 1, …, n-1 .$
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 (since ${\stackrel{^}{z}}_{0}$ is real, as is ${\stackrel{^}{z}}_{n/2}$ for $n$ even).
Alternatively, given a Hermitian sequence of $n$ complex data values ${z}_{j}$, this routine calculates their inverse (backward) discrete Fourier transform defined by
 $x^k = 1n ∑ j=0 n-1 zj × exp i 2πjk n , k= 0, 1, …, n-1 .$
The transformed values ${\stackrel{^}{x}}_{k}$ are real.
(Note the scale factor of $\frac{1}{\sqrt{n}}$ in the above definitions.)
A call of c06paf with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
c06paf uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983).
The same functionality is available using the forward and backward transform routine pair: c06pvf and c06pwf on setting ${\mathbf{n}}=1$. This pair use a different storage solution; real data is stored in a real array, while Hermitian data (the first unconjugated half) is stored in a complex array.

## 4References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23

## 5Arguments

1:     $\mathbf{direct}$ – Character(1)Input
On entry: if the forward transform as defined in Section 3 is to be computed, direct must be set equal to 'F'.
If the backward transform is to be computed, direct must be set equal to 'B'.
Constraint: ${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
2:     $\mathbf{x}\left({\mathbf{n}}+2\right)$ – 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 c06paf is called:
• if ${\mathbf{direct}}=\text{'F'}$, ${\mathbf{x}}\left(\mathit{j}\right)$ must contain ${x}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$;
• if ${\mathbf{direct}}=\text{'B'}$, ${\mathbf{x}}\left(2×\mathit{k}\right)$ and ${\mathbf{x}}\left(2×\mathit{k}+1\right)$ must contain the real and imaginary parts respectively of ${z}_{\mathit{k}}$, for $\mathit{k}=0,1,\dots ,n/2$. (Note that for the sequence ${z}_{k}$ to be Hermitian, the imaginary part of ${z}_{0}$, and of ${z}_{n/2}$ for $n$ even, must be zero.)
On exit:
• if ${\mathbf{direct}}=\text{'F'}$ and x is declared with bounds $\left(0:{\mathbf{n}}+1\right)$, ${\mathbf{x}}\left(2×\mathit{k}\right)$ and ${\mathbf{x}}\left(2×\mathit{k}+1\right)$ will contain the real and imaginary parts respectively of ${\stackrel{^}{z}}_{\mathit{k}}$, for $\mathit{k}=0,1,\dots ,n/2$;
• if ${\mathbf{direct}}=\text{'B'}$ and x is declared with bounds $\left(0:{\mathbf{n}}+1\right)$, ${\mathbf{x}}\left(\mathit{j}\right)$ will contain ${\stackrel{^}{x}}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of data values.
Constraint: ${\mathbf{n}}\ge 1$.
4:     $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least $3×{\mathbf{n}}+100$.
The workspace requirements as documented for c06paf may be an overestimate in some implementations.
On exit: ${\mathbf{work}}\left(1\right)$ contains the minimum workspace required for the current value of n with this implementation.
5:     $\mathbf{ifail}$ – 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).

## 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:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
$〈\mathit{\text{value}}〉$ is an invalid value of direct.
${\mathbf{ifail}}=3$
An internal error has occurred in this routine. Check the routine call and any array sizes. If the call is correct then please contact NAG for assistance.
${\mathbf{ifail}}=-99$
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.

## 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

c06paf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06paf 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.

## 9Further Comments

The time taken is approximately proportional to $n×\mathrm{log}\left(n\right)$, but also depends on the factorization of $n$. c06paf 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 c06paf with ${\mathbf{direct}}=\text{'F'}$), after expanding it from complex Hermitian form into a full complex sequence. It then performs an inverse transform using c06paf with ${\mathbf{direct}}=\text{'B'}$, and prints the sequence so obtained alongside the original data values.

### 10.1Program Text

Program Text (c06pafe.f90)

### 10.2Program Data

Program Data (c06pafe.d)

### 10.3Program Results

Program Results (c06pafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017