C06 Chapter Contents
C06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC06PAF

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

C06PAF calculates the discrete Fourier transform of a sequence of $n$ real data values or of a Hermitian sequence of $n$ complex data values.

## 2  Specification

 SUBROUTINE C06PAF ( DIRECT, X, N, WORK, IFAIL)
 INTEGER N, IFAIL REAL (KIND=nag_wp) X(N+2), WORK(*) CHARACTER(1) DIRECT

## 3  Description

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).

## 4  References

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

## 5  Parameters

1:     DIRECT – CHARACTER(1)Input
On entry: if the forward transform as defined in Section 3 is to be computed, then DIRECT must be set equal to 'F'.
If the backward transform is to be computed then DIRECT must be set equal to 'B'.
Constraint: ${\mathbf{DIRECT}}=\text{'F'}$ or $\text{'B'}$.
2:     X(${\mathbf{N}}+2$) – 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, then:
• 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 ${\stackrel{^}{z}}_{\mathit{k}}$, for $\mathit{k}=0,1,\dots ,n/2$. (Note that for the sequence ${\stackrel{^}{z}}_{k}$ to be Hermitian, the imaginary part of ${\stackrel{^}{z}}_{0}$, and of ${\stackrel{^}{z}}_{n/2}$ for $n$ even, must be zero.)
On exit:
• if ${\mathbf{DIRECT}}=\text{'F'}$ and X is declared with bounds $\left(0:{\mathbf{N}}+1\right)$ then ${\mathbf{X}}\left(2×\mathit{k}\right)$ 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 $\left(0:{\mathbf{N}}+1\right)$ then ${\mathbf{X}}\left(\mathit{j}\right)$ will contain ${x}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
3:     N – INTEGERInput
On entry: $n$, the number of data values. The total number of prime factors of N, counting repetitions, must not exceed $30$.
4:     WORK($*$) – 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:     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:
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{N}}\le 1$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{DIRECT}}\ne \text{'F'}$ or $\text{'B'}$.
${\mathbf{IFAIL}}=4$
 On entry, N has more than $30$ prime factors.
${\mathbf{IFAIL}}=5$
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 $n$. 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 $2$.

## 9  Example

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.

### 9.1  Program Text

Program Text (c06pafe.f90)

### 9.2  Program Data

Program Data (c06pafe.d)

### 9.3  Program Results

Program Results (c06pafe.r)