# NAG Library Routine Document

## 1Purpose

c06pcf calculates the discrete Fourier transform of a sequence of $n$ complex data values (using complex data type).

## 2Specification

Fortran Interface
 Subroutine c06pcf ( x, n, work,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Complex (Kind=nag_wp), Intent (Inout) :: x(n), work(*) Character (1), Intent (In) :: direct
#include nagmk26.h
 void c06pcf_ (const char *direct, Complex x[], const Integer *n, Complex work[], Integer *ifail, const Charlen length_direct)

## 3Description

Given a sequence of $n$ complex data values ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$, c06pcf calculates their (forward or backward) discrete Fourier transform (DFT) defined by
 $z^k = 1n ∑ j=0 n-1 zj × exp ±i 2πjk n , k= 0, 1, …, n-1 .$
(Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of c06pcf with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
c06pcf 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). If $n$ is a large prime number or if $n$ contains large prime factors, then the Fourier transform is performed using Bluestein's algorithm (see Bluestein (1968)), which expresses the DFT as a convolution that in turn can be efficiently computed using FFTs of highly composite sizes.

## 4References

Bluestein L I (1968) A linear filtering approach to the computation of the discrete Fourier transform Northeast Electronics Research and Engineering Meeting Record 10 218–219
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:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayInput/Output
On entry: if x is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the subroutine from which c06pcf is called, ${\mathbf{x}}\left(\mathit{j}\right)$ must contain ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
On exit: the components of the discrete Fourier transform. If x is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the subroutine from which c06pcf is called, ${\stackrel{^}{z}}_{k}$ is contained in ${\mathbf{x}}\left(k\right)$, for $0\le k\le n-1$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of data values.
Constraint: ${\mathbf{n}}\ge 1$.
4:     $\mathbf{work}\left(*\right)$ – Complex (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least $2×{\mathbf{n}}+15$.
The workspace requirements as documented for c06pcf may be an overestimate in some implementations.
On exit: the real part of ${\mathbf{work}}\left(1\right)$ 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, ${\mathbf{n}}<1$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{direct}}\ne \text{'F'}$ or $\text{'B'}$.
${\mathbf{ifail}}=4$
An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
${\mathbf{ifail}}=-99$
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

c06pcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pcf 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 $n$. c06pcf is faster if the only prime factors of $n$ are $2$, $3$ or $5$; and fastest of all if $n$ is a power of $2$. When the Bluestein's FFT algorithm is in use, an additional complex workspace of size approximately $8n$ is allocated.

## 10Example

This example reads in a sequence of complex data values and prints their discrete Fourier transform (as computed by c06pcf with ${\mathbf{direct}}=\text{'F'}$). It then performs an inverse transform using c06pcf with ${\mathbf{direct}}=\text{'B'}$, and prints the sequence so obtained alongside the original data values.

### 10.1Program Text

Program Text (c06pcfe.f90)

### 10.2Program Data

Program Data (c06pcfe.d)

### 10.3Program Results

Program Results (c06pcfe.r)

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