# NAG Library Routine Document

## 1Purpose

c06pkf calculates the circular convolution or correlation of two complex vectors of period $n$.

## 2Specification

Fortran Interface
 Subroutine c06pkf ( job, x, y, n, work,
 Integer, Intent (In) :: job, n Integer, Intent (Inout) :: ifail Complex (Kind=nag_wp), Intent (Inout) :: x(n), y(n), work(*)
#include nagmk26.h
 void c06pkf_ (const Integer *job, Complex x[], Complex y[], const Integer *n, Complex work[], Integer *ifail)

## 3Description

c06pkf computes:
• if ${\mathbf{job}}=1$, the discrete convolution of $x$ and $y$, defined by
 $zk = ∑ j=0 n-1 xj y k-j = ∑ j=0 n-1 x k-j yj ;$
• if ${\mathbf{job}}=2$, the discrete correlation of $x$ and $y$ defined by
 $wk = ∑ j=0 n-1 x-j y k+j .$
Here $x$ and $y$ are complex vectors, assumed to be periodic, with period $n$, i.e., ${x}_{j}={x}_{j±n}={x}_{j±2n}=\dots \text{}$; $z$ and $w$ are then also periodic with period $n$.
Note:  this usage of the terms ‘convolution’ and ‘correlation’ is taken from Brigham (1974). The term ‘convolution’ is sometimes used to denote both these computations.
If $\stackrel{^}{x}$, $\stackrel{^}{y}$, $\stackrel{^}{z}$ and $\stackrel{^}{w}$ are the discrete Fourier transforms of these sequences, and $\stackrel{~}{x}$ is the inverse discrete Fourier transform of the sequence ${x}_{j}$, i.e.,
 $x^k = 1n ∑ j=0 n-1 xj × exp -i 2πjk n , etc.,$
and
 $x~k = 1n ∑ j= 0 n- 1 xj × exp i 2πjk n ,$
then ${\stackrel{^}{z}}_{k}=\sqrt{n}.{\stackrel{^}{x}}_{k}{\stackrel{^}{y}}_{k}$ and ${\stackrel{^}{w}}_{k}=\sqrt{n}.{\stackrel{-}{\stackrel{^}{x}}}_{k}{\stackrel{^}{y}}_{k}$ (the bar denoting complex conjugate).

## 4References

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

## 5Arguments

1:     $\mathbf{job}$ – IntegerInput
On entry: the computation to be performed:
${\mathbf{job}}=1$
${z}_{k}=\sum _{j=0}^{n-1}{x}_{j}{y}_{k-j}$ (convolution);
${\mathbf{job}}=2$
${w}_{k}=\sum _{j=0}^{n-1}{\stackrel{-}{x}}_{j}{y}_{k+j}$ (correlation).
Constraint: ${\mathbf{job}}=1$ or $2$.
2:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayInput/Output
On entry: the elements of one period of the vector $x$. If x is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the subroutine from which c06pkf is called, ${\mathbf{x}}\left(\mathit{j}\right)$ must contain ${x}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
On exit: the corresponding elements of the discrete convolution or correlation.
3:     $\mathbf{y}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayInput/Output
On entry: the elements of one period of the vector $y$. If y is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the subroutine from which c06pkf is called, ${\mathbf{y}}\left(\mathit{j}\right)$ must contain ${y}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
On exit: the discrete Fourier transform of the convolution or correlation returned in the array x.
4:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of values in one period of the vectors x and y. The total number of prime factors of n, counting repetitions, must not exceed $30$.
Constraint: ${\mathbf{n}}\ge 1$.
5:     $\mathbf{work}\left(*\right)$ – 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 c06pkf 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.
6:     $\mathbf{ifail}$ – 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, ${\mathbf{job}}\ne 1$ or $2$.
${\mathbf{ifail}}=3$
An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
${\mathbf{ifail}}=4$
 On entry, n has more than $30$ prime factors.
${\mathbf{ifail}}=-99$
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

The results should be accurate to within a small multiple of the machine precision.

## 8Parallelism and Performance

c06pkf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pkf 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 $n$. c06pkf 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 the elements of one period of two complex vectors $x$ and $y$, and prints their discrete convolution and correlation (as computed by c06pkf). In realistic computations the number of data values would be much larger.

### 10.1Program Text

Program Text (c06pkfe.f90)

### 10.2Program Data

Program Data (c06pkfe.d)

### 10.3Program Results

Program Results (c06pkfe.r)

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