NAG Library Routine Document
C06EKF
1 Purpose
C06EKF calculates the circular convolution or correlation of two real vectors of period $n$. (No extra workspace is required.)
2 Specification
INTEGER 
JOB, N, IFAIL 
REAL (KIND=nag_wp) 
X(N), Y(N) 

3 Description
C06EKF computes:
 if ${\mathbf{JOB}}=1$, the discrete convolution of $x$ and $y$, defined by
 if ${\mathbf{JOB}}=2$, the discrete correlation of $x$ and $y$ defined by
Here $x$ and $y$ are real vectors, assumed to be periodic, with period $n$, i.e., ${x}_{j}={x}_{j\pm n}={x}_{j\pm 2n}=\dots \text{}$; $z$ 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
$\hat{x}$,
$\hat{y}$,
$\hat{z}$ and
$\hat{w}$ are the discrete Fourier transforms of these sequences, i.e.,
then
${\hat{z}}_{k}=\sqrt{n}.{\hat{x}}_{k}{\hat{y}}_{k}$ and
${\hat{w}}_{k}=\sqrt{n}.{\stackrel{}{\hat{x}}}_{k}{\hat{y}}_{k}$ (the bar denoting complex conjugate).
This routine calls the same auxiliary routines as
C06EAF and
C06EBF to compute discrete Fourier transforms, and there are some restrictions on the value of
$n$.
4 References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
5 Parameters
 1: JOB – INTEGERInput
On entry: the computation to be performed.
 ${\mathbf{JOB}}=1$
 ${z}_{k}={\displaystyle \sum _{j=0}^{n1}}{x}_{j}{y}_{kj}$ (convolution);
 ${\mathbf{JOB}}=2$
 ${w}_{k}={\displaystyle \sum _{j=0}^{n1}}{x}_{j}{y}_{k+j}$ (correlation).
Constraint:
${\mathbf{JOB}}=1$ or $2$.
 2: X(N) – REAL (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 C06EKF is called, then
${\mathbf{X}}\left(\mathit{j}\right)$ must contain
${x}_{\mathit{j}}$, for
$\mathit{j}=0,1,\dots ,n1$.
On exit: the corresponding elements of the discrete convolution or correlation.
 3: Y(N) – REAL (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 C06EKF is called, then
${\mathbf{Y}}\left(\mathit{j}\right)$ must contain
${y}_{\mathit{j}}$, for
$\mathit{j}=0,1,\dots ,n1$.
On exit: the discrete Fourier transform of the convolution or correlation returned in the array
X; the transform is stored in Hermitian form. If the components of the transform are:
where
${b}_{0}$ and
${b}_{n/2}$ when
$n$ is even then
${\mathbf{X}}\left(k+1\right)$ holds
${a}_{k}$ and
${\mathbf{X}}\left(nk+1\right)$ holds nonzero
${b}_{k}$ (see
Section 2.1.2 in the C06 Chapter Introduction).
 4: N – INTEGERInput
On entry:
$n$, the number of values in one period of the vectors
X and
Y. 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:
${\mathbf{N}}>1$.
 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:
 ${\mathbf{IFAIL}}=1$

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,  ${\mathbf{N}}\le 1$. 
 ${\mathbf{IFAIL}}=4$

On entry,  ${\mathbf{JOB}}\ne 1$ or $2$. 
7 Accuracy
The results should be accurate to within a small multiple of the machine precision.
The time taken is approximately proportional to $n\times \mathrm{log}n$, but also depends on the factorization of $n$. C06EKF is faster if the only prime factors of $n$ are $2$, $3$ or $5$; and fastest of all if $n$ is a power of $2$.
On the other hand, C06EKF is particularly slow if
$n$ has several unpaired prime factors, i.e., if the ‘squarefree’ part of
$n$ has several factors.
For such values of
$n$,
C06FKF (which requires additional real workspace) is considerably faster.
9 Example
This example reads in the elements of one period of two real vectors $x$ and $y$, and prints their discrete convolution and correlation (as computed by C06EKF). In realistic computations the number of data values would be much larger.
9.1 Program Text
Program Text (c06ekfe.f90)
9.2 Program Data
Program Data (c06ekfe.d)
9.3 Program Results
Program Results (c06ekfe.r)