Here $x$ and $y$ are complex vectors, assumed to be periodic, with period $n$, i.e., ${x}_{j}={x}_{j\pm n}={x}_{j\pm 2n}=\cdots \text{}$;$z$ 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, and $\stackrel{~}{x}$ is the inverse discrete Fourier transform of the sequence ${x}_{j}$, i.e.,
then ${\hat{z}}_{k}=\sqrt{n}.{\hat{x}}_{k}{\hat{y}}_{k}$ and ${\hat{w}}_{k}=\sqrt{n}.{\overline{\hat{x}}}_{k}{\hat{y}}_{k}$ (the bar denoting complex conjugate).
4References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
On entry: the elements of one period of the vector $x$. If x is declared with bounds $(0:{\mathbf{n}}-1)$ 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.
On entry: the elements of one period of the vector $y$. If y is declared with bounds $(0:{\mathbf{n}}-1)$ 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.
Note: the dimension of the array work
must be at least
$2\times {\mathbf{n}}+15$.
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 $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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 $\mathrm{-1}$, 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, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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.
9Further Comments
The time taken is approximately proportional to $n\times \mathrm{log}\left(n\right)$, 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.