% -*- mode: noweb; noweb-default-code-mode: R-mode; -*-
\documentclass[nojss]{jss}
\usepackage{dsfont}
\usepackage{bbm}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{algpseudocode}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\DeclareMathOperator*{\ketten}{K}
\newcommand{\Fmn}[2]{\ensuremath{\operatorname{{}_{#1}F_{#2}}}}
\newcommand{\ft}{\ensuremath{\Fmn{2}{1}}}
\newcommand{\fall}[2]{\left(#1\right)_{#2}}
\newcommand{\rise}[2]{\left(#1\right)^{#2}}
\newcommand{\ams}[1]{$\left(#1\right)$}
%% just as usual
\author{Robin K. S. Hankin\\Auckland University of Technology}
\title{Numerical evaluation of the Gauss hypergeometric function with
the \pkg{hypergeo} package}
%\VignetteIndexEntry{The hypergeo package}
%% for pretty printing and a nice hypersummary also set:
\Plainauthor{Robin K. S. Hankin}
\Plaintitle{The hypergeo package}
\Keywords{Hypergeometric functions, numerical evaluation, complex
plane, \proglang{R}, residue theorem}
\Plainkeywords{Hypergeometric functions, numerical evaluation, complex
plane, R, residue theorem}
\Abstract{This paper introduces the \pkg{hypergeo} package of R
routines, for numerical calculation of hypergeometric functions.
The package is focussed on efficient and accurate evaluation of the
hypergeometric function over the whole of the complex plane within
the constraints of fixed-precision arithmetic. The hypergeometric
series is convergent only within the unit circle, so analytic
continuation must be used to define the function outside the unit
circle. This short document outlines the numerical and conceptual
methods used in the package; and justifies the package philosophy,
which is to maintain transparent and verifiable links between the
software and AMS-55. The package is demonstrated in the context of
game theory. }
%% publication information
%% NOTE: This needs to filled out ONLY IF THE PAPER WAS ACCEPTED.
%% If it was not (yet) accepted, leave them commented.
%% \Volume{13}
%% \Issue{9}
%% \Month{September}
%% \Year{2004}
%% \Submitdate{2004-09-29}
%% \Acceptdate{2004-09-29}
%% The address of (at least) one author should be given
%% in the following format:
\Address{
Robin K. S. Hankin\\
Auckland University of Technology\\
New Zealand\\
E-mail: \email{hankin.robin@gmail.com}\\
}
%% need no \usepackage{Sweave.sty}
\SweaveOpts{}
\begin{document}
<<set_calculate_from_scratch, echo=FALSE,print=FALSE>>=
calculate_from_scratch <- FALSE
@
\section{Introduction}
The {\em geometric} series~$\sum_{k=0}^\infty t_k$ with~$t_k=z^k$ may
be characterized by its first term and the constant ratio of
successive terms~$t_{k+1}/t_k=z$, giving the familiar
identity~$\sum_{k=0}^\infty z^k=\left(1-z\right)^{-1}$. Observe that
while the series has unit radius of convergence, the right hand side
is defined over the whole complex plane except for~$z=1$ where it has
a pole. Series of this type may be generalized to a {\em
hypergeometric} series in which the ratio of successive terms is a
rational function of~$k$:
\[
\frac{t_{k+1}}{t_k}=\frac{P(k)}{Q(k)}
\]
where~$P(k)$ and~$Q(k)$ are polynomials. If both numerator and
denominator have been completely factored we would write
\[
\frac{t_{k+1}}{t_k} = \frac{(k+a_1)(k+a_2)\cdots(k+a_p)}{(k+b_1)(k+b_2)\cdots(k+b_q)(k+1)}z
\]
\noindent (the final term in the denominator is due to historical
reasons), and if we require~$t_0=1$ then we write
\begin{equation}\label{genhypergeo_definition}
\sum_{k=0}^\infty t_kz^k=
\Fmn{a}{b}\left[{
a_1, a_2, \ldots,a_p\atop
b_1, b_2, \ldots,b_q}
; z\right]
\end{equation}
when defined. An absent factor is indicated with a dash; thus
$\Fmn{0}{0}\left[\begin{array}{l}-\\-\end{array};z\right]=e^z$. In
most cases of interest one finds that~$p=2$, $q=1$ suffices.
Writing~$a,b,c$ for the two upper and one lower argument respectively,
the resulting function~$\ft\left(a,b;c;z\right)$ is known as {\em the}
hypergeometric function. Many functions of elementary analysis are of
this form; examples would include logarithmic and trigonometric
functions, Bessel functions, etc. For example,
$\ft\left(\frac{1}{2},1;\frac{3}{2};-z^2\right)=z^{-1}\operatorname{\arctan}
z$.
\citet{michel2008} state that physical applications are ``plethora''.
In addition, naturally-occuring combinatorial series frequently have a
sum expressible in terms of hypergeometric functions and an example
from the author's work in the field game theory is given below.
\subsection{Equivalent forms}
The hypergeometric function's series representation, namely
\begin{equation}\label{series}\tag{15.1.1}
\ft\left(a,b;c;z\right)=\sum_{k=0}^\infty\frac{\fall{a}{k}\fall{b}{k}}{\fall{c}{k}k!}z^k,\qquad
\fall{a}{k}=\Gamma(a+k)/\Gamma(a)
\end{equation}
\noindent has unit radius of convergence by the ratio test but the
integral form
\begin{equation}\label{integral}\tag{15.3.1}
\ft\left(a,b;c;z\right)=
\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{t=0}^1
t^{b-1}(1-t)^{c-b-1}(1-tz)^{-a}\,dt,
\end{equation}
\noindent due to Gauss, furnishes analytic continuation; it is usual
to follow Riemann and define a cut along the positive real axis
from~$1$ to~$\infty$ and specify continuity from below [NB: equations
with three-part numbers, as \ref{series} and \ref{integral} above,
are named for their reference in~\citet{abramowitz1965}]. This is
implemented as \code{f15.3.1()} in the package and exhibits
surprisingly accurate evaluation.
Gauss also provided a continued fraction form for the hypergeometric
function [implemented as~\code{hypergeo_contfrac()} in the package]
which has superior convergence rates for parts of the complex plane at
the expense of more complicated convergence
properties~\citep{cuyt2008}.
\section{The hypergeo package}
The \pkg{hypergeo} package provides some functionality for the
hypergeometric function; the emphasis is on fast vectorized
\proglang{R}-centric code, complex~$z$ and moderate real values for
the auxiliary parameters~$a,b,c$. The package is released under
GPL-2.
Observing the slow convergence of the series
representation~\ref{series}, the complex behaviour of the continued
fraction representation, and the heavy computational
expense of the integral representation~\ref{integral}, it is clear
that non-trivial numerical techniques are required for a production
package.
The package implements a generalization of the method
of~\citet{forrey1997} to the complex case. It utilizes the
observation that the ratio of successive terms approaches~$z$, and
thus the strategy adopted is to seek a transformation which reduces
the modulus of~$z$ to a minimum. \citeauthor{abramowitz1965} give the
following transformations:
\newcommand{\four}[4]{\frac{\Gamma\left(#1\right)\Gamma\left(#2\right)}{\Gamma\left(#3\right)\Gamma\left(#4\right)}}
\begin{align}
\ft\left(a,b;c;z\right)
&= \left(1-z\right)^{-a}\ft\left(a,c-b;c;\frac{z}{z-1}\right)\tag{15.3.4}\label{15.3.4}\\
&= \left(1-z\right)^{-b}\ft\left(a,c-a;c;\frac{z}{z-1}\right)\tag{15.3.5}\label{15.3.5}\\
&= \four{c}{c-a-b}{c-a}{c-b}\ft\left(a,b;a+b-c+1;1-z\right)\nonumber\\
&{}\qquad+ (1-z)^{c-a-b}\four{c}{a+b-c}{a}{b}\ft\left(c-a,c-b;c-a-b+1;1-z\right)\label{15.3.6}\tag{15.3.6}\\
&= \four{c}{b-a}{b}{c-a}\left(-z\right)^{-a}\ft\left(a,1-c+a;1-b+a;\frac{1}{z}\right)\nonumber\\
&{}\qquad+\four{c}{a-b}{a}{c-b}\left(-z\right)^{-b}\ft\left(b,1-c+b;1-a+b;\frac{1}{z}\right)\label{15.3.7}\tag{15.3.7}\\
&= (1-z)^{-a}\four{c}{b-a}{b}{c-a}\ft\left(a,c-b;a-b+1;\frac{1}{1-z}\right)\nonumber\\
&{}\qquad+(1-z)^{-b}\four{c}{a-b}{a}{c-b}\ft\left(b,c-a;b-a+1;\frac{1}{1-z}\right)\label{15.3.8}\tag{15.3.8}\\
&=\four{c}{c-a-b}{c-a}{c-b}z^{-a}\ft\left(a,a-c+1;a+b-c+1;1-\frac{1}{z}\right)\nonumber\\
&{}\qquad+\four{c}{a+b-c}{a}{b}(1-z)^{c-a-b}z^{a-c}\ft\left(c-a,1-a;c-a-b+1;1-\frac{1}{z}\right)\label{15.3.9}\tag{15.3.9}.
\end{align}
Observing that the
set~$\left\{z,\frac{z}{z-1},1-z,\frac{1}{z},\frac{1}{1-z},1-\frac{1}{z}\right\}$
forms a group under functional composition\footnote{It is the
anharmonic subgroup of the M\"{o}bius transformations, generated
by~$z\longrightarrow 1/z$ and~$z\longrightarrow 1-z$. It is
isomorphic to~$S_3$, the symmetric group on~3 elements.} we may
apply each of the transformations to the primary argument~$z$ and
choose the one of smallest absolute value to evaluate.
Given the appropriate transformation, the right hand side is evaluated
using direct summation. If~$\left|z\right|<1$, the series is
convergent by the ratio test, but may require a large number of terms
to achieve acceptable numerical precision. Summation is dispatched to
\code{genhypergeo_series()} which evaluates the generalized
hypergeometric function~\ref{genhypergeo_definition}; the \proglang{R}
implementation uses multiplication by repeatedly incremented upper and
lower indices~$a_i,b_i$.
%\begin{algorithmic}\label{alt}
% \State $\mathit{fac}\gets 1$
% \State $\mathit{temp}\gets\mathit{fac}$
% \State $\mathit{series}\gets\mathit{ZXCVXCVDFADF}$
% \While {$\mathit{series}\neq\mathit{temp}$}
% \State $\mathit{fac}\gets
% \mathit{fac}\times\frac{a_1\times\cdots\times a_p}{b_1\times\cdots\times b_q}\times z$
% \State $a_1\gets a_1+1,\ldots, b_q\gets b_q+1$
% \State $temp\gets \mathit{series}$
% \State $\mathit{series}\gets \mathit{series}+\mathit{fac}$
% \EndWhile
%\end{algorithmic}
%(lower indices~$b_i$ are appended with a ``$+1$'').
Thus for example if $(1-z)^{-1}$ is small in absolute value we would
use function \code{f13.3.8()}:
\begin{Schunk}
\begin{Sinput}
> require("hypergeo")
> f15.3.8
\end{Sinput}
\begin{Soutput}
function(A,B,C,z,tol=0,maxiter=2000){
jj <- i15.3.8(A,B,C)
jj[1]*(1-z)^(-A)*genhypergeo(U=c(A,C-B),L=A-B+1,z=1/(1-z),tol=tol,maxiter=maxiter) +
jj[2]*(1-z)^(-B)*genhypergeo(U=c(B,C-A),L=B-A+1,z=1/(1-z),tol=tol,maxiter=maxiter)
}
\end{Soutput}
\end{Schunk}
\noindent (slightly edited in the interests of visual clarity). This
is a typical internal function of the package and like all similar
functions is named for its equation number in~\cite{abramowitz1965}.
Note the helper function \code{i15.3.9()}, which calculates the Gamma
coefficients of the two hypergeometric terms in the identity. This
structure allows transparent checking of the code.
\subsection{Special cases}
The methods detailed above are not applicable for all values of the
parameters~$a,b,c$. If, for example, $c=a+b\pm m$, $m\in\mathbb{N}$
(a not uncommon case), then equation~\ref{15.3.6} is not useful
because each term has a pole; and it is numerically difficult to
approach the limit. In this case the package dispatches to
\code{hypergeo_cover1()} which uses~\ref{15.3.4} through~\ref{15.3.9}
but with~\ref{15.3.6} replaced with suitable limiting forms such as
\begin{equation}\tag{15.3.11}\label{15.3.11}
\ft\left(a,b;a+b+m;z\right)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}
\sum_{n=0}^\infty\frac{(a)_n(b)_n}{(n!)^2}\left[
2\psi(n+1)-\psi(a+n)-\psi(b+n)-\log(1-z)\right](1-z)^n,\qquad\pi<\left|\operatorname{\arg}(1-z)\right|<\pi,\left|1-z\right|<1
\end{equation}
(\citeauthor{abramowitz1965} give a similar expression for
negative~$m$). This equation is comparable to~\ref{15.3.6} in terms
of computational complexity but requires evaluation of the digamma
function~$\psi$. Equation~\ref{15.3.11} is evaluated in the package
using an algorithm similar to that for \code{genhypergeo_series()} but
includes a runtime option which specifies whether to
evaluate~$\psi\left(\cdot\right)$ \emph{ab initio} each time it is
needed, or to use the recurrence
relation~$\psi\left(z+1\right)=\psi\left(z\right)+1/z$ at each
iteration after the first. These two options appear to be comparable
in terms of both numerical accuracy and speed of execution, but
further work would be needed to specify which is preferable in this
context.
A similar methodology is used for the case~$b=a\pm m$,
$m=0,1,2,\ldots$ in which case the package dispatches to
\code{hypergeo_cover2()}.
However, the case~$c-a=0,1,2,\ldots$ is not covered
by~\cite{abramowitz1965} and the package dispatches
to~\code{hypergeo_cover3()} which uses formulae taken from the Wolfram
functions site~\citep{wolfram2014}. For example
\code{w07.23.06.0026.01()} gives a straightforwardly implementable
numerical expression for~$\Fmn{2}{1}$ as a sum of two {\em finite}
series and a generalized hypergeometric function~$\Fmn{3}{2}$ with
primary argument~$z^{-1}$.
In all these cases, the limiting behaviour is problematic. For
example, if~$a+b-c$ is close to, but not exactly equal to, an integer
then equation~\ref{15.3.11} is not applicable. The analytic value of
the hypergeometric function in these circumstances is typically of
moderate modulus, but both terms of equation~\ref{15.3.6} have large
amplitude and numerics are susceptible to cancellation errors.
\subsection{Critical points}
All the above methods fail when~$z=\frac{1}{2}\pm\frac{i\sqrt{3}}{2}$,
because none of the transformations~\ref{15.3.6}-\ref{15.3.9} change
the modulus of~$z$ from 1. The function is convergent at these points
but numerical evaluation is difficult. This issue does not arise in
the real case considered by~\citet{forrey1997}.
These points were considered by \cite{buhring1987} who presented a
computational method for these values; however, his method is not
suitable for finite-precision arithmetic (a brief discussion is
presented at \code{?buhring}) and the package employs either an
iterative scheme due to Gosper~\citep{mpmath}, or the residue theorem
if~$z$ is close to either of these points.
\section{Package testing suite}
The package comes with an extensive test suite in the \code{tests/}
directory. The tests fall into two main categories, firstly
comparison with either \proglang{Maple} or \proglang{Mathematica}
output (although~\cite{becken2000} caution that \proglang{Mathematica}
routines cannot be used as reference values); and secondly,
verification of identities which appear in AMS-55 as named equations.
\section{The package in use}
The \pkg{hypergeo} package offers direct numerical functionality to
the \proglang{R} user on the command line. One example from the
author's current work is in game theory. Consider a game in which a
player is given~$n$ counters each of which she must allocate into one
of two boxes, $A$ or $B$. At times $t = 1,2,3\ldots$ a box is
identified at random and, if it is not empty, a counter removed from
it; box~$A$ is chosen with probability~$p$ and box~$B$ with
probability~$1-p$. The object of the game is to remove all counters
as quickly as possible. If the player places~$a$ counters in box~$A$
and~$b$ in~$B$, then the probability mass function of removing the
final counter at time~$t=a+b+r$ is
<<loadpackages,echo=FALSE>>=
require("hypergeo")
require("elliptic")
@
\begin{equation}
p^a(1-p)^b\left[ {a+b+r-1 \choose a-1,
b+r}(1-p)^r+
{a+b+r-1 \choose a+r, b-1}p^r \right],\qquad r=0,1,2,\ldots.
\end{equation}
The two terms correspond to the final counter being removed from box~$A$
or~$B$ respectively. This PMF has expectation
\begin{align}
p^a(1-p)^b\left[
p {a+b\choose a+1,b-1}\,\ft\left(a+b+1,2;a+2;p\right)+\right.\nonumber\\ \left.
(1-p){a+b\choose a-1,b+1}\,\ft\left(a+b+1,2;b+2;1-p\right)
\right]\label{expectation}
\end{align}
with \proglang{R} idiom:
<<R_expectation>>=
expected <- function(a,b,p){
Re(
choose(a+b,b) * p^a * (1-p)^b * (
p *b/(1+a) * hypergeo(a+b+1,2,a+2, p) +
(1-p)*a/(1+b) * hypergeo(a+b+1,2,b+2,1-p) ))
}
@
Thus if~$p=0.8$ and given~$n=10$ counters we might wonder whether it
is preferable to allocate them~$(8,2)$ or~$(9,1)$:
<<useit>>=
c(expected(8,2,0.8),expected(9,1,0.8))
@
showing that the latter allocation is preferable in expectation.
The package is designed for use with \proglang{R} and
Figure~\ref{complexhypergeometricplot} shows the package being used to
visualize~$\ft\left(2,\frac{1}{2};\frac{2}{3};z\right)$ over a region of
the complex plane.
%% Thanks to Dario Strbenac for the following structure
<<hypergeo_figure_file,echo=FALSE>>=
png("hypergeometric_plot.png",width=800,height=800)
@
<<label=wp_figure_plot,echo=FALSE>>=
x <- seq(from=0,to=2,len=200)
y <- seq(from=-1,to=1,len=200)
z <- outer(x,1i*y,"+")
hz <- hypergeo(2,1/2,2/3,z)
par(pty='s')
view(x,y,hz,levels=seq(from=-4,to=4),xlab='Real',ylab='Imag')
@
<<label=wp_figure_close,echo=FALSE>>=
null <- dev.off()
@
\begin{figure}[htbp]
\begin{center}
\includegraphics{hypergeometric_plot.png}
\caption{View of the\label{complexhypergeometricplot}
function~$\ft\left(2,\frac{1}{2};\frac{2}{3};z\right)$ evaluated
over a part of the complex plane using the \pkg{hypergeo}
package. Function visualization following \cite{thaller1998}
and the \pkg{elliptic} package~\citep{hankin2006}; hue
corresponds to argument and saturation to modulus. Solid contour
lines correspond to real function values and dotted to imaginary
function values. Note the cut line along the real axis starting
at~$\left(1,0\right)$, made visible by an abrupt change in hue}
\end{center}
\end{figure}
\subsection{Conclusions and further work}
Evaluation of the hypergeometric function is hard, as evidenced by the
extensive literature concerning its numerical
evaluation~\citep{becken2000,michel2008,forrey1997,buhring1987}. The
\pkg{hypergeo} package is presented as a partial implementation,
providing reasonably accurate evaluation over a large portion of the
complex plane and covering moderate real values of the auxiliary
parameters~$a,b,c$. Difficulties arise when~$b-a$ or~$c-b-a$ become
close to, but not exactly, integers because the terms in
equations~\ref{15.3.6} through~\ref{15.3.9} become large and
cancellation errors become important.
Further work might include extension to complex auxiliary parameters
but \citeauthor{michel2008} caution that this is not straightforward.
\bibliography{hypergeometric}
\end{document}