Approximations to Pi via the Dedekind eta function
F.G. Garvan
Math Department
University of Florida
Gainesville,
FL 32611
J.M. Borwein
Math Department
Simon Fraser University
Burnaby
B.C, V5A 1S6
Canada
Abstract:

Arguably the most efficient algorithm
currently known for the extended precision calculation of $\pi$ is a
quartic iteration due to J.M. and P.B. Borwein. In their paper,
the Borwein's show how this iteration and others are intimately connected
to the work of Ramanujan. This connection is shown utilizing
their {\it alpha}function which is defined in terms of thetafunctions.
They are able to find $p$th order iterations based on this function
using modular equations for the thetafunctions. In this paper
we construct an infinite family of functions $\alpha_p$. Each $\alpha_p$
gives rise to a $p$th order iteration. For $p=4$ we obtain a quartic iteration
due to the Borweins but not the one that comes from the {\it alpha}function.
For $p=3$ we obtain a cubic iteration
due to the Borweins that does not come from the {\it alpha}function.
For $p=7$ we find a septic iteration that is analogous to
the cubic iteration. For $p=9$ we obtain a nonic (ninth order) iteration
that does not seem to come from iterating the cubic twice.
Our method depends on using the computer
and a symbolic algebra package to find and solve certain modular
equations.