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# Introduction

In Bailey, Borwein, and Borwein's paper [8] an overview of a method is given for constructing series and algorithms for finding rapid approximations for . Although Ramanujan did not know these algorithms many of the key ingredients are in his notebooks [13]. The algorithms depend crucially on the solvable forms of certain modular equations for the theta-functions due to Ramanujan. In [8] two algorithms are given -- one quartic and one quintic algorithm. In a related paper [5] a septic algorithm is sketched. In [4] a general method is given for constructing p-th order algorithms. These algorithms involve defining a sequence recursively and for which converges to to high order. In general, for us, p-th order convergence of a sequence to means that tends to and that

for some constant C>0. The proof of p-th order convergence depends crucially on identifying as the value of a certain function , which can be defined in terms of elliptic integrals or equivalently in terms of theta-functions.

Since [8] was first written, Borwein and Borwein [6] found an amazing cubic algorithm. See also [7]. This algorithm comes from a certain hypergeometric analog of elliptic integrals that was studied by Ramanujan. In this paper, we make an attempt to unify some of these results and find new algorithms. Instead of a fixed function , we define an infinite family of functions for p>1. Our goal is to construct for each p, a p-th order iteration which converges to , using the function .

In Section 2 we briefly describe the Borwein and Borwein -function. In Section 3 we define (for each p>1), in terms of Dedekind's eta function. We find that satisfies a nice modular transformation property, and a nice p-th order modular equation. In Section 4 we show how the results in Section 3 may be used to construct p-th order iterations which converge to . The method is illustrated with some MAPLE sessions. In Section 5 we give a brief overview of how our method relates to known quadratic, cubic and quartic Borwein and Borwein iterations. Details are given how the cases p=2, 4 relate to the quadratic and quartic algorithms. In Section 6 we show how the case p=3 gives the Borwein and Borwein cubic algorithm.

Our main goal in this paper is to somehow mimic the Borwein and Borwein cubic algorithm and obtain analogous higher order algorithms. In Section 7 we obtain an explicit solvable septic iteration which converges to . In Section 8 we obtain an explicit solvable nonic (ninth order) iteration which converges to . This nonic iteration does not appear to come from iterating the cubic twice.

Given the organic nature of this document, we hope, in a later version of this paper, to provide more complete details and improvements of the septic and nonic algorithms. In a later version we will also include some mixed order algorithms. For instance, although the case p=2 leads naturally to a quadratic iteration we may instead use it to construct a new cubic iteration.

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