Introduction

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.