On the Ramanujan AGM Fraction, I: The Real-Parameter Case

Abstract

The Ramanujan AGM continued fraction is a construct

{\small $${\cal R}_\eta(a,b) =\,\frac{a}{\displaystyle \eta+\frac{b^2}{\displaystyle \eta +\frac{4a^2}{\displaystyle \eta+\frac{9b^2}{\displaystyle \eta+{}_{\ddots}}}}}$$}

enjoying attractive algebraic properties, such as a striking arithmetic-geometric mean (AGM) relation and elegant connections with elliptic-function theory. But the fraction also presents an intriguing computational challenge. Herein we show how to rapidly evaluate ${\cal R}$ for any triple of positive reals $a,b,\eta$. Even in the problematic scenario when $a \approx b$ certain transformations allow rapid evaluation. In this process we find, for example, that when $a\eta = b\eta = $ a rational number, ${\cal R}_\eta$ is essentially an $L$-series that can be cast as a finite sum of fundamental numbers. We ultimately exhibit an algorithm that yields $D$ good digits of ${\cal R}$ in $O(D)$ iterations where the implied big-$O$ constant is independent of the positive-real triple $a,b,\eta$. Finally, we address the evidently profound theoretical and computational dilemmas attendant on complex parameters, indicating how one might extend the AGM relation for complex parameter domains.