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

Abstract

The Ramanujan AGM continued fraction is a construct 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 R for any triple of positive reals a, b,η. Even in the problematic scenario when a ≈ b certain transformations allow rapid evaluation. In this process we find, for example, that when a η = b η = a rational number, R η 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 R in O(D) iterations where the implied big-O constant is independent of the positive-real triple a, b,η. 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.