More quadratically converging algorithms for 𝜋

Abstract

We present a quadratically converging algorithm for π \pi based on a formula of Legendre’s for complete elliptic integrals of modulus sin ⁡ ( π / 12 ) \sin (\pi /12) and the arithmetic-geometric mean iteration of Gauss and Legendre. Precise asymptotics are provided which show this algorithm to be (marginally) the most efficient developed to date. As such it provides a natural computational check for the recent large-scale calculations of π \pi .