More Quadratically Converging Algorithms for $\pi$

We present a quadratically converging algorithm for or based on a formula of Legendre's for complete elliptic integrals of modulus sin(7r/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 7T. 1. The Algorithms. The arithmetic-geometric mean of Gauss and Legendre is defined, for k E (0,1], by (1) a ,,1 + 2 b ,= a Cn+1 = 2 (an-bn) with ao:= 1, bo:= 1-k2 :=k', co:= k.Thecommonlimitof {an} and {bn} we call AGM(k'). The remarkable utility of the above iteration stems from two observations. Firstly, O b < bn+I < a +, < an and c 2+1=c,/4a"+1 which show that both sequences converge quadratically. Secondly, their common limit can be expressed in terms of complete elliptic integrals of the first kind K:= K(k), that is, (2) 2AGM(k') lo 1/-2dt K. Complete elliptic integrals of the second kind E:= E(k):= f g12 -ksi dt can also be calculated from the arithmetic-geometric mean iteration. Precisely, (3) (K -E)IK = 1/2( C2 + 2 C2 + +* 2 nC2 + . This powerful tool for computing elliptic integrals can be used to derive algorithms for S as follows. Let E':= E(k') and K':= K(k'). (These are the complete elliptic integrals in the conjugate modulus k' = 1 k2.) Then, Legendre's formula relating these quantities is (4) EK' + E'K-KK' = Received May 24, 1983; revised January 23, 1984 and September 25, 1984. 1980 Mathematics Subject Classification. Primary 65D20, 65C25, 32A25, 1OF05.