More quadratically converging algorithms for $\pi$

Authors:
J. M. Borwein and P. B. Borwein

Journal:
Math. Comp. **46** (1986), 247-253

MSC:
Primary 65D20

DOI:
https://doi.org/10.1090/S0025-5718-1986-0815846-6

MathSciNet review:
815846

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Abstract | References | Similar Articles | Additional Information

Abstract: We present a quadratically converging algorithm for $\pi$ based on a formula of Legendre’s for complete elliptic integrals of modulus $\sin (\pi /12)$ 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 $\pi$.

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Additional Information

Keywords:
<IMG WIDTH="18" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\pi$">,
arithmetic-geometric mean iteration,
high-precision calculation

Article copyright:
© Copyright 1986
American Mathematical Society