Time- and Space-Efficient Evaluation of Some Hypergeometric Constants

Howard Cheng 1 Guillaume Hanrot 2 Emmanuel Thomé 2, * Eugene Zima 3 Paul Zimmermann 2
* Corresponding author
2 CACAO - Curves, Algebra, Computer Arithmetic, and so On
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : The currently best known algorithms for the numerical evaluation of hypergeometric constants such as $\zeta(3)$ to $d$ decimal digits have time complexity $O(M(d) \log^2 d)$ and space complexity of $O(d \log d)$ or $O(d)$. Following work from Cheng, Gergel, Kim and Zima, we present a new algorithm with the same asymptotic complexity, but more efficient in practice. Our implementation of this algorithm improves slightly over existing programs for the computation of $\pi$, and we announce a new record of 2 billion digits for $\zeta(3)$.
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Howard Cheng, Guillaume Hanrot, Emmanuel Thomé, Eugene Zima, Paul Zimmermann. Time- and Space-Efficient Evaluation of Some Hypergeometric Constants. International Symposium on Symbolic and Algebraic Computation - ISSAC'07, Jul 2007, Waterloo, Canada. ACM, pp.85-91, 2007, Proceedings of the 2007 international symposium on Symbolic and algebraic computation. 〈10.1145/1277548.1277561〉. 〈inria-00177850〉

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