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inria-00126428, version 2

## Time- and Space-Efficient Evaluation of Some Hypergeometric Constants

Howard Cheng () a1, Guillaume Hanrot () b2, Emmanuel Thomé () b2, Eugene Zima () c3, Paul Zimmermann () b2

N° RR-6105 (2007)

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 $\zeta(3)$.

• a –  University of Lethbridge
• b –  INRIA
• c –  Wilfrid Laurier University
• 1:  Department of Mathematics and Computer Science
• University of Lethbridge
• 2:  CACAO (INRIA Lorraine - LORIA)
• CNRS : UMR7503 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)
• 3:  Department of physics and computer science
• Wilfrid Laurier University
• Domain : Computer Science/Symbolic Computation
• Keywords : Hypergeometric constants – binary splitting – sieve
• Internal note : RR-6105
• Available versions :  v1 (2007-01-25) v2 (2007-01-25)

• inria-00126428, version 2
• oai:hal.inria.fr:inria-00126428
• From:
• Submitted on: Thursday, 25 January 2007 14:24:04
• Updated on: Thursday, 25 January 2007 14:24:36