Section: New Results

Multiple-precision arithmetic

In [25] , Pascal Molin showed that the error function erf can be computed very efficiently using a formula involving an integeral of a form appropriate for fast evaluation using the trapezoidal scheme. A rigorous analysis of the scheme in this context allows to get precise bounds on the various errors terms, and therefore to give a proven compexity result for the multiple-precision evaluation of erf . The good theoretical behaviour is confirmed by an implementation in Pari.

Together with David Harvey (New York University), P. Zimmermann studied the short division of long integers, i.e., the division of a $2n$-bit integer by an $n$-bit integer where only the integer quotient is wanted, or an approximation of it. They gave detailed algorithms with rigorous errors bounds, and implemented them in GNU MPFR. Using Harvey's integer middle product code, they obtain a speedup of up to 10% with respect to the best known implementation [20] .

With Guillaume Melquiond (Proval project-team, INRIA Saclay), and Prof. W. Georg Nowak (Institute of Mathematics, Vienna), P. Zimmermann worked on the numerical approximation of the Masser-Gramain constant, following some work of Gramain and Weber in 1985. This work disproves a conjecture of Gramain, and enables one to determine the following approximation of that constant:

This work has been completed in 2011  [12] .

The article “The Great Trinomial Hunt” has been published in the Notices of the AMS [7] .