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## Re: [Axiom-developer] Bernoulli puzzle

**From**: |
daly |

**Subject**: |
Re: [Axiom-developer] Bernoulli puzzle |

**Date**: |
Mon, 20 Oct 2014 07:19:34 -0500 |

Excellent, thank you.
>*PS. This is essentially using the recursive definition given*
>*in Wikipedia. Wikipedia calls this inefficient, but AFAICS*
>*due to memoization it is much better than Akiyama-Tanigawa*
>*presented in Wikipedia.*
I read several papers on the Bernoulli function. Axiom references
John Brillhart's "On the Euler and Bernoulli polynomials" which
was his Berkeley PhD thesis but I can't find a copy online anywhere.
There were a couple papers which were an efficiency contest between
Mathematica and Sage. Sage uses Pari's implementation by default
which gives excellent results. Pari uses
\[ |B_n| = \frac{2n!}{(2\pi)^n}\zeta(n) \]
with floats but you have to completely control the precision.
Wolfram published
http://blog.wolfram.com/2008/04/29/today-we-broke-the-bernoulli-record-from-the-analytical-engine-to-mathematica
Sage claims that bernoulli(10^5) takes about 11 seconds.
Tim