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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