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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

Sage claims that bernoulli(10^5) takes about 11 seconds.


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