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Zbl 0755.11006
Fillebrown, Sandra
Faster computation of Bernoulli numbers. (English)
[J] J. Algorithms 13, No.3, 431-445 (1992). ISSN 0196-6774

The author presents an algorithm, based on the classical formula $$B\sb{2k}=(-1)\sp{k+1}2(2k!)\zeta(2k)(2\pi)\sp{-2k},$$ to compute the $2k$th Bernoulli number $B\sb{2k}$, defined by $X/(e\sp X- 1)=\sum\sb{n\geq 1}B\sb n X\sp n/n!$. The space requirement of this algorithm is ${\cal O}(n\log\log n)$ bits and it involves ${\cal O}(n\sp 2\log\sp 2n\log\log n)$ bit operations. The algorithm can also be efficiently generalized to compute all Bernoulli numbers up to some point. This slightly improves on the previous algorithms of {\it S. Chowla} and {\it P. Hartung} [Acta Arith. 22, 113-115 (1972; Zbl 0244.10008)] and of {\it D. E. Knuth} and {\it T. J. Buckholtz} [Math. Comput. 21, 663-688 (1967; Zbl 0178.044)].
[A.Granville (Athens / Georgia)]

MSC 2000:: *11B68 Bernoulli numbers, etc.
11Y16 Algorithms
68Q25 Analysis of algorithms and problem complexity

Keywords: algorithm; Bernoulli numbers

Citations: Zbl 0244.10008; Zbl 0178.044

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