# Fast computation of Bernoulli, Tangent and Secant numbers

(with Richard Brent)

Computational and Analytical Mathematics, Springer Proceedings in Mathematics & Statistics, Vol. 50, 2013, 127–142 (DOI).

arXiv preprint (August 2011).

## Abstract

We consider the computation of Bernoulli, Tangent (zag), and Secant (zig or Euler) numbers. In particular, we give asymptotically fast algorithms for computing the first *n* such numbers in *O*(*n*^{2} (log *n*)^{2 + o(1)}) bit-operations. We also give very short in-place algorithms for computing the first *n* Tangent or Secant numbers in *O*(*n*^{2}) integer operations. These algorithms are extremely simple, and fast for moderate values of *n*. They are faster and use less space than the algorithms of Atkinson (for Tangent and Secant numbers) and Akiyama and Tanigawa (for Bernoulli numbers).

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