Fast computation of Bernoulli, Tangent and Secant numbers

TL;DR

This paper introduces new, faster algorithms for computing Bernoulli, Tangent, and Secant numbers, significantly improving efficiency and space usage over previous methods, especially for moderate n.

Contribution

It presents asymptotically fast algorithms and simple in-place methods for computing these special numbers, outperforming prior algorithms in speed and space efficiency.

Findings

Algorithms run in O(n^2.(log n)^(2+o(1))) bit-operations.

In-place algorithms operate in O(n^2) integer operations.

New methods are faster and require less space than previous algorithms.

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).