Fast Computation of Bernoulli, Tangent and Secant Numbers

Richard P. Brent; David Harvey

doi:10.1007/978-1-4614-7621-4_8

Fast Computation of Bernoulli, Tangent and Secant Numbers

Richard P. Brent, David Harvey

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

9 Citations (Scopus)

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 O(n²(logn)^2+o(1)). We also give very short in-place algorithms for computing the first n Tangent or Secant numbers in O(n²) 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).

Original language	English
Title of host publication	Computational and Analytical Mathematics
Subtitle of host publication	In Honor of Jonathan Borwein's 60th Birthday
Publisher	Springer New York LLC
Pages	127-142
Number of pages	16
ISBN (Print)	9781461476207
DOIs	https://doi.org/10.1007/978-1-4614-7621-4_8
Publication status	Published - 2013

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	50
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

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10.1007/978-1-4614-7621-4_8

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title = "Fast Computation of Bernoulli, Tangent and Secant Numbers",

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 O(n2(logn)2+o(1)). We also give very short in-place algorithms for computing the first n Tangent or Secant numbers in O(n2) 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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Fast Computation of Bernoulli, Tangent and Secant Numbers. / Brent, Richard P.; Harvey, David.
Computational and Analytical Mathematics: In Honor of Jonathan Borwein's 60th Birthday. Springer New York LLC, 2013. p. 127-142 (Springer Proceedings in Mathematics and Statistics; Vol. 50).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - 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 O(n2(logn)2+o(1)). We also give very short in-place algorithms for computing the first n Tangent or Secant numbers in O(n2) 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).

AB - 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 O(n2(logn)2+o(1)). We also give very short in-place algorithms for computing the first n Tangent or Secant numbers in O(n2) 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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KW - Secant number

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Fast Computation of Bernoulli, Tangent and Secant Numbers

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