A multimodular algorithm for computing Bernoulli numbers
نویسنده
چکیده
We describe an algorithm for computing Bernoulli numbers. Using a parallel implementation, we have computed Bk for k = 10 8, a new record. Our method is to compute Bk modulo p for many small primes p, and then reconstruct Bk via the Chinese Remainder Theorem. The asymptotic time complexity is O(k2 log k), matching that of existing algorithms that exploit the relationship between Bk and the Riemann zeta function. Our implementation is significantly faster than several existing implementations of the zeta-function method.
منابع مشابه
THE AKIYAMA-TANIGAWA ALGORITHM FOR CARLITZ’S q-BERNOULLI NUMBERS
We show that the Akiyama-Tanigawa algorithm and Chen’s variant for computing Bernoulli numbers can be generalized to Carlitz’s q-Bernoulli numbers. We also put these algorithms in the larger context of generalized Euler-Seidel matrices.
متن کاملFast computation of Bernoulli, Tangent and Secant numbers
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(n2(logn)2+o(1)) bit-operations. 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 fo...
متن کاملThe Akiyama-Tanigawa algorithm for Bernoulli numbers
A direct proof is given for Akiyama and Tanigawa’s algorithm for computing Bernoulli numbers. The proof uses a closed formula for Bernoulli numbers expressed in terms of Stirling numbers. The outcome of the same algorithm with different initial values is also briefly discussed. 1 The Algorithm In their study of values at non-positive integer arguments of multiple zeta functions, S. Akiyama and ...
متن کاملExtended Zeilberger’s Algorithm for Identities on Bernoulli and Euler Polynomials
We present a computer algebra approach to proving identities on Bernoulli and Euler polynomials by using the extended Zeilberger’s algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the re...
متن کاملAlgorithms for Bernoulli and Allied Polynomials
We investigate some algorithms that produce Bernoulli, Euler and Genocchi polynomials. We also give closed formulas for Bernoulli, Euler and Genocchi polynomials in terms of weighted Stirling numbers of the second kind, which are extensions of known formulas for Bernoulli, Euler and Genocchi numbers involving Stirling numbers of the second kind.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Math. Comput.
دوره 79 شماره
صفحات -
تاریخ انتشار 2010