A multimodular algorithm for computing Bernoulli numbers

@article{Harvey2010AMA,
  title={A multimodular algorithm for computing Bernoulli numbers},
  author={D. Harvey},
  journal={Math. Comput.},
  year={2010},
  volume={79},
  pages={2361-2370}
}
  • D. Harvey
  • Published 2010
  • Mathematics, Computer Science
  • Math. Comput.
  • We describe an algorithm for computing Bernoulli numbers. Using a parallel implementation, we have computed B(k) for k = 10^8, a new record. Our method is to compute B(k) modulo p for many small primes p, and then reconstruct B(k) via the Chinese Remainder Theorem. The asymptotic time complexity is O(k^2 log(k)^(2+epsilon)), matching that of existing algorithms that exploit the relationship between B(k) and the Riemann zeta function. Our implementation is significantly faster than several… CONTINUE READING

    Tables and Topics from this paper.

    Fast computation of Bernoulli, Tangent and Secant numbers
    • 13
    • PDF
    A subquadratic algorithm for computing the n-th Bernoulli number
    • 5
    • PDF
    Irregular primes to 163 million
    • 34
    • PDF
    Modern Computer Arithmetic
    • 176
    • PDF
    Computing Bernoulli and Tangent Numbers
    Irregular primes to two billion
    • 7
    • PDF
    Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture
    • 2
    • PDF
    An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only
    • 2
    • Highly Influenced
    • PDF
    A Two Pronged Progress in Structured Dense Matrix Multiplication
    • 2
    • PDF

    References

    Publications referenced by this paper.
    SHOWING 1-10 OF 25 REFERENCES
    Faster Computation of Bernoulli Numbers
    • 17
    Modular multiplication without trial division
    • 2,383
    • PDF
    On Schönhage's algorithm and subquadratic integer gcd computation
    • 55
    • PDF
    A classical introduction to modern number theory
    • 2,249
    Cyclotomic Fields I and II
    • 221
    Modern computer algebra
    • 881
    • PDF
    Schnelle Multiplikation großer Zahlen
    • 698
    Modular Forms, A Computational Approach
    • 270
    • PDF