An FFT extension of the elliptic curve method of factorization
An FFT extension of the elliptic curve method of factorization
Simplification and optimization transformations of chains of recurrences
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Efficient rational number reconstruction
Journal of Symbolic Computation
Fast Multiple-Precision Evaluation of Elementary Functions
Journal of the ACM (JACM)
On accelerated methods to evaluate sums of products of rational numbers
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Computational strategies for the Riemann zeta function
Journal of Computational and Applied Mathematics - Special issue on numerical analysis in the 20th century vol. 1: approximation theory
Acceleration of Euclidean Algorithm and Rational Number Reconstruction
SIAM Journal on Computing
Fast Multiprecision Evaluation of Series of Rational Numbers
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
Maximal quotient rational reconstruction: an almost optimal algorithm for rational reconstruction
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
P-adic reconstruction of rational numbers
ACM SIGSAM Bulletin
Time-and space-efficient evaluation of some hypergeometric constants
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
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Many important constants, such as e and Apéry's constant ζ(3), can be approximated by a truncated hypergeometric series. The evaluation of such series to high precision has traditionally been done by binary splitting followed by fixed-point division. However, the numerator and the denominator computed by binary splitting usually contain a very large common factor. In this paper, we apply standard computer algebra techniques including modular computation and rational reconstruction to overcome the shortcomings of the binary splitting method. The space complexity of our algorithm is the same as a bound on the size of the reduced numerator and denominator of the series approximation. Moreover, if the predicted bound is small, the time complexity is better than the standard binary splitting approach. Our approach allows a series to be evaluated to a higher precision without additional memory. We show that when our algorithm is applied to compute ζ(3), the memory requirement is significantly reduced compared to the binary splitting approach.