Time- and Space-Efficient Evaluation of Some Hypergeometric Constants

The currently best known algorithms for the numerical evaluation of hypergeometric constants such as $\zeta(3)$ to $d$ decimal digits have time complexity $O(M(d) \log^2 d)$ and space complexity of $O(d \log d)$ or $O(d)$. Following work from Cheng, Gergel, Kim and Zima, we present a new algorithm with the same asymptotic complexity, but more efficient in practice. Our implementation of this algorithm improves slightly over existing programs for the computation of $\pi$, and we announce a new record of 2 billion digits for $\zeta(3)$

Chains of Recurrences - a method to expedite the evaluation of closed-form functions

Chains of Recurrences (CR's) are introduced as an effective method to evaluate functions at regular intervals. Algebraic properties of CR's are examined and an algorithm that constructs a CR for a given function is explained. Finally, an implementation of the method in MAXIMA/Common Lisp is discussed. 1 Introduction Given a closed-form function G(x), a common computational task is to evaluate the function at a number of points in an interval. More precisely, given a starting point x0 and an increment h, the task is to compute G(x0 + ih) for i = 0; 1; : : : ; n \Gamma 1. Such computations occur frequently in practice: plotting curves of functions, computing finite sums and products, calculating integrals, and solving differential equations. Straightforward evaluations of G at all n points can be very inefficient and may sometimes even become the bottleneck of a given system. The SIG [5] graphing system is such an example. One way to speed up this type of evaluation is to compute the ..