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 $\zeta(3)$

On computational properties of chains of recurrences

ezima @ scg.uwaterloo.ca Backward and mixed chains of recurrences are introduced. A complete set of chains of recurrences manipulation tools is described. Applications of these tools, related to the safety and numeric stability of chained computations are given. 1

Mixed representation of polynomials oriented towards fast parallel shift

In this paper we consider the form of polynomial represen-tation useful in problems connected with performing poly-nomial shift. We propose basic parallel algorithms suited for SIMD architecture to perform the shift in O(1) time if we have 0(ra2) Processor Elements available, and the shift haa to be performed repeatedly. Proposed algorithms are easy to generalize to multivariate polynomials shift. The possibility of applying these algorithms to polynomials with coefficients from non-commutative rings is discussed as well as the bit-wise complexity of the algorithm.

Waterloo Workshop on Computer Algebra

This book discusses the latest advances in algorithms for symbolic summation, factorization, symbolic-numeric linear algebra and linear functional equations. It presents a collection of papers on original research topics from the Waterloo Workshop on Computer Algebra (WWCA-2016), a satellite workshop of the International Symposium on Symbolic and Algebraic Computation (ISSAC’2016), which was held at Wilfrid Laurier University (Waterloo, Ontario, Canada) on July 23–24, 2016.   This workshop and the resulting book celebrate the 70th birthday of Sergei Abramov (Dorodnicyn Computing Centre of the Russian Academy of Sciences, Moscow), whose highly regarded and inspirational contributions to symbolic methods have become a crucial benchmark of computer algebra and have been broadly adopted by many Computer Algebra systems

On Accelerated Methods to Evaluate Sums of Products of Rational Numbers

In this paper we consider the problem of fast computation of sums of n-ary products of rational numbers, for large n. We present improvements to the standard binary splitting algorithm which are due to numerous factors, including changing the standard arbitrary precision integer representation to one that is more suitable for such computations, unrolling, and chains of recurrences techniques. For the computation of i(3) to 640000 decimal digits, we achieve a speedup factor of 2.65 over the standard binary splitting algorithm, which compares favorably to the ideal case in which the numerator and the denominator can be reduced by their greatest common divisor at no cost. If asymptotically fast multiplication is not available (as in the Java Development Kit), a speedup of an order of magnitude is easily obtained. Categories and Subject Descriptors I.1.1 [Symbolic and Algebraic Manipulation]: Expressions and Their Representation---Representations (general and polynomial) General Terms Representation of integers, binary splitting, chains of recurrences 1