Space-efficient evaluation of hypergeometric series

• Authors: Howard Cheng;Barry Gergel;Ethan Kim;Eugene Zima
• Venue: ACM SIGSAM Bulletin
• Year: 2005

We consider the evaluation of the truncated hypergeometricseries[EQUATION]to high precision, where a, b, p, andq are polynomials with integer coefficients, anda(n),b(n),p(n),q(n) have bit lengthO(log n). We also assume thatthe series is linearly convergent, so that thenth term of (1) isO(c-n)with c > 1. These series are commonlyused in the high precision evaluation of elementary functions andother constants, including the exponential function, logarithms,trigonometric functions, and constants such as theApéry's constant ζ(3) [9, 10]."Binary splitting" is an approach that has been independentlydiscovered and used by many authors in the computation of (1) [2,3, 4, 5, 8, 10, 12]. Binary splitting computes the numerator anddenominator of the rational numberS(N). The decimalrepresentation of S(N) isthen computed by fixed-point division of the numerator by thedenominator. The binary splitting approach takes advantage of thespecial form of the series (1) to obtain a denominator that isrelatively small (of size O(Nlog N)). It also takes advantage of fast integermultiplication to obtain a time complexity ofO((logN)2M(N)),where M(N) =O(N log Nlog log N) is the complexity of integermultiplication of two N-bit integers [16]. Thespace complexity of the algorithm isO(N logN), the size of the computed numerator anddenominator.Typically, the numerator and denominator computed by binarysplitting have large common factors. For example, in thecomputation of 640000 digits of ζ(3), as much as 86% ofthe size of the computed numerator and denominator can beattributed to their common factor [7]. Empirically, we haveobserved that the size of the reduced numerator and denominator isO(N) instead ofO(N log N)as computed by binary splitting. The additional digits computed notonly slow down the final division but also require more memory tobe used during the computation. For computing a large number ofdecimal digits, either the computation cannot be done at all orsome data would have to be swapped out of memory, increasing thecomputation time dramatically.In this poster, we study the application of well-knowntechniques in computer algebra to the evaluation of (1). If a boundon the size of the reduced numerator anddenominator is known, we can compute the image ofS(N) in (1) under anappropriately chosen modulus. Fast rational number reconstructioncan then be applied to recover the reduced numerator anddenominator [13, 14, 15, 17]. We show how to apply our techniquesto the computation of ζ(3), including the prediction ofthe size of the reduced numerator and denominator. In particular,we obtain the desired O(N)bound on the size of reduced numerator and denominator, which is aninteresting result by itself. The techniques used in the analysiscan be applied to similar hypergeometric series.