Nullspace computation over rational function fields for symbolic summation

Abstract

Symbolic summation methods, either based on the WZ-Fasenmyer paradigm [2, 11, 10] or Karr’s algorithm [6], reduce the given summation problem to finding vectors in the nullspace of a matrix over a rational function field with one or more variables. Direct methods to find such vectors use Gaussian elimination with heuristic pivoting strategies, but, due to intermediate expression swell, can fail to produce results for even moderate sized inputs. In this poster we focus on the case of one variable, and consider two related approaches that use homomorphic imaging and Chinese remaindering to compute a nullspace of a matrix over Q(x). We report on a preliminary implementation of these methods in the open source computer mathematics system Sage [8] and comment on their performance on actual input matrices obtained from the Mathematica implementation of Wegshaider’s algorithm [10]. While conceptually simple, these approaches make essential use of many asymptotically fast methods for integers and polynomials, including not only multiplication, but also radix conversion, interpolation, rational function and number reconstruction, etc. The single problem of computing a nullspace over Q(x) thus provides a good test of, and should motivate the further development of, highly optimized libraries such as GMP [3], FFLAS [1], FLINT [4] and zn poly [5].