4 research outputs found
Refined Holonomic Summation Algorithms in Particle Physics
An improved multi-summation approach is introduced and discussed that enables
one to simultaneously handle indefinite nested sums and products in the setting
of difference rings and holonomic sequences. Relevant mathematics is reviewed
and the underlying advanced difference ring machinery is elaborated upon. The
flexibility of this new toolbox contributed substantially to evaluating
complicated multi-sums coming from particle physics. Illustrative examples of
the functionality of the new software package RhoSum are given.Comment: Modified Proposition 2.1 and Corollary 2.
Hypergeometric summation algorithms for high order finite elements
High order finite elements are usually defined by means of certain orthogonal polynomials. The performance of iterative solution methods depends on the condition number of the system matrix, which itself depends on the chosen basis functions. The goal is now to design basis functions minimizing the condition number, and which can be computed efficiently. In this paper we demonstrate the application of recently developed computer algebra algorithms for hypergeometric summation to derive cheap recurrence relations allowing a simple implementation for fast basis function evaluation