Complexity estimates for two uncoupling algorithms

Abstract

Uncoupling algorithms transform a linear differential system of first order into one or several scalar differential equations. We examine two approaches to uncoupling: the cyclic-vector method (CVM) and the Danilevski-Barkatou-Zürcher algorithm (DBZ). We give tight size bounds on the scalar equations produced by CVM, and design a fast variant of CVM whose complexity is quasi-optimal with respect to the output size. We exhibit a strong structural link between CVM and DBZ enabling to show that, in the generic case, DBZ has polynomial complexity and that it produces a single equation, strongly related to the output of CVM. We prove that algorithm CVM is faster than DBZ by almost two orders of magnitude, and provide experimental results that validate the theoretical complexity analyses.

Large Scale Analytic Calculations in Quantum Field Theories

“…Sometimes it is also useful to consider, in addition, the related system of linear difference equations. One may decouple these systems using the algorithms implemented in the packages [28,29], as e.g. Zürcher's algorithm [30].…”

Refined Holonomic Summation Algorithms in Particle Physics

“…Lemma 2.1 specializes to one linear recurrence (and does not treat a system in the multivariate sequence case). However, it dispenses the user to work with Gröbner bases and expensive uncoupling procedures [70,32] that are needed in the standard approaches [34,44]. In particular, the constraints (14) and (16) have been worked out explicitly which will be the basis for further explorations.…”

“…More precisely, we utilized and refined the Sigma-approach that has been developed in [54,57] to unite Karr's Π Σ -field setting with the holonomic system approach: one can solve the parameterized telescoping problem in terms of elements from a Π Σ -field together with summation objects which are solutions of inhomogeneous linear difference equations. In particular, a refined tactic has been worked out for the well-known holonomic approach [34] that finds recurrences without Gröbner basis computations or expensive uncoupling algorithms [70,32]. This efficient and flexible approach has been applied to derive the first alternative proof [20] of Stembridge's TSPP theorem [64].…”

Computer algebra tools for Feynman integrals and related multi-sums

“…1. Uncouple the system by using, e.g., Zürcher's algorithm [29,71] implemented in the package OreSys [35]. Usually 8 one gets one scalar linear differential equation in one of the unknown functions, say f 1 (x, ε) where the right-hand side can be given in terms of a power series whose coefficients are given explicitly in terms of indefinite nested sums.…”