212 research outputs found
Computer algebra tools for Feynman integrals and related multi-sums
In perturbative calculations, e.g., in the setting of Quantum Chromodynamics
(QCD) one aims at the evaluation of Feynman integrals. Here one is often faced
with the problem to simplify multiple nested integrals or sums to expressions
in terms of indefinite nested integrals or sums. Furthermore, one seeks for
solutions of coupled systems of linear differential equations, that can be
represented in terms of indefinite nested sums (or integrals). In this article
we elaborate the main tools and the corresponding packages, that we have
developed and intensively used within the last 10 years in the course of our
QCD-calculations
Trusting Computations: a Mechanized Proof from Partial Differential Equations to Actual Program
Computer programs may go wrong due to exceptional behaviors, out-of-bound
array accesses, or simply coding errors. Thus, they cannot be blindly trusted.
Scientific computing programs make no exception in that respect, and even bring
specific accuracy issues due to their massive use of floating-point
computations. Yet, it is uncommon to guarantee their correctness. Indeed, we
had to extend existing methods and tools for proving the correct behavior of
programs to verify an existing numerical analysis program. This C program
implements the second-order centered finite difference explicit scheme for
solving the 1D wave equation. In fact, we have gone much further as we have
mechanically verified the convergence of the numerical scheme in order to get a
complete formal proof covering all aspects from partial differential equations
to actual numerical results. To the best of our knowledge, this is the first
time such a comprehensive proof is achieved.Comment: N° RR-8197 (2012). arXiv admin note: text overlap with
arXiv:1112.179
A Refined Difference Field Theory for Symbolic Summation
In this article we present a refined summation theory based on Karr's
difference field approach. The resulting algorithms find sum representations
with optimal nested depth. For instance, the algorithms have been applied
successively to evaluate Feynman integrals from Perturbative Quantum Field
Theory.Comment: Uses elseart.cls and yjsco.st
- …