On computational applications of the Levi-Civita field

Abstract

Abstract In this paper, we study the computational applications of the Levi-Civita field whose elements are functions from the additive abelian group of rational numbers to the real numbers field, with left-finite support. After reviewing the algebraic and order structures of the Levi-Civita field, we introduce the Tulliotools library which implements the Levi-Civita field in the C++ programming language. We show that this software can replicate the results of (Shamseddine, 2015) by finding high order derivatives of certain functions faster than commercial software. We show how a similar method can be used to compute numerical sequences using generating functions and we compare this method with a number of conventional approaches. Finally, we show how the ability to quickly and accurately compute high order derivatives can be combined with Darboux’s formula to preform numerical integration. We compare the performance of this new approach to numerical integration with more conventional approaches as well as commercial software and show promising results with regards to both speed and accuracy.