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.

在本文中，我们研究Levi-Civita字段的计算应用，该字段的元素是从有理数字的加性阿贝尔群到实数字段的函数，并带有左有限支持。在回顾了Levi-Civita字段的代数和阶结构之后，我们介绍了Tulliotools库，该库以C ++编程语言实现了Levi-Civita字段。我们证明，该软件可以比商业软件更快地找到某些功能的高阶导数，从而复制（Shamseddine，2015）的结果。我们展示了如何使用生成函数使用相似的方法来计算数字序列，并将此方法与许多常规方法进行比较。最后，我们展示了如何快速准确地计算高阶导数的能力可以与Darboux公式结合起来进行数值积分。我们将这种新的数值积分方法与更常规的方法以及商业软件的性能进行了比较，并在速度和准确性方面都显示出了令人鼓舞的结果。