Evaluation, analysis, and the state-of-applications of Taylor expansion

Abstract

Contemporarily, Taylor expansion is a pretty powerful mathematical tool widely used in various fields for approximating complex functions with simpler polynomial approximations. It has applications in fields inducing physics, engineering, and computer science, ranging from numerical methods for solving differential equations to optimization algorithms and machine learning scenarios. Despite its extensively use, there are still open questions and challenges related to its application, including accuracy, convergence, computational cost, and numerical stability, etc. This study will provide a comprehensive summary as well as analysis of the existing literature on the application of Taylor expansion, identifies the current state-of-the-art, and proposes future research directions. The paper aims to address the limitations and challenges of Taylor expansion, improve its accuracy and efficiency, and explore new applications. The findings of this research can benefit researchers, practitioners, and engineers interested in utilizing Taylor expansion in their fields. Overall, these results contribute to the advancement of the field of Taylor expansion-based approximations.