Taylor series expansion using matrices: An implementation in MATLAB®

Abstract

Taylor series expansions are widely used in engineering approximations, for instance, to develop finite differences schemes or numerical integration methods. This technical note presents a novel technique to generate, display and manipulate Taylor series expansion by using matrices. The resulting approach allows algebraic manipulation as well as differentiation in a very intuitive manner in order to experiment with different numerical schemes, their truncation errors and their structures, while avoiding manual calculation errors. A detailed explanation of the mathematical procedure to generate a matrix form of the Taylor series expansion for a function of two variables is presented along with the algorithm of an implementation in MATLAB®. Example cases of different orders are tabulated to illustrate the generation and manipulation capabilities of this technique. Additionally, an extended application is developed to determine the modified equations of finite difference schemes for partial differential equations, with one-dimensional examples of the wave equation and the heat equation using explicit and implicit schemes.