A Posteriori Error Estimation and Adaptive Taylor Series Methods for Nonlinear Function Approximation

The Taylor approximation theorem is a fundamental tool in numerical analysis, providing a local polynomial representation of smooth functions. In practical computations, a function f is approximated by a finite Taylor polynomial $P_n$, and controlling the resulting truncation error is of central importance. In this paper, we introduce two novel a posteriori error estimation techniques for Taylor polynomial approximations. The proposed estimators are fully computable and do not require prior bounds on the $(n+1)$st derivatives of f. We prove that the estimators converge to the exact error both pointwise and in the $L^2$-norm as $n\to \infty$, and we establish their asymptotic sharpness through effectivity analysis. Based on these results, we develop two adaptive algorithms that automatically determine the minimal degree n required to achieve a prescribed tolerance, either at a specific point or over a domain. We further extend the analysis to multivariate functions and show that analogous estimators and effectivity properties hold in higher dimensions. Numerical experiments are presented to validate the theoretical results and demonstrate the practical performance of the proposed methods.