Error Bounds on Derivatives during Simulations

TL;DR

This paper proves the correctness and establishes error bounds for a real-time, past-dependent numerical differentiation algorithm that handles irregular data and higher derivatives, improving practical derivative estimation during simulations.

Contribution

It provides a rigorous proof of correctness and explicit error bounds for Bard's past-dependent differentiation algorithm, including formulas for coefficients and related corollaries.

Findings

Proved the correctness of Bard's differentiation algorithm.

Derived explicit error bounds for the method.

Presented formulas for coefficients used in the algorithm.

Abstract

The methods commonly used for numerical differentiation, such as the "center-difference formula" and "four-points formula" are unusable in simulations or real-time data analysis because they require knowledge of the future. In Bard'11, an algorithm was shown that generates formulas that require knowledge only of the past and present values of $f(t)$ to estimate $f'(t)$. Furthermore, the algorithm can handle irregularly spaced data and higher-order derivatives. That work did not include a rigorous proof of correctness nor the error bounds. In this paper, the correctness and error bounds of that algorithm are proven, explicit forms are given for the coefficients, and several interesting corollaries are proven.