Error Bounds on Derivatives during Simulations
For researchers and engineers needing real-time derivative estimation in simulations or data analysis, this work provides rigorous theoretical guarantees for a previously unproven algorithm.
This paper proves the correctness and error bounds of an algorithm for numerical differentiation that uses only past and present data, enabling real-time derivative estimation. Explicit coefficient forms and corollaries are provided.
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.