Determining the Rolle function in Lagrange interpolatory approximation
J. S. C. Prentice
TL;DR
This work addresses how to determine the Rolle function that governs the error in Lagrange interpolation by formulating and solving an appropriate initial‑value problem. By computing the Rolle function $\varphi(x)$, one can express the interpolation error as $\Delta(x;P_n)=\frac{f^{(n+1)}(\xi(x))}{(n+1)!}Q_n(x)$ and, in principle, reconstruct the error or its leading term. The authors demonstrate numerical procedures to obtain $\varphi(x)$ and provide examples showing how knowledge of $\varphi(x)$ enables a polynomial correction to $P_n(x)$, dramatically improving accuracy (e.g., from $P_1(x)$ to an augmented polynomial $P_8^{\Delta}(x)$). They also discuss the notion of implied Rolle numbers at interpolation nodes and outline a practical device for improving Lagrange interpolation using the corrected remainder term. The approach promises broader applicability, including extensions to Hermite interpolation.
Abstract
We determine the Rolle function in Lagrange polynomial approximation using a suitable differential equation. We then propose a device for improving the Lagrange approximation by exploiting our knowledge of the Rolle function.