Determining the Rolle function in Hermite interpolatory approximation by solving an appropriate differential equation

Technical Summary: Determining the Rolle Function in Hermite Interpolatory Approximation

Problem Statement
The paper addresses the challenge of improving the accuracy of Hermite interpolatory approximations. While the standard Hermite polynomial $H_{2n+1}(x)$ interpolates a function $f(x)$ and its derivative $f'(x)$ at $n+1$ nodes, the pointwise error $\Delta(x | H_{2n+1})$ is governed by a term involving the $(2n+2)$-th derivative of $f$ evaluated at an unknown point $\xi(x)$, known as the Rolle function. The core problem is that $\xi(x)$ is generally unknown, preventing the direct calculation or correction of the error term. The author seeks a method to determine $\xi(x)$ numerically to facilitate a more accurate approximation of $f(x)$.

Methodology
The proposed methodology extends a technique previously applied to Lagrange interpolation to the Hermite case. The approach involves the following steps:

1. Derivation of a Differential Equation: Starting from the standard error formula for Hermite interpolation, the author differentiates the error term with respect to $x$. By applying the chain rule and isolating the derivative of the Rolle function, $\frac{d\xi}{dx}$, a first-order ordinary differential equation (ODE) is derived. This ODE relates $\frac{d\xi}{dx}$ to the known functions $f(x)$, $H_{2n+1}(x)$, and their derivatives, as well as the unknown $\xi(x)$.
2. Initial Value Problem Formulation: To solve the ODE, an initial value $\xi_z = \xi(x_z)$ is required at a point $x_z$ close to (but not at) an interpolation node. Since the error is zero at the nodes themselves, $\xi$ cannot be determined there. The author selects a point $x_z$ near a node and solves the algebraic error equation numerically to find a valid initial $\xi_z$.
3. Numerical Solution: The derived ODE is solved numerically using a seventh-order Runge-Kutta (RK) method. The paper notes that multiple mathematical solutions for the initial value may exist; the correct one is selected based on the constraint that $\xi(x)$ must lie strictly within the interval defined by the interpolation nodes ($x_0 < \xi(x) < x_n$).
4. Error Correction: Once $\xi(x)$ is determined, the term $f^{(2n+2)}(\xi(x))$ in the error formula becomes known. The author approximates this term using a polynomial $H_\xi(x)$ (via least-squares fitting or cubic splines). This polynomial is multiplied by the nodal factor $\prod (x-x_k)^2$ and divided by $(2n+2)!$ to form an error polynomial $E(x)$.
5. Refined Approximation: The final, improved approximation is constructed by adding the error polynomial to the original Hermite polynomial: $f(x) \approx H_{2n+1}(x) + E(x)$.

Key Results
The methodology was tested on the function $f(x) = e^x \sin x$ over the nodes $\{0, 3\pi/2\}$ (where $n=1$).

• Rolle Function Determination: The numerical solution successfully generated a Rolle function $\xi(x)$ that satisfied the interval constraints. The numerical solution of the ODE was verified by comparing the left-hand and right-hand sides of the error equation, showing them to be essentially indistinguishable.
• Accuracy Improvement:
  • The original Hermite polynomial $H_3(x)$ exhibited a maximum approximation error of 7.04.
  • By adding an error polynomial $E(x)$ derived from a least-squares fit of degree 11, the maximum error was reduced to $7.3 \times 10^{-8}$.
  • Using a cubic spline to generate the approximation of the derivative term further reduced the maximum error to approximately $10^{-12}$.
• Preservation of Interpolatory Properties: The paper verifies that the corrected approximation $H_{2n+1}(x) + E(x)$ retains the original Hermite interpolatory properties, matching both the function values and derivative values at the nodes.
• Numerical Integration: The paper briefly demonstrates that integrating the corrected polynomial over the interval yields a significantly more accurate result ($\sim 3 \times 10^{-12}$ error) compared to integrating the original Hermite polynomial ($\sim 0.7$ error).

Significance and Claims
The paper claims that determining the Rolle function via an initial-value problem allows for the explicit approximation of the pointwise error in Hermite interpolation. The primary significance lies in the ability to construct a new polynomial approximation that is orders of magnitude more accurate than the standard Hermite polynomial, while strictly maintaining the original interpolation conditions.

The author notes that the accuracy of the final result is currently limited by the precision of the RK method used to solve for $\xi(x)$, suggesting that future work could focus on error control within the RK solver. The paper concludes that this technique, building on previous work in Lagrange interpolation, demonstrates substantial potential for improving numerical approximation quality.