Determining the Rolle function in Hermite interpolatory approximation by solving an appropriate differential equation
For numerical analysts and practitioners using Hermite interpolation, this method offers a way to reduce approximation error, though it is incremental as it builds on existing error analysis.
The paper presents a method to improve Hermite interpolation accuracy by solving a differential equation derived from the error term, then adding a polynomial approximation of the error to the original interpolant. An example shows significant accuracy improvements.
We determine the pointwise error in Hermite interpolation by numerically solving an appropriate differential equation, derived from the error term itself. We use this knowledge to approximate the error term by means of a polynomial, which is then added to the original Hermite polynomial to form a more accurate approximation. An example demonstrates that improvements in accuracy are significant.