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We determine the pointwise error in Hermite interpolation by numericallysolving an appropriate differential equation, derived from the error termitself. We use this knowledge to approximate the error term by means of apolynomial, which is then added to the original Hermite polynomial to form amore accurate approximation. An example demonstrates that improvements inaccuracy are significant.

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