An Error Term and Uniqueness for Hermite-Birkhoff Interpolation
Involving only Function Values and/or First Derivative Values
Walt Finden
This paper discusses various aspects of Hermite-Birkhoff interpolation
that involve prescibed values of a function and/or its first derivative.
An algorithm is given that finds the unique polynomial that satisfies the
given conditions if it exists. A mean value type error term is developed
which illustrates the ill-conditioning present when trying to find a solution
to a problem that is close to a problem that does not have a unique solution.
The interpolants we consider and the associated error
term may be useful in the development of continuous approximations for
ordinary differential equations that allow asymptotically correct
defect control. Expressions in the algorithm are also useful in
determining whether certain specific types of problems have unique solutions.
This is useful, for example, in strategies involving approximations to
solutions of boundary value problems by collocation.