We show how to derive error estimates between a function and its interpolating polynomial and between their corresponding derivatives. The derivation is based on a new definition of well-poisedness for the interpolation set, directly connecting the accuracy of the error estimates with the geometry of the points in the set. This definition is equivalent to the boundedness of Lagrange polynomials, but it provides new geometric intuition. Our approach extracts the error bounds for all of the derivatives using the same analysis; the error bound for the function values is then derived a posteriori. We also develop an algorithm to build a set of well-poised interpolation points or to modify an existing set to ensure its well-poisedness. We comment on the optimal geometries corresponding to the best possible well-poised sets in the case of linear interpolation.
Preprint 03-09, Department of Mathematics, University of Coimbra, Portugal, April 2003