Chapter 4 - Newton Interpolation and Difference Triangles

Abstract

The principal focus of this chapter is the divided difference, which provides the dual functionals for the Newton basis. The Newton basis allows us to use Homer's method for fast polynomial evaluation, and the divided difference generates the coefficients for the polynomial interpolant relative to the Newton basis. This chapter develops three approaches to the divided difference—computational: a recurrence based on difference quotients; theoretical: the highest-order coefficient of the polynomial interpolant; axiomatic: a system of four properties (symmetry, linearity, cancellation, and differentiation) that completely characterize the divided difference. This chapter employs these different approaches to derive properties, formulas, and identities for the divided difference. It also considers fast forward differencing, a technique for fast polynomial evaluation at equally spaced parameter values. Interpolation is a classical topic in approximation theory and numerical analysis, computer graphics and computer-aided design often deal with approximation as well as with interpolation. Many of the topics encountered in interpolation, including dynamic programming procedures, up and down recurrence, basis functions, dual functionals, divided differences, rational schemes, and tensor product, triangular, lofted, and Boolean sum surfaces are also relevant to approximation theory.