Chapter 2 - Lagrange Interpolation and Neville's Algorithm

Abstract

This chapter is designed to study the Lagrange Interpolation and Neville's Algorithm. It encounters majority of the central ideas of this discipline: existence and uniqueness theorems; dynamic programming procedures; pyramid algorithms; up and down recurrences; basis functions; blends of overlapping data; rational schemes; tensor product, triangular, lofted, and Boolean sum surfaces; along with the use of barycentric coordinates to represent points in the domain of triangular surface patches. This chapter investigates methods for generating polynomial curves and surfaces to go through a finite collection of points in affine space. It begins with schemes for curves and later extend these techniques to surfaces. One core tenet of approximation theory and numerical analysis is that all polynomial bases are not equal. This chapter illustrates that the Lagrange basis, and not the standard monomial basis, is most suited both for point interpolation and for polynomial multiplication.