Interpolation

Abstract

The chapter begins with polynomial interpolation in which we seek a polynomial of specified degree that agrees with given data. The first approach is to use our knowledge of solving linear systems of equations to find the Lagrange interpolation polynomial by solving the Vandermonde system for the coefficients. However that is both inefficient and because of ill-conditioning subject to computational error. The use of the Lagrange basis polynomials is equivalent to transforming that system to a diagonal form, and is a more practical approach. Difference schemes allow both a more efficient use of the data, including adding new data points. Part of the significance of difference representations lies in the ability to recenter the data so that local data assumes greater importance relative to more distant data points. It also allows us to halt computation (effectively reducing the degree of the polynomial) so that the interpolation can be local. The final topic for this chapter is spline interpolation. Here the basic idea is to use low degree polynomials which connect as smoothly as possible as we move through the data. The example we focus on is cubic spline interpolation where the resulting function retains continuity in its slope and curvature at each data point. As always, practical problem-solving and implementation in Python are addressed.