Introduction to Polynomial Interpolation
Two types of problems frequently arise: (a) You have a function which is known exactly at a set of points and wish to develop an interpolating function. The interpolant may be exact at the set of points, or it may be a "best fit" in some sense. You might be concerned about having good representation of the first derivative as well as a good function. (b) You have experimental data that is to be fit with a mathematical model. The data has experimental error so that there is some uncertainty. You wish to derive the parameters in the model as well as the uncertainty in the determination of those parameters.
The first problem is addressed in this chapter, and the second is addressed elsewhere (link). The interpolation schemes are: global polynomials as powers of x that go through a fixed number of points; orthogonal polynomials of x that give a best fit; rational polynomials that are ratios of polynomials; piecewise polynomials derived with forward differences (points to the right) and backward differences (points to the left); splines; and finite elements.