One of the oldest problems in mathematics—and, at the same time, one of the most applied—is the problem of constructing an approximation to a given function f from among simpler functions, typically (but not always) polynomials. A slight variation of this problem is that of constructing a smooth function from a discrete set of data points.
In this chapter we will study both of these problems and develop several methods for solving them. We start with a more general treatment of an idea we first saw in Chapter 2.
The basic interpolation problem can be posed in one of two ways:
Note that in the first case we are trying to fit a polynomial to the data, and in the second we are trying to approximate a given function with the interpolating polynomial.
While the two cases are in fact different, we can always consider the first one to be a special case of the second (by taking yi = f(xi), for each i), so we will present most ...