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# INTERPOLATION AND APPROXIMATION

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.

# 4.1 LAGRANGE INTERPOLATION

The basic interpolation problem can be posed in one of two ways:

1. Given a set of nodes {xi, 0 ≤ in} and corresponding data values {yi, 0 ≤ in}, find the polynomial pn(x) of degree less than or equal to n, such that

2. Given a set of nodes {xi, 0 ≤ in} and a continuous function f(x), find the polynomial pn(x) of degree less than or equal to n, such that

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 ...

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