Obviously, polynomials are useful functions for fitting data. When you were a kid you drew ``connect the dots'' pictures, but you didn't realize that you were fitting a data set with ``linear splines''. Now you might, or at least you can be so informed, and not terribly disturbed....
Polynomials are useful partly because of the
Weierstrass Approximation Theorem: Suppose that 
. For each
, there exists a polynomial P(x) such that
When using the secant method, we fitted a line to two points and used that to approximate a root. Müller's method involved fitting a quadratic polynomial to three points, and we profitably used our knowledge of linear algebra to do so. That's the sort of ``interpolation'' that we want to engage in throughout the present chapter.
While Taylor series polynomials are very valuable throughout numerical analysis, this is one time when we want to chuck them into the garbage. They're not particularly useful for interpolating functions, because they focus their information at a point, and we're interested in fitting points over space: we want to distribute error, not focus it at a point.
Generally, we eschew the use of high degree polynomials for the interpolation of data, for several reasons:
- They tend to wiggle more than we like;
- Their derivatives (while easy to calculate) also wiggle alot;
- small errors in coefficients can result in dramatic changes in the polynomial.
-
Lagrange interpolating polynomial: the unique polynomial of
degree n that interpolates (that is, passes through) the n+1 data points
.
There is a very interesting way of deriving the Lagrange interpolating
polynomial (linear algebra aside!). Write the degree polynomial as a
sum of degree polynomials, as follows:
Theorem 3.2: Let 
be n+1 distinct numbers, and f
is a function defined at those numbers, having values
. Then there is a unique polynomial of degree n passing
through those points, and it is given by
where
