Finding All the Roots : Sturm ’ s Theorem

In our last lecture, we studied two root-finding methods that each took in a polynomial f(x) and an interval [a, b], and returned a root of that function on that interval. This was great for the problem we asked at the start of the class — how to find a root of a quintic polynomial — but is not necessarily so great for many other problems we may want to study. For example, we may want to find a root of a degree-6 polynomial! In this situation, we can’t obviously use either of the methods we’ve examined earlier: we have no obvious way to find an interval on which an even-order polynomial changes sign. Also, we may often want to find not just one root, but all of the roots of a given polynomial! Our earlier methods only give us one root, which is not necessarily very useful to us in practice. In this lecture we’re going to study Sturm’s theorem, a tool that helps with both of these problems.