Example
Suppose we wish to find the number of roots in some range for the polynomial . So and . Using polynomial long division to divide by gives the remainder, and upon multiplying this remainder by −1 we obtain . Next dividing by and multiplying the remainder by −1, we obtain . And dividing by and multiplying the remainder by −1, we obtain . This completes the chain of Sturm polynomials.
To find the number of roots between −∞ and ∞, first evaluate and at −∞ and note the sequence of signs of the results: + − + + −, which contains 3 sign changes (+ to −, then − to +, then + to −). The same procedure for +∞ gives the sign sequence + + + − −, which contains just 1 sign change. Hence the number of roots of the original polynomial between −∞ and ∞ is 3 − 1 = 2. That this is correct can be seen by noting that can be factored as, where it is readily verifiable that has the two roots −1 and 1 while has no real roots. In more complicated examples in which there is no advance knowledge of the roots because factoring is either impossible or impractical, one can experiment with various finite bounds for the range to be considered, thus narrowing down the locations of the roots.
Read more about this topic: Sturm's Theorem
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