11453

Sturm's Theorem for Polynomials

Let be the number of real roots of an algebraic equation with real coefficients whose real roots are simple over an interval and are not or . Then , the difference between the number of sign changes of the Sturm chain evaluated at and at .
By subdividing an interval until every subinterval contains at most one root, one can locate subintervals containing all the real roots in the original interval.

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The Sturm chain of a polynomial is the sequence of polynomials:
,
where
Here and are the polynomial quotient and remainder of . The chain ends when the polynomial is a constant.
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