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Sturm's Theorem for Polynomials

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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 .

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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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Contributed by: Izidor Hafner (January 2017)
Open content licensed under CC BY-NC-SA

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The Sturm chain of a polynomial is the sequence of polynomials:

,

where

p2(x)= q1(x)p1(x)-p0(x),&IndentingNewLine;p3(x)=q2(x)p2(x)-p1(x),&IndentingNewLine;…&IndentingNewLine;ps(x)=qs-1(x)ps-1(x)-ps-2(x).

Here and are the polynomial quotient and remainder of . The chain ends when the polynomial is a constant.

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