Sturm's Theorem

Sturm's Theorem

In mathematics, the Sturm's sequence of a polynomial p is a sequence of polynomials associated to p and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of p located in an interval in terms of the number of changes of signs of the values of the Sturm's sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of p.

Whereas the fundamental theorem of algebra readily yields the overall number of complex roots, counted with multiplicity, Sturm's theorem yields the number of distinct real roots and locates them in intervals. By subdividing the intervals containing some roots, it allows eventually to isolate the roots in arbitrary small intervals each containing exactly one root. This yields a symbolic root finding algorithm, that is available in most computer algebra systems, although some more efficient methods are now usually preferred (see below).

Sturm's sequence and Sturm's theorems are named after Jacques Charles François Sturm.

Read more about Sturm's TheoremSturm Chains, Statement, Example, Proof, History Section and Other Related Methods, Generalized Sturm Chains

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