About: Sturm's theorem

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In mathematics, the Sturm sequence of a univariate polynomial p is a sequence of polynomials associated with 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 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. The Sturm sequence and Sturm's theorem are named after Jacques Charles François Sturm, who discovered the theorem in 1829.

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rdfs:label	Sturm's theorem (en)
rdfs:comment	In mathematics, the Sturm sequence of a univariate polynomial p is a sequence of polynomials associated with 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 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. The Sturm sequence and Sturm's theorem are named after Jacques Charles François Sturm, who discovered the theorem in 1829. (en)
sameAs	Sturm's theorem Sturm's theorem
dbp:wikiPageUsesTemplate	dbt:= dbt:Cbignore dbt:Citation dbt:Cite_book dbt:Cite_journal dbt:Dead_link dbt:Div_col dbt:Div_col_end dbt:Math dbt:Mvar dbt:Reflist dbt:Short_description dbt:Slink dbt:Space dbt:Void
Subject	Articles containing proofs Computer algebra Real algebraic geometry Theorems about polynomials Theorems in real analysis
gold:hypernym	Sequence
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Sturm's_theorem?oldid=1070543027&ns=0
Wikipage page ID	1221919 (xsd:integer)
page length (characters) of wiki page	18245 (xsd:nonNegativeInteger)
Wikipage revision ID	1070543027 (xsd:integer)
has abstract	In mathematics, the Sturm sequence of a univariate polynomial p is a sequence of polynomials associated with 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 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, it does not provide a procedure for calculating them. Sturm's theorem counts the number of distinct real roots and locates them in intervals. By subdividing the intervals containing some roots, it can isolate the roots into arbitrarily small intervals, each containing exactly one root. This yields the oldest real-root isolation algorithm, and arbitrary-precision root-finding algorithm for univariate polynomials. For computing over the reals, Sturm's theorem is less efficient than other methods based on Descartes' rule of signs. However, it works on every real closed field, and, therefore, remains fundamental for the theoretical study of the computational complexity of decidability and quantifier elimination in the first order theory of real numbers. The Sturm sequence and Sturm's theorem are named after Jacques Charles François Sturm, who discovered the theorem in 1829. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Sturm's_theorem
is Wikipage redirect of	Sturm's Theorem Sturm Chain Sturm chain Sturm sequence Sturm theorem Sturmian Sequence
is Link from a Wikipage to another Wikipage of	List of theorems Anna Johnson Pell Wheeler Bisection method Budan's theorem Carl Wilhelm Borchardt Jacques Charles François Sturm James Lockhart (banker) Joseph Fourier Christoffel–Darboux formula Hopf bifurcation List of polynomial topics Edward Routh Derivation of the Routh array Internal rate of return Laguerre polynomials Euclidean algorithm
is known for of	Jacques Charles François Sturm

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