Difference between revisions of "RouthHurwitz criterion"
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Revision as of 19:54, 24 March 2012
Hurwitz criterion
A necessary and sufficient condition for all the roots of a polynomial
with real coefficients and , to have negative real parts. It consists of the following: All principal minors , , of the Hurwitz matrix are positive (cf. Minor). Here is the matrix of order whose th row has the form
where, by definition, if or (the Hurwitz condition or the Routh–Hurwitz condition). This criterion was obtained by A. Hurwitz [1] and is a generalization of the work of E.J. Routh (see Routh theorem).
A polynomial satisfying the Hurwitz condition is called a Hurwitz polynomial, or, in applications of the Routh–Hurwitz criterion in the stability theory of oscillating systems, a stable polynomial. There are other criteria for the stability of polynomials, such as the Routh criterion, the Liénard–Chipart criterion, and methods for determining the number of real roots of a polynomial are also known.
References
[1]  A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt" Math. Ann. , 46 (1895) pp. 273–284 
[2]  F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian) 
Comments
See also Routh theorem.
RouthHurwitz criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=RouthHurwitz_criterion&oldid=13913