Hurwitz Criterion

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Hurwitz Criterion


a criterion that makes it possible to determine when all the roots of the polynomial

p(z) ‗ aozn + a1zn-1 + . . . + an-1z + an

have negative real parts. For example, for the polynomials with real coefficients a0 > 0, a1, . . . ,an, the Hurwitz criterion states the following: in order that all the roots of the polynomial p(z) have negative real parts, it is necessary and sufficient that for all k = 1,2, ... , n the following inequality is valid:

This criterion was discovered by the German mathematician A. Hurwitz in 1895. It is used primarily to determine the stability of the solutions of a system of differential equations with constant coefficients.


Kurosh, A. G. Kurs vysshei algebry, 9th ed. Moscow-Leningrad. 1968.
Chetaev. N. G. Ustoichivost’ dvizheniia. Moscow-Leningrad, 1946.
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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