Hurwitz Criterion

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

p(z) ‗ a_ozⁿ + a₁z^n-1 + . . . + a_n-1z + a_n

have negative real parts. For example, for the polynomials with real coefficients a₀ > 0, a₁, . . . ,a_n, 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.

REFERENCES

Kurosh, A. G. Kurs vysshei algebry, 9th ed. Moscow-Leningrad. 1968.
Chetaev. N. G. Ustoichivost’ dvizheniia. Moscow-Leningrad, 1946.

单词	hurwitz criterion
释义	Hurwitz Criterion Hurwitz Criterion a criterion that makes it possible to determine when all the roots of the polynomial p(z) ‗ a_ozⁿ + a₁z^n-1 + . . . + a_n-1z + a_n have negative real parts. For example, for the polynomials with real coefficients a₀ > 0, a₁, . . . ,a_n, 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. REFERENCES Kurosh, A. G. Kurs vysshei algebry, 9th ed. Moscow-Leningrad. 1968. Chetaev. N. G. Ustoichivost’ dvizheniia. Moscow-Leningrad, 1946.
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