Routh-Hurwitz criterion

Hurwitz criterion

A necessary and sufficient condition for all the roots of a polynomial

$$f(x)=a_0x^n+a_1x^{n-1}+\dotsb+a_n,$$

with real coefficients and $a_0>0$, to have negative real parts. It consists of the following: All principal minors $\Delta_i$, $i=1,\dotsc,n$, of the Hurwitz matrix $H$ are positive (cf. Minor). Here $H$ is the matrix of order $n$ whose $i$-th row has the form

$$a_{2-i},a_{4-i},\dotsc,a_{2n-i},$$

where, by definition, $a_k=0$ if $k<0$ or $k>n$ (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 $f(x)$ 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.