# Routh-Hurwitz criterion

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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)