A modification of the Routh–Hurwitz criterion, which reduces all calculations in it to the calculation of the principal minors of only even (or only odd) orders of a Hurwitz matrix.
Suppose one is given a polynomial
let be its Hurwitz matrix (cf. Routh–Hurwitz criterion); let be its principal minor of order , .
The Liénard–Chipart criterion: Any of the following four conditions is necessary and sufficient in order that all roots of a polynomial (*) with real coefficients have negative real parts:
The criterion was established by A. Liénard and H. Chipart .
|||A. Liénard, H. Chipart, "Sur la signe de la partie réelle des racines d'une équation algébrique" J. Math. Pures Appl. , 10 (1914) pp. 291–346|
|||F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)|
Liénard-Chipart criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Li%C3%A9nard-Chipart_criterion&oldid=23382