Liénard-Chipart criterion

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

$$f(x)=a_0x^n+a_1x^{n-1}+\dotsb+a_n,\quad a_0>0;\label{*}\tag{*}$$

let $H$ be its Hurwitz matrix (cf. Routh–Hurwitz criterion); let $\Delta_i$ be its principal minor of order $i$, $i=1,\dotsc,n$.

The Liénard–Chipart criterion: Any of the following four conditions is necessary and sufficient in order that all roots of a polynomial \eqref{*} with real coefficients have negative real parts:

1) $a_n>0,a_{n-2}>0,\dotsc,\Delta_1>0,\Delta_3>0,\dotsc$;

2) $a_n>0,a_{n-2}>0,\dotsc,\Delta_2>0,\Delta_4>0,\dotsc$;

3) $a_n>0,a_{n-1}>0,a_{n-3}>0,\dotsc,\Delta_1>0,\Delta_3>0,\dotsc$;

4) $a_n>0,a_{n-1}>0,a_{n-3}>0,\dotsc,\Delta_2>0,\Delta_1>0,\dotsc$.

The criterion was established by A. Liénard and H. Chipart [1].

References

[1]	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
[2]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)