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Liénard-Chipart criterion

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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

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

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

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

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

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

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

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

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)
How to Cite This Entry:
Liénard-Chipart criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Li%C3%A9nard-Chipart_criterion&oldid=33372
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article