Difference between revisions of "LiénardChipart criterion"
From Encyclopedia of Mathematics
Ulf Rehmann (talk  contribs) m (moved Liénard–Chipart criterion to LienardChipart criterion: ascii title) 
Ulf Rehmann (talk  contribs) m (moved LienardChipart criterion to LiénardChipart criterion: accented title) 
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Revision as of 09:54, 26 March 2012
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:
1) ;
2) ;
3) ;
4) .
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:
LiénardChipart criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Li%C3%A9nardChipart_criterion&oldid=23382
LiénardChipart criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Li%C3%A9nardChipart_criterion&oldid=23382
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics  ISBN 1402006098. See original article