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Difference between revisions of "Liénard-Chipart criterion"

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A modification of the [[Routh–Hurwitz criterion|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.
 
A modification of the [[Routh–Hurwitz criterion|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
 
Suppose one is given a polynomial
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$f(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n,\quad a_0>0;\tag{*}$$
  
let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587802.png" /> be its Hurwitz matrix (cf. [[Routh–Hurwitz criterion|Routh–Hurwitz criterion]]); let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587803.png" /> be its principal minor of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587805.png" />.
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let $H$ be its Hurwitz matrix (cf. [[Routh–Hurwitz criterion|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 (*) with real coefficients have negative real parts:
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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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587806.png" />;
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1) $a_n>0,a_{n-2}>0,\dots,\Delta_1>0,\Delta_3>0,\dots$;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587807.png" />;
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2) $a_n>0,a_{n-2}>0,\dots,\Delta_2>0,\Delta_4>0,\dots$;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587808.png" />;
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3) $a_n>0,a_{n-1}>0,a_{n-3}>0,\dots,\Delta_1>0,\Delta_3>0,\dots$;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058780/l0587809.png" />.
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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 [[#References|[1]]].
 
The criterion was established by A. Liénard and H. Chipart [[#References|[1]]].

Latest revision as of 18:28, 23 September 2014

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=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