# 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 | ||

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

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

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

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

− | 4) | + | 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 19:28, 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