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Difference between revisions of "Routh-Hurwitz criterion"

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''Hurwitz criterion''
 
''Hurwitz criterion''
  
 
A necessary and sufficient condition for all the roots of a polynomial
 
A necessary and sufficient condition for all the roots of 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/r/r082/r082760/r0827601.png" /></td> </tr></table>
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$$f(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n,$$
  
with real coefficients and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082760/r0827602.png" />, to have negative real parts. It consists of the following: All principal minors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082760/r0827603.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082760/r0827604.png" />, of the Hurwitz matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082760/r0827605.png" /> are positive (cf. [[Minor|Minor]]). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082760/r0827606.png" /> is the matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082760/r0827607.png" /> whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082760/r0827608.png" />-th row has the form
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with real coefficients and $a_0>0$, to have negative real parts. It consists of the following: All principal minors $\Delta_i$, $i=1,\dots,n$, of the Hurwitz matrix $H$ are positive (cf. [[Minor|Minor]]). Here $H$ is the matrix of order $n$ whose $i$-th row has the form
  
<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/r/r082/r082760/r0827609.png" /></td> </tr></table>
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$$a_{2-i},a_{4-i},\dots,a_{2n-i},$$
  
where, by definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082760/r08276010.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082760/r08276011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082760/r08276012.png" /> (the Hurwitz condition or the Routh–Hurwitz condition). This criterion was obtained by A. Hurwitz [[#References|[1]]] and is a generalization of the work of E.J. Routh (see [[Routh theorem|Routh theorem]]).
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where, by definition, $a_k=0$ if $k<0$ or $k>n$ (the Hurwitz condition or the Routh–Hurwitz condition). This criterion was obtained by A. Hurwitz [[#References|[1]]] and is a generalization of the work of E.J. Routh (see [[Routh theorem|Routh theorem]]).
  
A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082760/r08276013.png" /> satisfying the Hurwitz condition is called a Hurwitz polynomial, or, in applications of the Routh–Hurwitz criterion in the stability theory of oscillating systems, a stable polynomial. There are other criteria for the stability of polynomials, such as the Routh criterion, the [[Liénard–Chipart criterion|Liénard–Chipart criterion]], and methods for determining the number of real roots of a polynomial are also known.
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A polynomial $f(x)$ satisfying the Hurwitz condition is called a Hurwitz polynomial, or, in applications of the Routh–Hurwitz criterion in the stability theory of oscillating systems, a stable polynomial. There are other criteria for the stability of polynomials, such as the Routh criterion, the [[Liénard–Chipart criterion|Liénard–Chipart criterion]], and methods for determining the number of real roots of a polynomial are also known.
  
 
====References====
 
====References====

Latest revision as of 18:25, 23 September 2014

Hurwitz criterion

A necessary and sufficient condition for all the roots of a polynomial

$$f(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n,$$

with real coefficients and $a_0>0$, to have negative real parts. It consists of the following: All principal minors $\Delta_i$, $i=1,\dots,n$, of the Hurwitz matrix $H$ are positive (cf. Minor). Here $H$ is the matrix of order $n$ whose $i$-th row has the form

$$a_{2-i},a_{4-i},\dots,a_{2n-i},$$

where, by definition, $a_k=0$ if $k<0$ or $k>n$ (the Hurwitz condition or the Routh–Hurwitz condition). This criterion was obtained by A. Hurwitz [1] and is a generalization of the work of E.J. Routh (see Routh theorem).

A polynomial $f(x)$ satisfying the Hurwitz condition is called a Hurwitz polynomial, or, in applications of the Routh–Hurwitz criterion in the stability theory of oscillating systems, a stable polynomial. There are other criteria for the stability of polynomials, such as the Routh criterion, the Liénard–Chipart criterion, and methods for determining the number of real roots of a polynomial are also known.

References

[1] A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt" Math. Ann. , 46 (1895) pp. 273–284
[2] F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)


Comments

See also Routh theorem.

How to Cite This Entry:
Routh-Hurwitz criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Routh-Hurwitz_criterion&oldid=33371
This article was adapted from an original article by E.N. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article