# 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 | ||

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

− | with real coefficients and | + | with real coefficients and $a_0>0$, to have negative real parts. It consists of the following: All principal minors $\Delta_i$, $i=1,\dotsc,n$, 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 |

− | + | $$a_{2-i},a_{4-i},\dotsc,a_{2n-i},$$ | |

− | where, by definition, | + | 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 | + | 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 12:37, 14 February 2020

*Hurwitz criterion*

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

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

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

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

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://encyclopediaofmath.org/index.php?title=Routh-Hurwitz_criterion&oldid=22997