Hurwitz polynomial

A Hurwitz polynomial (after Adolf Hurwitz ) is a real polynomial whose zeros all have a really negative real part .

Definition and necessary condition

A real polynomial (all ) ${\displaystyle a_{i}\in \mathbb {R} }$

${\displaystyle N(s)=\sum _{i=0}^{n}a_{i}s^{i}=a_{n}s^{n}+a_{n-1}s^{n-1}+...+a_{0}}$

is called a Hurwitz polynomial if:

${\displaystyle N(r_{i})=0\,\Rightarrow \,\mathrm {Re} \,r_{i}<0,\quad i=1,\ldots ,n.}$

In the case of a 1st or 2nd degree polynomial ( ) it can be shown that the coefficients of the normalized Hurwitz polynomial ( ) must be positive. Conversely, a normalized polynomial with real coefficients, in which a coefficient is less than or equal to zero, must have a zero that does not have a real negative real part. The condition that the coefficients are positive is therefore necessary and also sufficient. ${\displaystyle n\leq 2}$ ${\displaystyle a_{n}=1}$

For (a third or higher degree polynomial) a new sufficient and necessary condition is required: the Hurwitz determinant . ${\displaystyle n\geq 3}$

Hurwitz criterion

In the following we assume that the guide coefficient is positive. If this is not the case in the original polynomial, it can be achieved by multiplying the polynomial with . The zeros of the polynomial do not change. The determinant of the Hurwitz matrix , the so-called Hurwitz determinant , is first formed from the coefficients of the polynomial . The Hurwitz matrix is a matrix corresponding to the coefficients . (see below) ${\displaystyle a_{n}}$ ${\displaystyle -1}$ ${\displaystyle a_{0},\ldots ,a_{n}}$ ${\displaystyle a_{0},\ldots ,a_{n}}$ ${\displaystyle n\times n}$

${\displaystyle H={\begin{vmatrix}a_{n-1}&a_{n-3}&a_{n-5}&a_{n-7}&\ldots \\a_{n-0}&a_{n-2}&a_{n-4}&a_{n-6}&\ldots \\0&a_{n-1}&a_{n-3}&a_{n-5}&\ldots \\0&a_{n-0}&a_{n-2}&a_{n-4}&\ldots \\\ldots &\ldots &\ldots &\ldots &\ldots \\\end{vmatrix}}}$

Non-existent coefficients are therefore expressed by a zero. The polynomial is a Hurwitz polynomial if and only if all "northwestern sub-determinants" (also called main minors ) are positive. The matrix is ​​then positive definite .

In the example, the northwest sub-determinants for the case are : ${\displaystyle n=3}$

${\displaystyle {\begin{aligned}H_{1}&={\begin{vmatrix}a_{n-1}\end{vmatrix}}&&={\begin{vmatrix}a_{2}\end{vmatrix}}&&=a_{2}>0\\[2mm]H_{2}&={\begin{vmatrix}a_{n-1}&a_{n-3}\\a_{n-0}&a_{n-2}\\\end{vmatrix}}&&={\begin{vmatrix}a_{2}&a_{0}\\a_{3}&a_{1}\\\end{vmatrix}}&&=a_{2}a_{1}-a_{0}a_{3}>0\\[2mm]H_{3}&={\begin{vmatrix}a_{n-1}&a_{n-3}&a_{n-5}\\a_{n-0}&a_{n-2}&a_{n-4}\\0&a_{n-1}&a_{n-3}\\\end{vmatrix}}&&={\begin{vmatrix}a_{2}&a_{0}&0\\a_{3}&a_{1}&0\\0&a_{2}&a_{0}\\\end{vmatrix}}&&=a_{0}H_{2}>0\end{aligned}}}$

(Development after the 3rd column)

With our preliminary considerations on the necessary condition, the additional requirement arises . This is not fulfilled for. ${\displaystyle n=3}$ ${\displaystyle a_{2}a_{1}>a_{0}a_{3}}$ ${\displaystyle a_{0}=a_{1}=a_{2}=a_{3}=1}$

This procedure ("move" and "fill in") is repeated until a square matrix is ​​created. ${\displaystyle n\times n}$

Other definitions of the Hurwitz matrix can be found in the literature. The coefficients are often named differently. Hurwitz himself included the polynomial in his publication . ${\displaystyle a_{0}x^{n}+a_{1}x^{n-1}+...+a_{n}}$

Another notation for the Hurwitz determinant is:

${\displaystyle H_{z}={\begin{vmatrix}a_{1}&a_{0}&0&0&0&...&0\\a_{3}&a_{2}&a_{1}&a_{0}&0&...&0\\a_{5}&a_{4}&a_{3}&a_{2}&a_{1}&...&0\\\vdots &&&&&&\vdots \\a_{2z-1}&a_{2z-2}&...&...&...&...&a_{z}\\\end{vmatrix}}}$

application

Hurwitz polynomials are used in systems theory to examine a time-continuous system for asymptotic stability : If the denominator of the system function is a Hurwitz polynomial, the system is asymptotically stable.

• See also : Root theorem of Vieta

literature

• Adolf Hurwitz: Conditions under which an equation only has roots with negative real parts . In: Mathematische Annalen No. 46 , Leipzig 1895, pp. 273–285

Web links

• ETH website "Hurwitz Memorial Lecture Series"