- Hurwitz polynomial
In

mathematics , a**Hurwitz polynomial**, named afterAdolf Hurwitz , is apolynomial whose coefficients are positivereal number s and whose zeros are located in the left half-plane of the complex plane, that is, the real part of every zero is negative. One sometimes uses the term**Hurwitz polynomial**simply as a (real or complex) polynomial with all zeros in the left-half plane (i.e., a Hurwitzstable polynomial ).**Examples**A simple example of a Hurwitz polynomial is the following:

:$x^2\; +\; 2x\; +\; 1.$

The only real solution is −1, as it factors to:

:$(x+1)^2.$

**Properties**For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive. For all of a polynomial's roots to lie in the left half-plane, it is necessary and sufficient that the polynomial in question pass the

Routh-Hurwitz stability criterion .A given polynomial can be tested to be Hurwitz or not by using the continued fraction expansion technique.

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