In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose coefficients are positive real numbers 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 Hurwitz stable polynomial).
[ Examples
A simple example of a Hurwitz polynomial is the following:
- x2 + 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 thee continued fraction expansion technique.