Hurwitz polynomial
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A Hurwitz polynomial 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). These polynomials are named after Adolf Hurwitz.
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.
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.
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