Hurwitz polynomial
In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose roots (zeros) are located in the left half-plane of the complex plane or on the imaginary axis, that is, the real part of every root is zero or negative. Such a polynomial must have coefficients that are positive real numbers. The term is sometimes restricted to polynomials whose roots have real parts that are strictly negative, excluding the axis (i.e., a Hurwitz stable polynomial).
A polynomial function P(s) of a complex variable s is said to be Hurwitz if the following conditions are satisfied:
Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems. Whether a polynomial is Hurwitz can be determined by solving the equation to find the roots, or from the coefficients without solving the equation by the Routh–Hurwitz stability criterion.
A simple example of a Hurwitz polynomial is the following:
The only real solution is −1, as it factors to
In general, all second-degree polynomials with positive coefficients are Hurwitz. This follows directly from the quadratic formula:
where, if the determinant b^2-4ac is less than zero, then the polynomial will have two complex-conjugate solutions with real part -b/2a, which is negative for positive a and b. If it is equal to zero, there will be two coinciding real solutions at -b/2a. Finally, if the determinant is greater than zero, there will be two real negative solutions, because for positive a, b and c.
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