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 imaginary 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.
Examples.
A simple example of a Hurwitz polynomial is:
The only real solution is −1, because it factors as
In general, all quadratic polynomials with positive coefficients are Hurwitz.
This follows directly from the quadratic formula:
where, if the discriminant "b"2−4"ac" is less than zero, then the polynomial will have two complex-conjugate solutions with real part −"b"/2"a", which is negative for positive "a" and "b".
If the discriminant is equal to zero, there will be two coinciding real solutions at −"b"/2"a". Finally, if the discriminant is greater than zero, there will be two real negative solutions,
because formula_4 for positive "a", "b" and "c".
Properties.
For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive (except for quadratic polynomials, which also doesn't imply sufficiency). A necessary and sufficient condition that a polynomial is Hurwitz is that it passes the Routh–Hurwitz stability criterion. A given polynomial can be efficiently tested to be Hurwitz or not by using the Routh continued fraction expansion technique.