# 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.^{[1]} 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).^{[2]}^{[3]}

A polynomial function *P*(*s*) of a complex variable *s* is said to be Hurwitz if the following conditions are satisfied:

- 1.
*P*(*s*) is real when*s*is real.

- 2. The roots of
*P*(*s*) have real parts which are zero or negative.

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

[edit]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 for positive *a*, *b* and *c*.

## Properties

[edit]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 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.

## References

[edit]**^**Kuo, Franklin F. (1966).*Network Analysis and Synthesis, 2nd Ed*. John Wiley & Sons. pp. 295–296. ISBN 0471511188.**^**Weisstein, Eric W (1999). "Hurwitz polynomial".*Wolfram Mathworld*. Wolfram Research. Retrieved July 3, 2013.**^**Reddy, Hari C. (2002). "Theory of two-dimensional Hurwitz polynomials".*The Circuits and Filters Handbook, 2nd Ed*. CRC Press. pp. 260–263. ISBN 1420041401. Retrieved July 3, 2013.

- Wayne H. Chen (1964)
*Linear Network Design and Synthesis*, page 63, McGraw Hill.