Editing Hurwitz polynomial

{{short description|Polynomial whose complex roots have non-positive real parts}}

In [[mathematics]], a '''Hurwitz polynomial''' (named after German mathematician  [[Adolf Hurwitz]]) is a [[polynomial]]  whose [[root of a function|roots]] (zeros) are located in the left half-plane of the [[complex plane]] or on the [[imaginary axis]], that is, the [[complex number|real part]] of every root is zero or negative.<ref name="Kuo">{{cite book   
  | last = Kuo
  | first = Franklin F.
  | title = Network Analysis and Synthesis, 2nd Ed.
  | publisher = John Wiley & Sons
  | date = 1966
  | pages = 295–296
  | isbn = 0471511188}}</ref>  Such a polynomial must have [[coefficient]]s that are positive [[real number]]s. 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]]).<ref name=" Weisstein">{{cite web
  | last =  Weisstein
  | first = Eric W
  | title = Hurwitz polynomial
  | work = Wolfram Mathworld
  | publisher = Wolfram Research
  | date = 1999
  | url = http://mathworld.wolfram.com/HurwitzPolynomial.html
  | access-date = July 3, 2013}}</ref><ref name="Reddy">{{cite conference
  | first = Hari C.
  | last = Reddy
  | title = Theory of two-dimensional Hurwitz polynomials
  | book-title = The Circuits and Filters Handbook, 2nd Ed.
  | pages = 260–263
  | publisher = CRC Press
  | date = 2002
  | url = https://books.google.com/books?id=SmDImt1zHXkC&dq=hurwitz+polynomial&pg=PA262
  | isbn = 1420041401
  | access-date = July 3, 2013}}</ref>

A polynomial function {{math|''P''(''s'')}} of a [[complex variable]] {{mvar|s}} is said to be Hurwitz if the following conditions are satisfied:

# {{math|''P''(''s'')}} is real when {{mvar|s}} is real.
# The roots of {{math|''P''(''s'')}} have real parts which are zero or negative.

Hurwitz polynomials are important in [[control system|control systems theory]], because they represent the [[Characteristic polynomial#Characteristic equation|characteristic equations]] of [[Stability theory|stable]] [[linear system]]s.  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:

:<math>x^2 + 2x + 1.</math>

The only real solution is −1, because it factors as

:<math>(x+1)^2.</math>

In general, all [[quadratic polynomial]]s with positive coefficients are Hurwitz.
This follows directly from the [[quadratic formula]]:
:<math>x=\frac{-b\pm\sqrt{b^2-4ac\ }}{2a}.</math>
where, if the discriminant ''b''<sup>2</sup>−4''ac'' is less than zero, then the polynomial will have two [[complex conjugate|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 <math>\sqrt{b^2-4ac} < b</math> 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 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 ==
{{reflist}}
* Wayne H. Chen (1964) ''Linear Network Design and Synthesis'', page 63, [[McGraw Hill]].

{{DEFAULTSORT:Hurwitz Polynomial}}
[[Category:Polynomials]]