In mathematics, a Hurwitz polynomial, named after

Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...

, is a

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

whose roots (zeros) are located in the left half-plane of the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...

or on the imaginary axis, that is, the

real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

of every root is zero or negative. Such a polynomial must have

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...

s that are positive

real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

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 In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either: * all its roots lie in the open left half-plane, or * all its roots lie in the open unit disk. The ...

). A polynomial function ''P''(''s'') of a

complex variable Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...

''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 system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstracti ...

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 In control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system. A stable system is one ...

Examples

A simple example of a Hurwitz polynomial is: :

x^2 + 2x + 1.

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

(x+1)^2.

In general, all

quadratic polynomial In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomia ...

s with positive coefficients are Hurwitz. This follows directly from the

quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, g ...

: :

x=\frac.

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

\sqrt < b

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

. A given polynomial can be efficiently tested to be Hurwitz or not by using the Routh continued fraction expansion technique.

References

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

McGraw Hill McGraw Hill is an American educational publishing company and one of the "big three" educational publishers that publishes educational content, software, and services for pre-K through postgraduate education. The company also publishes referen ...

. {{DEFAULTSORT:Hurwitz Polynomial Polynomials