mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Hurwitz polynomial (named after German mathematician

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

) is a

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

whose

roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...

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

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

or on the

imaginary axis An imaginary number is the product of a real number and the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is ...

, 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 form ...

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

coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...

s that are positive

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

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. Th ...

). A polynomial function 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 algebraic g ...

is said to be Hurwitz if the following conditions are satisfied: # is real when is real. # The roots of have real parts which are zero or negative. Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of

stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...

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 abstractio ...

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 the control theory, control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stable polynomial, stability of a linear time-invariant system, linear time-invarian ...

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 function of a single variable is a function of the form :f(x)=ax^2+bx+c,\quad a \ne 0, where is its variable, and , , and are coefficients. The expression , especially when treated as an object in itself rather tha ...

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

quadratic formula In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions. Given a general quadr ...

: :

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 education science company that provides educational content, software, and services for students and educators across various levels—from K-12 to higher education and professional settings. They produce textbooks, ...

. {{DEFAULTSORT:Hurwitz Polynomial Polynomials