In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Routh–Hurwitz theorem gives a test to determine whether a given polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 is Hurwitz-stable

Stable polynomial

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 first condition defines Hurwitz stability and the second one Schur stability. Stable polynomials arise in various mathematical fields, for example in...

. It was proved

Derivation of the Routh array

The Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial...

 in 1895 and named after Edward John Routh and Adolf Hurwitz

Adolf Hurwitz

Adolf Hurwitz was a German mathematician.-Early life:He was born to a Jewish family in Hildesheim, former Kingdom of Hannover, now Lower Saxony, Germany, and died in Zürich, in Switzerland. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed...

.

Notations

Let f(z) be a polynomial (with complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 coefficients) of degree n with no roots on the imaginary line

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

 (i.e. the line Z=ic where i is the imaginary unit

Imaginary unit

In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

 and c is a real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

). Let us define (a polynomial of degree n) and (a nonzero polynomial of degree strictly less than n) by , respectively the real and imaginary parts of f on the imaginary line.

Furthermore, let us denote by:

• p the number of roots of f in the left half-plane (taking into account multiplicities);
• q the number of roots of f in the right half-plane (taking into account multiplicities);
• the variation of the argument of f(iy) when y runs from -∞ to +∞;
• w(x) is the number of variations of the generalized Sturm chain obtained from and by applying the Euclidean algorithm;
• is the Cauchy index
  Cauchy index
  In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an interval. By the Routh-Hurwitz theorem, we have the following interpretation: the Cauchy index of...
  
   of the rational function
  Rational function
  In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
  
   r over the real line.

Statement

With the notations introduced above, the Routh–Hurwitz theorem states that:

From the first equality we can for instance conclude that when the variation of the argument of f(iy) is positive, then f(z) will have more roots to the left of the imaginary axis than to its right.
The equality p-q=w(+∞)-w(-∞) can be viewed as the complex counterpart of Sturm's theorem

Sturm's theorem

In mathematics, Sturm's theorem is a symbolic procedure to determine the number of distinct real roots of a polynomial. It was named for Jacques Charles François Sturm...

. Note the differences: in Sturm's theorem, the left member is p+q and the w from the right member is the number of variations of a Sturm chain (while w refers to a generalized Sturm chain in the present theorem).

Routh–Hurwitz stability criterion

We can easily determine a stability criterion using this theorem as it is trivial that f(z) is Hurwitz-stable

Stable polynomial

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 first condition defines Hurwitz stability and the second one Schur stability. Stable polynomials arise in various mathematical fields, for example in...

 iff

IFF

IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...

p − q = n. We thus obtain conditions on the coefficients of f(z) by imposing w(+∞) = n and w(−∞) = 0.