The Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial. Central to the field of control systems design, the Routh–Hurwitz theorem and Routh array emerge by using the Euclidean algorithm and Sturm's theorem in evaluating Cauchy indices.

The Cauchy index

Given the system: Assuming no roots of f(x) = 0 lie on the imaginary axis, and letting then we have Expressing f(x) in polar form, we have where and from (2) note that where Now if the ith root of f(x) = 0 has a positive real part, then ( using the notation y=(RE[y],IM[y]) ) and and Similarly, if the ith root of f(x)=0 has a negative real part, and and From (9) to (11) we find that when the ith root of f(x) has a positive real part, and from (12) to (14) we find that when the ith root of f(x) has a negative real part. Thus, So, if we define then we have the relationship and combining (3) and (17) gives us Therefore, given an equation of f(x) of degree n we need only evaluate this function \Delta to determine N, the number of roots with negative real parts and P, the number of roots with positive real parts. In accordance with (6) and Figure 1, the graph of vs \theta, varying x over an interval (a,b) where and are integer multiples of \pi, this variation causing the function \theta(x) to have increased by \pi, indicates that in the course of travelling from point a to point b, \theta has "jumped" from +\infty to -\infty one more time than it has jumped from -\infty to +\infty. Similarly, if we vary x over an interval (a,b) this variation causing \theta(x) to have decreased by \pi, where again \theta is a multiple of \pi at both x = ja and x = jb, implies that has jumped from -\infty to +\infty one more time than it has jumped from +\infty to -\infty as x was varied over the said interval. Thus, is \pi times the difference between the number of points at which jumps from -\infty to +\infty and the number of points at which jumps from +\infty to -\infty as x ranges over the interval provided that at, is defined. In the case where the starting point is on an incongruity (i.e., i = 0, 1, 2, ...) the ending point will be on an incongruity as well, by equation (17) (since N is an integer and P is an integer, \Delta will be an integer). In this case, we can achieve this same index (difference in positive and negative jumps) by shifting the axes of the tangent function by \pi/2, through adding \pi/2 to \theta. Thus, our index is now fully defined for any combination of coefficients in f(x) by evaluating over the interval (a,b) = when our starting (and thus ending) point is not an incongruity, and by evaluating over said interval when our starting point is at an incongruity. This difference, \Delta, of negative and positive jumping incongruities encountered while traversing x from -j\infty to +j\infty is called the Cauchy Index of the tangent of the phase angle, the phase angle being \theta(x) or \theta'(x), depending as \theta_a is an integer multiple of \pi or not.

The Routh criterion

To derive Routh's criterion, first we'll use a different notation to differentiate between the even and odd terms of f(x): Now we have: Therefore, if n is even, and if n is odd: Now observe that if n is an odd integer, then by (3) N+P is odd. If N+P is an odd integer, then N-P is odd as well. Similarly, this same argument shows that when n is even, N-P will be even. Equation (15) shows that if N-P is even, \theta is an integer multiple of \pi. Therefore, is defined for n even, and is thus the proper index to use when n is even, and similarly is defined for n odd, making it the proper index in this latter case. Thus, from (6) and (23), for n even: and from (19) and (24), for n odd: Lo and behold we are evaluating the same Cauchy index for both:

Sturm's theorem

Sturm gives us a method for evaluating. His theorem states as follows: Given a sequence of polynomials where:

1. If f_k(x) = 0 then, , and
2. for and we define V(x) as the number of changes of sign in the sequence for a fixed value of x, then: A sequence satisfying these requirements is obtained using the Euclidean algorithm, which is as follows: Starting with f_1(x) and f_2(x), and denoting the remainder of by f_3(x) and similarly denoting the remainder of by f_4(x), and so on, we obtain the relationships: or in general where the last non-zero remainder, f_m(x) will therefore be the highest common factor of. It can be observed that the sequence so constructed will satisfy the conditions of Sturm's theorem, and thus an algorithm for determining the stated index has been developed. It is in applying Sturm's theorem (28) to (29), through the use of the Euclidean algorithm above that the Routh matrix is formed. We get and identifying the coefficients of this remainder by c_0, -c_1, c_2, -c_3, and so forth, makes our formed remainder where Continuing with the Euclidean algorithm on these new coefficients gives us where we again denote the coefficients of the remainder f_4(\omega) by d_0, -d_1, d_2, -d_3, making our formed remainder and giving us The rows of the Routh array are determined exactly by this algorithm when applied to the coefficients of (20). An observation worthy of note is that in the regular case the polynomials f_1(\omega) and f_2(\omega) have as the highest common factor and thus there will be n polynomials in the chain. Note now, that in determining the signs of the members of the sequence of polynomials that at the dominating power of \omega will be the first term of each of these polynomials, and thus only these coefficients corresponding to the highest powers of \omega in, and f_m(x), which are a_0, b_0, c_0, d_0, ... determine the signs of f_1(x), f_2(x), ..., f_m(x) at. So we get that is, V(+\infty) is the number of changes of sign in the sequence a_0\infty^n,, , ... which is the number of sign changes in the sequence a_0, b_0, c_0, d_0, ... and ; that is V(-\infty) is the number of changes of sign in the sequence , , , ... which is the number of sign changes in the sequence a_0, -b_0, c_0, -d_0, ... Since our chain a_0, b_0, c_0, d_0, ... will have n members it is clear that since within if going from a_0 to b_0 a sign change has not occurred, within going from a_0 to -b_0 one has, and likewise for all n transitions (there will be no terms equal to zero) giving us n total sign changes. As and, and from (18) , we have that and have derived Routh's theorem - The number of roots of a real polynomial f(z) which lie in the right half plane is equal to the number of changes of sign in the first column of the Routh scheme. And for the stable case where P = 0 then by which we have Routh's famous criterion: In order for all the roots of the polynomial f(z) to have negative real parts, it is necessary and sufficient that all of the elements in the first column of the Routh scheme be different from zero and of the same sign.