Derivation of The Routh Array - Sturm's Theorem

Sturm's Theorem

Sturm gives us a method for evaluating . His theorem states as follows:

Given a sequence of polynomials where:

1) If then, and

2) for

and we define as the number of changes of sign in the sequence for a fixed value of, then:

A sequence satisfying these requirements is obtained using the Euclidean algorithm, which is as follows:

Starting with and, and denoting the remainder of by and similarly denoting the remainder of by, and so on, we obtain the relationships:

or in general

where the last non-zero remainder, 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 (26), through the use of the Euclidean algorithm above that the Routh matrix is formed.

We get

and identifying the coefficients of this remainder by, 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 by, ,

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 (19). An observation worthy of note is that in the regular case the polynomials and have as the highest common factor and thus there will be 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 will be the first term of each of these polynomials, and thus only these coefficients corresponding to the highest powers of in, and, which are, ... determine the signs of, ..., at .

So we get that is, is the number of changes of sign in the sequence, ... which is the number of sign changes in the sequence, ... and ; that is is the number of changes of sign in the sequence, ... which is the number of sign changes in the sequence, ...

Since our chain, ... will have members it is clear that since within if going from to a sign change has not occurred, within going from to one has, and likewise for all transitions (there will be no terms equal to zero) giving us total sign changes.

As and, and from (17), we have that and have derived Routh's theorem -

The number of roots of a real polynomial 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 then by which we have Routh's famous criterion:

In order for all the roots of the polynomial 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.