An elementary proof of the classic Routh method for counting the number of left half-plane and right half-plane zeros of a real coefficient polynomial Pn(s) of degree n is given. Such a proof refers to the polynomials Pi(s) of degree i⩽n formed from the entries of the rows of order i and i-1 of the relevant Routh array. In particular, it is based on the consideration of an auxiliary polynomial Pi(s; q), linearly dependent on a real parameter q, which reduces to either polynomial Pi(s) or to polynomial Pi-1(s) for particular values of q. In this way, it is easy to show that i-1 zeroes of Pi(s) lie in the same half-plane as the zeros of Pi(s), and the remaining zero lies in the left or in the right half-plane according to the sign of the ratio of the leading coefficients of Pi(s) and Pi-1(s). By successively applying this property to all pairs of polynomials in the sequence, starting from Po(s) and P1(s), the standard rule for determining the zero distribution of Pn(s) is immediately derived
A simple proof of the Routh test
VIARO, Umberto
1999-01-01
Abstract
An elementary proof of the classic Routh method for counting the number of left half-plane and right half-plane zeros of a real coefficient polynomial Pn(s) of degree n is given. Such a proof refers to the polynomials Pi(s) of degree i⩽n formed from the entries of the rows of order i and i-1 of the relevant Routh array. In particular, it is based on the consideration of an auxiliary polynomial Pi(s; q), linearly dependent on a real parameter q, which reduces to either polynomial Pi(s) or to polynomial Pi-1(s) for particular values of q. In this way, it is easy to show that i-1 zeroes of Pi(s) lie in the same half-plane as the zeros of Pi(s), and the remaining zero lies in the left or in the right half-plane according to the sign of the ratio of the leading coefficients of Pi(s) and Pi-1(s). By successively applying this property to all pairs of polynomials in the sequence, starting from Po(s) and P1(s), the standard rule for determining the zero distribution of Pn(s) is immediately derivedI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
