The resultant iteration for determining the stability of a polynomial

Abstract:

The square root iteration ${x_{i + 1}} = ({x_i} + x_i^{ - 1})/2$ is known to be convergent to $+ 1$ from starting values ${x_0}$ in the open right half-plane, and to $- 1$ from starting values in the open left half-plane. The resultant of the quadratic ${z^2} - 2wz + 1$ and a given polynomial $P(z)$, of proper degree n, is a polynomial $R(w)$ whose zeros, expressed in terms of the zeros ${z_i}$ of $P(z)$, are just ${w_i} = ({z_i} + z_i^{ - 1})/2$. The construction of $R(w)$ is thus equivalent to n independent applications of the square root iteration, with unknown starting values ${z_i}$. Repetition of this construction generates a sequence of resultants whose zeros are independently convergent to either $+ 1$ or $- 1$ according as the initial zero of $P(z)$ lies in the open right or left half-plane. When the given polynomial, and each of the resultants, is written as a linear combination of the $n + 1$ polynomials ${(z + 1)^n},{(z + 1)^{n - 1}}(z - 1),{(z + 1)^{n - 2}}{(z - 1)^2}, \ldots ,(z + 1){(z - 1)^{n - 1}},{(z - 1)^n}$. the sequence of resultants is ultimately convergent to one of the elements ${(z + 1)^{n - p}}{(z - 1)^p}$ of this basis, whence it follows that $P(z)$ has p zeros with positive real part and $n - p$ zeros with negative real part, and the Hurwitz stability problem for $P(z)$ is solved. By an application of the principle of argument it is, in general, possible to determine p at a finite stage of the resultant iteration. When formulated in terms of this basis, the resultant iteration becomes formally identical with the root-squaring process. This fact may be exploited to establish the properties of the resultant iteration, and to show that root-squaring may be applied to solve the Schur stability problem for a given polynomial. Although precise results are not available, numerical results and the general properties of this iteration suggest that, for the purpose at hand, it is unaffected by the progressive deterioration of condition that sometimes occurs in other applications of root-squaring.