A Routh Zero-Location Testing Method for Integer Polynomials With Any Singularity Profile

Abstract

This article presents an integer-preserving algorithm to determine the zero location (ZL) with respect to the imaginary axis for any polynomial with Gaussian-integer coefficients. It extends the so-called optimal integer Routh stability test, previously limited to normal conditions, to integer polynomials with any singularity profile. The primary advantage of integer arithmetic lies in its inherent error-free quality. This characteristic is particularly crucial for the general ZL case because it involves workarounds for abnormal and singular cases that are triggered by correct identification of the value 0—an occurrence that can be obscured by rounding errors in floating-point arithmetic. Running the algorithm with integers, prevents the risk of false acceptance or rejection of these workarounds. Since, in practice, any noninteger polynomial can be scaled into an integer polynomial, the new algorithm paves the way for an error-free solution to the classical Routh–Hurwitz problem in its most general possible setting.