Adaptive Root Estimation Algorithms

Abstract

This paper considers adaptive factorization of polynomials with real, slowly time-varying coefficients. A new Gauss-Newton type algorithm is presented, analyzed and compared to existing algorithms for adaptive root estimation. The method is based on the ideas behind the algorithm of Starer and Nehorai (1989) for adaptive root estimation, but uses a different polynomial factorization. The polynomial is factored into second order polynomial terms instead of single order root terms. In the analysis of the algorithms it is shown that the factorization into second order terms is the key to the new algorithms ability to track roots freely over the complex plane, for example when the number of complex conjugate roots and real roots are unknown. The algorithm presented is also shown to be four times faster than the one of Starer and Nehorai and requires approximately 3.5 n 2 flops to update the roots of an n th order polynomial.