The recursive companion matrix root tracker

A new sequential O(n²) polynomial factorization algorithm that updates all roots of an nth-order polynomial with real time-varying coefficients simultaneously and efficiently in response to coefficient perturbations is introduced. The algorithm is based on a variant of sequential orthogonal iteration and exploits the special structure of the coefficient companion matrix. All internal operations are based on real passive Givens plane rotations and real matrix-vector multiplications. The algorithm is unconditionally stable and requires no initial guess of the root values. Numerical examples are presented to demonstrate the performance of the algorithm. Comparisons are made to the Starer and Nehorai (1991) root tracking algorithm