Polynomial factorization algorithms for adaptive root estimation

The authors present two new polynomial factorization algorithms suitable for use in adaptive signal processing applications where it is required to track the movements of roots. Their distinguishing feature is that they provide methods for updating the roots optimally (and efficiently) in response to coefficient perturbations. This is useful, for example, for online estimation and tracking of time-varying roots. A Gauss-Newton type algorithm that requires approximately 2n² floating-point operations to update all roots of an nth-order polynomial is presented. An eigenvalue-based method that requires approximately 6n floating-point operations per individual root update is also proposed. Close to the true root vector, the rate of convergence of both algorithms is quadratic or faster. Unlike available direct pole estimation algorithms, the methods proposed here are not restricted to estimation of the roots of ARMA (autoregressive moving average)-like processes, but can be used in any situation where online estimates of polynomial coefficients are available