Adaptive algorithms for tracking roots of spectral polynomials

The authors propose two fast adaptive algorithms, namely Newton's gradient algorithm and the modified Rayleigh-quotient adaptive algorithm. These methods work in association with adaptive eigensubspace algorithms for tracking the zeros of a nonstationary spectrum polynomial. Newton's gradient algorithm is developed under a linearly constrained minimization procedure, whereas the modified Rayleigh-quotient adaptive technique is derived from the Rayleigh-quotient calculating procedure for the eigenstructure of the companion matrix of the spectrum polynomial. For an Nth-order polynomial, the adaptive algorithm has requires computational complexity O(N). The adaptive algorithms operate independently for each zero and have better tracking and computational complexity than the direct rooting method or the zero-sensitive adaptive algorithm.