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a b s t r a c tMatrix methods are increasingly popular for polynomial root-finding. The idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial. The algorithms also solve secular equation. QR algorithm is the most customary method for eigen-solving, but we explore the inverse Rayleigh quotient iteration instead, which turns out to be competitive with the most popular root-finders because of its excellence in exploiting matrix structure. To advance the iteration we preprocess the matrix and incorporate Newton's linearization, repeated squaring, homotopy continuation techniques, and some heuristics. The resulting algorithms accelerate the known numerical root-finders for univariate polynomial and secular equations, and are particularly well suited for the acceleration by using parallel processing. Furthermore, even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots.

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We specify some initial assumptions that guarantee rapid refinement of a rough initial approximation to the inverse of a Cauchy-like matrix, by means of our new modification of Newton's iteration, where the input, output, and all the auxiliary matrices are represented with their short generators defined by the associated scaling operators. The computations are performed fast since they are confined to operations with short generators of the given and computed matrices. Because of the known correlations among various structured matrices, the algorithm is immediately extended to rapid refinement of rough initial approximations to the inverses of Vandermonde-like, ChebyshevVandermonde-like, and Toeplitz-like matrices, where again the computations are confined to operations with short generators of the involved matrices.