Abstract. We relate polynomial computations with operations involving infinite band Toeplitz matrices and show applications to the numerical solution of Markov chains, of nonlinear matrix equations, to spectral factorizations and to the solution of finite Toeplitz systems. In particular two matrix versions of Graeffe's iteration are introduced and their convergence properties are analyzed. Correlations between Graeffe's iteration for matrix polynomials and cyclic reduction for block Toeplitz matrices are pointed out. The paper contains a systematic treatment of known topics and presentation of new results, improvements and extensions.

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