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Summary. We introduce a class C n of n × n structured matrices which includes three well-known classes of generalized companion matrices: tridiagonal plus rank-one matrices (comrade matrices), diagonal plus rank-one matrices and arrowhead matrices. Relying on the structure properties of C n , we show that if A ∈ C n then A = RQ ∈ C n , where A = QR is the QR decomposition of A. This allows one to implement the QR iteration for computing the eigenvalues and the eigenvectors of any A ∈ C n with O(n) arithmetic operations per iteration and with O(n) memory storage. This iteration, applied to generalized companion matrices, provides new O(n 2 ) flops algorithms for computing polynomial zeros and for solving the associated (rational) secular equations. Numerical experiments confirm the effectiveness and the robustness of our approach.

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 extension

In this paper we address the problem of efficiently computing all the eigenvalues of a large N × N Hermitian matrix modified by a possibly non Hermitian perturbation of low rank. Previously proposed fast adaptations of the QR algorithm are considerably simplified by performing a preliminary transformation of the matrix by similarity into an upper Hessenberg form. The transformed matrix can be specified by a small set of parameters which are easily updated during the QR process. The resulting structured QR iteration can be carried out in linear time using linear memory storage. Moreover, it is proved to be backward stable. Numerical experiments show that the novel algorithm outperforms available implementations of the Hessenberg QR algorithm already for small values of N.

Several computational and structural properties of Bezoutian matrices expressed with respect to the Bernstein polynomial basis are shown. The exploitation of such properties allows the design of fast algorithms for the solution of Bernstein-Bezoutian linear systems without never making use of potentially ill-conditioned reductions to the monomial basis. In particular, we devise an algorithm for the computation of the greatest common divisor (GCD) of two polynomials in Bernstein form. A series of numerical tests are reported and discussed, which indicate that Bernstein-Bezoutian matrices are much less sensitive to perturbations of the coe cients of the input polynomials compared to other commonly used resultant matrices generated after having performed the explicit conversion between the Bernstein and the power basis.