A multiprecision algorithm for the solution of polynomials and polynomial eigenvalue problems

Many applications of the real world are modelled by matrix polynomials [EQUATION] where A_i are m x m matrices, see for instance [2], [6]. A computational task encountered in this framework is computing the eigenvalues of P(x), that is, the solutions of the polynomial equation det P(x) = 0. This task is generally accomplished by reducing P(x) to a linear pencil of the kind xL -- K for suitable matrices K, L of size mn, and to solving the eigenvalue problem (λL -- K)v = 0 by means of standard numerical algorithms. A wide literature exists on this approach, we refer the reader to [7] for an example.