Rational Approximations from Power Series of Vector-valued Meromorphic Functions with an Application to Matrix Eigenvalue Problems

Let F(z) be a vector-valued function, F : e _ eN, which is analytic at z= 0 and meromorphic .. in a neighbourhood ofz = 0, and let its Maclaurin series be given. In this work we develop vector­ valued rational approximation procedures for F(z) by applying vector extrapolation methods to the sequence of partial sums of its Maclaurin series. We analyze some of the algebraic and analytic properties of the rational approximations thus obtained, and show that they are akin to Pade approximants. In particular, we prove a Koenig type theorem concerning their poles and a de Montessus type thorem concerning their uniform convergence. We show how optimal approximations to multiple poles and to Laurent exp.ansions about these poles can be constructed. We exploit these developments to devise bona fide generalizations of the classical power method. These generalizations can be used to obtain simultaneously several of the largest eigenvalues and corresponding eigenvectors of ~rbitrary matrices which mayor may not be diagonalizable. We provide interesting constructions for both simple and defective eigenvalues and their corresponding eigenvectors, along with their complete convergence theory. This is made possible by the observation that vectors obtained by power iterations with a matrix are actually coefficients of the Maclaurin series of a vector-valued rational function whose poles are reciprocals of eigenvalues of the matrix being considered.