Rational Approximations from Power Series of Vector-Valued Meromorphic Functions

Abstract

Let F(z) be a vector-valued function, F: C → CN, which is analytic at z = 0 and meromorphic in a neighbourhood of z = 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 Padé approximants. In particular, we prove a Koenig-type theorem concerning their poles and a de Montessus-type theorem concerning their uniform convergence. We show how "optimal" approximations to multiple poles and to Laurent expansions about these poles can be constructed. Extensions of the procedures above and the accompanying theoretical results to functions defined in arbitrary linear spaces is also considered. One of the most interesting and immediate applications of the results of this work is to the matrix eigenvalue problem. In a new work we exploit the developments of the present work to devise bona fide generalizations of the classical power method that are especially suitable for very large and sparse matrices. These generalizations can be used to approximate simultaneously several of the largest distinct eigenvalues and corresponding eigenvectors and invariant subspaces of arbitrary matrices, which may or may not be diagonalizable, and are very closely related with known Krylov subspace methods.