Technical Report CS0664

Revised and Enlarged version: Let \ $F(z)$\ be a vector-valued function, \ $F:C \rightarrow C^{N}$, which is analytic at \mbox{$z = 0$}\ and meromorphic in a neighbourhood of \mbox{$z = 0$}, and let its Maclaurin series be given. In this work we develop vector-valued rational approximation procedures for \mbox{$F(z)$}\ 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 Pad\'{e} 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. We exploit these developments 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 obtain simultaneously several of the largest distinct 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. Extensions of the procedures above and the accompanying theoretical results to functions defined in arbitrary linear spaces is also considered.