Let F(z) be a vectorvalued function, F:\mbox{\boldmath C} \rightarrow
\mbox{\boldmath 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 vectorvalued 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 Pad\'{e} 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 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.
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 forthcoming paper 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.
