Application of Vector-Valued Rational Approximations to the Matrix Eigenvalue Problem and Connections with Krylov Subspace Methods

Let F(z) be a vector-valued function F C CN, which is analytic at z 0 and meromorphic in a neighborhood of z 0, and let its Maclaurin series be given. In a recent work [J. Approx. Theory, 76 (1994), pp. 89-111] by the author, vector-valued rational approximation procedures for F(z) that are based on its Maclaurin series, were developed, and some of their convergence properties were analyzed in detail. In particular, a Koenig-type theorem concerning their poles and a de Montessus-type theorem concerning their uniform convergence in the complex plane were given. With the help of these theorems it was shown how optimal approximations to the poles of F(z) and the principal parts of the corresponding Laurent series expansions can be obtained. In this work we use these rational approximation procedures in conjunction with power iterations to develop bona fide generalizations of the power method for an arbitrary N N matrix that may or may not be diagonalizable. These generalizations can be used to obtain simultaneously several of the largest distinct eigenvalues and corresponding eigenvectors and other vectors in the invariant subspaces. We provide interesting constructions for both nondefective and defective eigenvalues and the corresponding invariant subspaces, and present a detailed convergence theory for them. 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 the reciprocals of some or all of the nonzero eigenvalues of the matrix being considered, while the coefficients in the principal parts of the Laurent expansions of this rational function are vectors in the corresponding invariant subspaces. In addition, it is shown that the generalized power methods of this work are equivalent to some Krylov subspace methods, among them the methods of Arnoldi and Lanczos. Thus, the theory of the present work provides a set of completely new results and constructions for these Krylov subspace methods. At the same time this theory suggests a new mode of usage for these Krylov subspace methods that has been observed to possess computational advantages over their common mode of usage in some cases. We illustrate some of the theory and conclusions derived from it with numerical examples.