Approximation theorem for the first eigenpair of single birth processes

Abstract:

      The explicit solution to the Poisson equation corresponding to the Q-matrix of a single birth process is obtained, thus the explicit inversion (if exists) is presented directly. As an application, inspired by the inverse power method, combining the explicit inversion with Collatz-Wielandt formula, a powerful approximation theorem for the maximal eigenpair corresponding to this kind of infinitesimal generator is presented. Different from the classical acceleration method using some fixed shift in the iteration, the shift in each iteration step is varying and the sequence formed by these shifts is strictly monotone and increases to the eigenvalue needed, which effectively reduces the number of iterations. Some examples are studied to illustrate the power of these results.Approximation theorem for the first eigenpair of single birth processes