Algorithms and Merit Functions for the Principal Eigenvalue

Abstract

This paper describes an efficient algorithm for finding the principal eigenvalue and a corresponding positive eigenvector of a positive matrix. It is based on the use of a merit function for the problem. A separately quasi-convex function defined on the positive unit simplex cross the positive numbers is described whose minimizers yield the eigenvalue and a corresponding normalized eigenvector. The algorithm provides explicit formulae for descent directions at each stage which yield strict descent. A relative error estimate for the principal eigenvalue is described and it is used in the stopping condition for the algorithm.