Monotone convergence of Newton-like iteration for a structured nonlinear eigen-problem

Abstract

A structured eigen-problem Ax+F(x)=λx is studied in this paper, where in applications A∈Rn×n is an irreducible Stieltjes matrix. Under certain restrictions, this problem has a unique positive solution. We show that, starting from a multiple of the positive eigenvector of A, the Newton-like algorithm for this eigen-problem is well defined and converges monotonically. Numerical results illustrate the effectiveness of this Newton-like method.