Global monotone convergence of Newton-like iteration for a nonlinear eigen-problem

TL;DR

This paper proves that a Newton-like iterative method converges monotonically to the unique positive solution of a nonlinear eigen-problem involving an irreducible Stieltjes matrix, supported by numerical evidence.

Contribution

It establishes the global monotone convergence of a Newton-like method for a specific class of nonlinear eigen-problems, which was not previously known.

Findings

Convergence is monotone under certain conditions.

Numerical results confirm the theoretical convergence.

The method effectively finds the positive solution.

Abstract

The nonlinear eigen-problem $ Ax+F(x)=\lambda x$ is studied where $A$ is an $n\times n$ irreducible Stieltjes matrix. Under certain conditions, this problem has a unique positive solution. We show that, starting from a multiple of the positive eigenvector of $A$, the Newton-like iteration for this problem converges monotonically. Numerical results illustrate the effectiveness of this Newton-like method.