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

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