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Some sufficient conditions are proposed in this paper such that the nonlinear eigenvalue problem with an irreducible singular M-matrix has a unique positive eigenvector. Under these conditions, the Newton-SOR iterative method is proposed for numerically solving such a positive eigenvector and some convergence results on this iterative method are established for the nonlinear eigenvalue problems with an irreducible singular M-matrix, a nonsingular M-matrix, and a general M-matrix, respectively. Finally, a numerical example is given to illustrate that the Newton-SOR iterative method is superior to the Newton iterative method

Positive eigenvector of nonlinear eigenvalue problem with a singular M-matrix and Newton-SOR iterative solution