Research On Theory And Algorithms Of Nonlinear Eigenvalue Problem With Z-matrix

Nonlinear equations are widely arise in the fields of mathematics,physics,chemistry and biology.In recent years,some PDEs,stemming from practical problems of physics,chemistry,biology,engineering and economics,have been discretized into nonlinear eigenvalue problems with special matrix structures.With the development of science and technology,the numerical solution of nonlinear eigenvalue problem has very important theoretical significance and application value.Therefore,the study of nonlinear eigenvalue problem has become one of the hot topics of many scholars.In this paper,we study the theory and algorithms of nonlinear eigenvalue problem with Z-matrix structure.We mainly research the efficient algorithm of positive solution of nonlinear eigenvalue equation and its convergence theory.The details as follows:Firstly,considering the case of nonlinear eigenvalue equation with singular M-matrix,some conditions are proposed firstly such that the positive solution of the nonlinear eigenvalue equation exists and is unique.Then,the Newton iterative method is applied and improved to derive a new method called the Newton-SOR iterative method for solving this equation.Finally,the convergence results of the two methods are established.Secondly,the nonlinear eigenvalue equation with a Z-matrix but not an M-matrix is investigated to present a sufficient condition for the existence and uniqueness of the positive solution of the nonlinear eigenvalue equation.Under this condition,the Newton and Newton-SOR iterative methods are applied to this equation respectively,and the convergence of the two methods is given.Thirdly,numerical experiments show that the two methods are very effective,but the Newton-SOR iterative method needs fewer conditions of initial value than the Newton iterative method.