Further Discussion On The Singular Problem Of Newton's Method

Newton's method is a classical method for solving nonlinear equations.Due to its advantages of fast convergence speed and self-adjustment,it attracts many scholars to constantly improve its research,many effective algorithms are obtained by many scholars on the basis of this.The singular problem of Newton's method is one of the important problems of Newton's method research.In this paper,based on the semi-convergence theory of singular linear equations,the second kind of singular problems appearing in the iterative process of Newton's method are studied.Firstly,the simple iterative method and the relaxed iterative method for the singular problems of Newton's method are constructed respectively by the semi-convergence theory of simple iteration method and extrapolation iteration method of the singular linear systems,and the corresponding semi-convergence theorem and its proof are given.Secondly,the block relaxation iterative method(Newton-BSOR)for the singular problem of Newton's method and the singular problem of p-cycle is established severally,which is based on the theory of partitioned matrix and the semi-convergence theorem of singular p-cyclic linear equations,and the corresponding semi-convergence theorem and its proof are given.Thirdly,the full rank decomposition algorithm and the full rank decomposition relaxation iteration algorithm for the singular problems of Newton's method are constructed,respectively,by using the full rank decomposition theory of singular matrix,and the corresponding semi-convergence theorem is given.At the end of the paper,the algorithm flow of Newton's singular problem for solving nonlinear equations is given for various semi-convergence algorithms of Newton's method singularity problem.The numerical examples are given to illustrate the rationality of the algorithm.