Some Discussion For Singular Problems In Newton’s Method

Studying numerical methods for solving systems of nonlinear equations is animportant research field in nonlinear problems. Newton’s method is the corealgorithm in solving systems of nonlinear equations. However, it will have losen itseffectiveness when the Jacobian matrix of the function is singular in solution points orin iterative processes.In this paper, the singular problems in Newton’s method are discussed andstudied using tools of matrix splitting and Moore-Penrose inverse. Firstly, on the onehand the alternating iterative method is used to construct Newton-alternating iterativemethod and its corresponding relaxation iterative method, and their convergence isproved. On the other hand, the P-regular splittings are used to construct P-regularsplitting-alternating iterative method, and its semiconvergence is proved. What’s more,P-regular parallel multisplittings iterative method, nonnegative parallel multisplittingsiterative method and P-regular splitting-alternating iterative method are used forsolving singular problems by combining them with parallel mutisplittings method.The process of the algorithm and a relevant numerical example are given.Secondly, Moore-Penrose inverse is used to construct Newton’ method which isfor solving a class of singular nonlinear systems of equations, and its convergence isdiscussed, which includes the local convergence criteria, semilocal convergencecriteria and the radius of convergence ball is also obtained. The numerical exampleindicates the effectiveness of the algorithm.Lastly, inexact methods of Newton’s method constructed with Moore-Penroseinverse are given. First, inexact Gauss-Newton method and inexactLevenberg-Marquardt method are deduced by taking an approximate solution of theleast squares solution of Newton equations. Second, the inexact method is constructedby taking an approximate matrix of Moore-Penrose inverse of Jacobian matrix, and itsconvergence is proved. Third, the inexact method is constructed by using localinformation instead of the whole information of the Jacobian matrix, and theconvergence is proved. The numerical example also indicates its superiority in solvinglarge system of equations.