The Study Of Modified Newton Method For Nonlinear Equations

In this paper, we investigate the behavior of the modified Newton method for nonlinear equations.Nowadays, the science of nonlinearity is an important research direction in the development of science. The nonlinearity exists widely in the material world and the social life. It relates to many fields including the natural science, engineering and economics, and so on. However, the numerical operations of nonlinear equations are indispensable to the science of nonlinearity.Generally speaking, we can not solve the solution of the nonlinear equations exactly. We often search its approximate solution by numerical methods. Because of the nonlinearity of the operator F , the research faces a certain difficulty. At present, we usually adopt the iteration to solve the nonlinear equations. Several iterations, which are adopted frequently, are simple Newton method, Newton method and its transmutations, numerical extension method, parameter differential method, and so on.It is well known that a damped or relaxation Newton's method will sometimes solve a system of nonlinear equations when the full Newton's method cannot. This happens, for example, when only a poor initial approximation to the solution is known, Newton method is difficult to converge to the exact solution.When Newton's method is applied to solve nonlinear equations, certain situation occurs. Though the initial value is near the solution, the iterative result is far away from the solution or converges to others. In order to avoid this case, this paper adds a modified term to Newton method and proves its convergence and convergent rate theoretically. Further, we consider the continuous case. Moreover, we solve its initial value problem by weakly A-stable method, investigate the computational format, and analyze its convergence. Finally, its validity is testified by numerical example.