Deformed Newton Iteration With The Approximation Of Inverse Operators

Nonlinear system has been one of the key research fields of modern mathe-matics. For solving nonlinear equations in Banach space iterative method is undoubtedly the most practical one, and specifically, Newton iteration is the most classical among all the iterative methods. As a result that the deformation of most iterative methods are obtained based on the Newton iteration method.This paper is also based on the Newton iterative method, and it analyzes the convergence and error estimate of the deformation. The biggest advantage of deformation in this paper is that, in the premise of not affecting the convergence speed, matrix multiplication is used to approach the inverse operators, so that it can avoid the complex calculation of inverse operator and greatly improve the computational efficiency. This paper is divided into four chapters:The first chapter mainly introduced the related background knowledge for this paper.The second chapter has given one deformations of the Newton iteration, and the detailed analysis of the convergence and the error estimation of the algorithm.The third chapter has made expansion which is based on the deformation of the Ulm-type iterative method, and it also has given us the detailed analysis of the convergence of the algorithm in this chapter.The fourth chapter applied the new iterative method to two practical exam-ples, and compared them to the Newton method to show the superiority of the deformation.