The Convergence Orders Of Hermite And Lagrange Type Newton Methods And Their Optimal Formats

Newton iterative method is one of the most commonly used and most important numerical methods in the calculation of nonlinear equations.This paper mainly studies several new deformed iterative formats of Newton iteration methods.The article can be divided into four parts.In the first chapter of the introduction,we mainly introduced the research background and existing research results of Newton method,then introduced the basic concepts such as convergence order and efficiency index,and briefly introduced the method of inverse interpolation.In the second chapter,we constructed a multi-node Hermitian deformed Newton iteration format by using the inverse interpolation technique,and obtained the equation satisfied by the convergence order of the multi-node Hermitian Newton method in the general form.It is also proved that the convergence order of the multi-node Hermitian Newton method increases monotonously and approaches 3 with the increase of the number of nodes.Then,we used the efficiency index as a criterion to get the corresponding optimal iterative format of Hermite Newton method under different calculations.In the third ehapter,we mainly study the Lagrange type Newton method.Similar to Chapter 2,by means of inverse interpolation techniques and Lagrange interpolation residual formula,we proved that the convergence order of multi-node Lagrange Newton method increases monotonically and tends to 2 with the number of nodes.At the same time,the optimal iterative format of Lagrangian Newton method is further discussed.In the fourth chapter,we also discussed a new class of Newton iteration methods without derivatives.We got a more general expression of the new iterative format,and proved that the convergence order of this new iterative format is at least 4 when the parameters satisfy the conditions.Finally,we summarized and looked forward to the full text.