Convergence Analysis Of A Two-step Combined Method

With the rapid development of science and technology,the theory of solving nonlinear equations is increasingly applied by experts in the fields of mathematics,computer science,physics science and etc.Nonlinear science also plays an increasingly important role in life.In this thesis,solving nonlinear equations is studied,and a convergent sequence is obtained by two-step iterative method to approximate the solution of the original equation.The details are as followsIn the first chapter,the background of the two-step iterative method and its preparatory knowledge are introduced,including the affine radius H???lder condition with L average,iterative convergence condition,convergence order,some basic concepts and the related knowledge of this thesis.At the end,the main results of the paper are given.In the second chapter,the local convergence of the two-step iterative method is discussed under the condition that the first order Fréchet derivative and the second order Fréchet derivative satisfy the central affine H???lder condition with L average and Lipschitz condition with L average,respectively,and the local convergence conditions are obtained.At the same time,it is proved that the R order of convergence of this method is at least ?1+p?/2+??1+p?²/4+p²?^1/2.In the last chapter,we study the semi-local convergence theorem of the two-step iterative method under the condition that the first order Fréchet derivative satisfies? condition.The semi-local convergence theorem of the two-step iterative method is obtained.In the last section,we present an application of our method.