Newton Iteration. Weak Conditions And Deformation Of The Halley Iteration

When we use mathematical methods to study natural and social phenomena or to solve engineering technique problems, we often regard the solutions of most practical problems as the ones of nonlinear equations in form ofF(x) = 0 (1)in Banach space. For a long time, we have been keeping on the study on the uniqueness of the solution of equation, the conditions of convergence and error estimates for various iterative methods. There are the well-known second-order Newton's iteration, third-order Halley's iteration, Chebyshev's iteration, Super-Halley's iteration and their deformations, and so on. This dissertation consists of four chapters, mainly makes an analysis on the convergence of Newton iteration under two types of weaker conditions and deformed Halley method under Kantorovich conditions.In chapter 1, we summarize not only various corrections of Lipschitz condition since Kantorovich condition was put forward, but also the convergence theorem of Newton's method and the uniqueness of the solution of the equation.In chapter 2, we discuss Lipschitz condition which is satisfied by the second Frechet-derivative of operator through the use of recurrence relations, so as to make it meaningful in general and get the convergence theorem.In chapter 3, we study the same problem by majorizing operator technique and obtain some Kantorovich-type theorem, which makes Lipschitz condition more universal.In chapter 4, we introduce a new two-step method which is from the famous one-point iteration of order three called Halley method to approximate a solution of a nonlinear equation in Banach space. Under the same Lipschitz condition as for Newton's method, we give a result on the existence of a unique solution for the nonlinear equation by using a technique based on a new system of recurrence relations.The main contents are as follows.Let F be a Frechet differentiable operator defined on a non-empty convex set ft included in a Banach space X and with values in another Banach space Y, the Newton iterationis used to approximate a solution of the equation (1). Studying the uniqueness of the solution of the equation (1) and the convergence of Newton's method, we often discuss Lipschitz condition which is satisfied by F' or F". In chapters two and three, we generalize Lipschitz condition which is satisfied by F" and obtain the results, respectively.In the second chapter, we get the following theorem by using recurrence relations:Theorem 2.1 Suppose that the operator F satisfies the following conditons:(1) Let X, Y be Banach spaces and F: X--> Y be a nonlinear twice Prechet differentiable operator in an open convex domain ft;(2) There exists a point ft where the operator F'(X0)^-1 is defined, F'(x₀)^-1 (3) where the function w(z) is a continuous and nondecreasing real function for z > 0, and such that (4) There exists at least one positive root of the equationWe denote the smaller positive root of the equation by R;Then, the sequence {xn} produced by Newton's method (2) with the initial value X0 is well defined and converges to a solution x of the equation (1). And x is unique inB(X0,R).In chapter three, we obtain the following theorem by means of a majorizing function:Theorem 3.1 Let X, Y be Banach spaces, F: X --> Y be a nonlinear twice Frechet differentiable operator in an open convex domain ft, and ft. Supposewhere R satisfiesrRand L(w) is a positive nondecreasing function in [0, .R]. Then the sequence {xn} generated by (2) starting from X0 converges to the unique solution x of F(x) inB(X0, t} U B(X0, t), where 0 < t are two positive zeros of the functionMoreover, we haveandwhere {tn} is the sequence produced by Newton's iteration for h(t) with to = 0. In addition, Halley's iteration is also a common iterative method:LF(x) = F'(x)^-1F"(x)F'(x)^-1F(x)x_n+1 =x_n-(I-(1/2)L_F(x_n))^-1F'(xn)^-1F(xn), In chapter four, a deformed Halley's iter...