Some Theories About Iterative Methods For Solving Nonlinear Equations

This dissertation mainly study the theoretic analysis of solving nonlinear equations f(x) = 0 in Banach space, especially on the local and semi-local convergenceof Newton's method, inexact Newton method and Newton-like method.Newton's methodis the usual method for solving nonlinear equations. Given a sufficiently good initial guess x₀, Newton's method is convergent to the root of the equations quickly. Moreover, the computation of every step only has relation to the previous step, so the error isn't progenitive and self-corrective. Hence Newton's method is an important method in both theorem and application, and is popular with the numerical researchers. Since Newton's method was proposed, there has been a great many theoretic production, mainly concerning the local convergence, especially the size of the convergent ball and the unique ball; the semi-local convergence, especially the Mysovskii type theorem and the Kantorovich type theorem; as well as the global convergence. Kantorovich theorem is a beautiful and powerful semi-local convergence theorem and attracts studiers to improve and develop its condition and conclusion. Wang([13]) also introduces a semilocalconvergence of Newton's method, which is so recapitulative and elegant that combines Kantorovich type condition with Smale type condition. Here we give a new application and show that it can conclude and improve the result in Argyros([14]). It is stated as the following theorem.Theorem 0.1 If f satisfies the following conditions, which are proposed by Argyros in ([14]): (4)p₂(s)≤0, where P₂(r) is defined by:and s is a root of p'₂(r).Then f satisfies the conditions in Wang([13]):(â… )For (?)x∈S(x₀,δ) and (?)x'∈S(x,δ-ρ(x)), there is(â…¡)Letδ₀ be such that∫₀^δ0L(u) du =1 and b=∫₀^δ0 L(u)udu. Assume(â…¢) t^*≤δ, where t^* is the least positive root of h(t), andSince we have to solve a linear equations f'(x_k)â–³_k=-f(x_k) (often called Newton equations) every step, once the number of the unknowns is great, it will take us high cost in solving it by a direct method such as elimination method. For this reason, Dembo-Eisenstat-Steihaug([59])introduce inexact Newton method, i.e.which computes an approximate solution to the Newton equations. We not only introduce some related local and semi-local convergence results proposed by many researchers, but also give a new semi-local convergence theorem when f' satisfies some weak Lipschitz condition. As special cases of our main result we re-obtain some well-known convergence theorems for Newton methods.定理0.2 Let f : D (?) X→Y be Frechet differentiable on S(x₀,δ) (?) D. Suppose x₀∈D is a given initial guess such that f'(x₀)^-1 exists. Let L(u) be a positive nondecreasing function in [0,δ],ρ(x) =‖x- x₀‖,ρ(xx') =ρ(x) +‖x' - x‖≤δ. Assume f'(x₀)^-1f' satisfy the center Lipschitz condition in the inscribed sphere with the average of L, i.e.For 0<η₀<1/2 andη_k<2η₀,the residual r_k such thatSuppose s₀≤b andwhereσ_k:=α_k/(1-η_kα_k)and v_k:=σ_k+1／σ_k.as well aswhereδ₀ is defined by∫₀^δ₀L(u)du=1-2η₀,b=∫₀^δ₀uL(u)du／(1-η₀),t₀^* is theleast positive root ofφ₀(t), andThen, inexact Newton sequence {x_k}(k≥0) remains in S(x₀,t₀^*) and convergentto a solution x^* of f(x) = 0.when we solve the Newton equations, the calculation of f'(x_k) is sometimes difficult and with high cost. For improving the computing efficiency, we often use nonsingular operator A(x_k) to approximate f'(x_k). That is Newton-like method, i.e.We discuss its local and semi-local convergence and give a new semi-local convergencetheorem derived from inexact Newton method.定理0.3 Let f: D (?) X→Y be Frechet differentiable on S(x₀,δ) (?) D and A(x) is an approximation to f'(x). Suppose A(x₀) is nonsingular for some initial guess x₀∈S(x₀,r₀), where r₀∈[0, r]. Assume where x∈S(x₀, r) and‖x - x₀‖+‖x - x'‖≤r, and the following:(â… ) There exists nonnegative constants s₀,α,βand 0≤u₀ < 1 such thatand(â…¡) There hold for some a≥max{1,α+β}(â…¢) S(x₀,t^*) (?) S(x₀,r₀), for the least positive root t^* ofφ(t):Then the Newton-like sequence {x_k} {k≥0) remains in S(x₀,t^*) and convergent to a solution x^* of f(x) = 0.