Convergence Of The Variants Of A Chebyshev-Halley Iteration Family

Information development is very rapid in modern society, iterative methods for solv-ing nonlinear problems are also to be synchronized with the rapid information develop-ment. So, how to increase the speed of iteration, determine the convergence range and reduce the iteration cost take a key part in the computational mathematics. This thesis explored the semi-local convergence of Chebyshev-Halley type methods, expanded the convergence range and accelerated the convergence speed. The specific contents are as follows:In Chapter 1, we introduced Halley method, Chebyshev method and Chebyshev-Halley type methods, which include the basic concepts, order of convergence, convergence criteria and some relevant results in Banach space. We also presented the layout of this thesis.In Chapter 2, under condition ‖F"(x)‖≤ω(‖x‖), we discussed the convergence of the variants of a Chebyshev-Halley iteration family. The convergence criterion and semi-local convergence were obtained. Finally, the impact of the change of parameter a on the convergence radius was analyzed, so a kind of choosing criterion for the parameter was provided. The technique used is to decompose the Chebyshev-Halley type iterative into two parts, the first part is simple Newton iteration which can be solved by the literature knowledge directly, and the second part is the key part, which was investigated by using a monotonic sequence to reduce the number of iterations and obtain the convergence result.In Chapter 3, we demonstrated the Chebyshev-Halley type iterative methods under the central Lipschitz condition,‖F’(xα,0)-1[F"(y)-F"(xα,0)]‖≤L‖Fy-xα,0‖F, (?)y ∈Ω. The convergence and errors were analysed by using recursive sequences. At last, by combining Newton like methods and Chebyshev-Halley type iterative methods in the cases when the first order derivative exists but the second order derivative does not exist, we studied the computational cost reduction and the simplification of iteration condition.