The Analysis Of Semilocal Convergence On Simplified Newton Method And Chebyshev Method Under Affine Transformation Conditions

This thesis mainly concerns the semilocal convergence and error estimations of simplified Newton method and Chebyshev method to solve the nonlinear equation(s) F(x)= 0, under two affine transformation conditions by using the approach of majorizing functions and majorizing sequences. It enlarges the application of affine transformation conditions, and weakens some relevant convergent conditions and improves some results. The contents are as follows:Chapter 1 introduces the background and status of Newton-type iterations, includ-ing some iterative schemes and some convergence conditions. Also, it presents relevant preliminary knowledge, such as convergence of iteration, the termination condition, con-vergence order, convergence efficiency and relevant knowledge in Banach space. At last it shows the structure of the thesis.Chapter 2 firstly introduces the concepts and classification of affine transformations. It then points out that the two kinds of iterative schemes studied in the thesis satisfy affine invariant conditions. In this chapter the concepts of majorizing function and majorizing sequence have also been concerned.In Chapter 3, according to the conclusion that affine contravariant condition can be applied to the simplified Newton method, we study the semilocal convergence and error estimation of simplified Newton method under affine contravariantγ- condition. There-fore the application of the affine contravariant condition is further improved. Moreover, the iterative algorithm is based on the control of the residuals.In Chapter 4, we study the semilocal convergence and error estimation of Chebyshev method under an affine covariant condition we have introduced above by adopting the majorizing sequence generated by the abstract majorizing function. Through the method, a clear relationship between the majorizing function and the nonlinear operator is shown. In this method, the affine covariant condition is a little weaker than the present L-average Lipschitz condition. The advantage is that the abstract majorizing function only requires one zero point while that in Wang Xinghua's L-average condition needs two zero points. However, it can still guarantee the third order convergence for Chebyshev method and gain the error estimation and its unique ball. Especially, the main results we gain can weaken the conditions of the convergent results in some literatures and improve the results in some literatures, that is, affine covariant Lipschitz conditions and affine covariantγ-condition are weakened and the relevant results are generalized. At the end of this chapter we state that the local convergence of Chebyshev method, or even the local and semilocal convergence of the family of Halley—Chebyshev method can be studied by using the similar way as used in this chapter.