The New Criteria Of Convergence For Several Iterate Schemes

Nonlinear problems is one of the main fields in modern mathematical research,the numerical solution of the nonlinear equations is a very important direction of the nonlinear problems.Because of its extensive practical background and important theoretical value,the numerical solution of the nonlinear equations is always focused by many numerical analysis researchers.In this paper,the numerical solution of the nonlinear equations is studied,under the point estimates or weak condition the dominating sequence is constructed,at the same time,the existence and the convergence theorem is obtained and its proof is given.First,by analyzing and summarizing the achievement of domestic and foreign researchers in this domain,the article expounds the significance and practical background of solving the nonlinear equations under the point estimates or weak condition .Simultaneously the main content of the article is given. Second,the convergence theorem of Newton method and Newton-Moser method is proved for solving the nonlinear equations with nondifferential terms under weak condition,the corresponding proof is also given,the convergence theorem of King-Werner method is discussed for solving the nonlinear equations with nondifferential terms under the point estimates,the existence and convergence theorem of Parallel-Newton method is proved for solving the equations with nondifferential terms under the point estimates,the proof is given.Third,by means of Smale,s point estimates theorem,the existence and the convergence theorem of Broyden,s iteration is given, the method has less calculation and the speed of super linear convergence for solving the nonlinear equationswith nondifferential terms under the point estimates or weak condition. the corresponding proof is alse given.Fourth,the convergence theorem of generalized chord Newton iteration is proved for solving the nonlinear equations with nondifferential terms under the pointestimates.