Three Iteration Methods For Solving Singular Nonlinear Equation

Supposing B is a Banach space, F is a differentiable operator from B to B , the study of solving singular nonlinear equation: F ( x ) = 0 has great necessary and practical meanings in nature science and social science, for example in the boundary layer theory, the elasticity theory, reaction diffusion system, saddle point, fold point, nonlinear optical and so on. Aiming at nonsingular problems, many researchers did thorough systematic research in it, and a lot of research results was obtained. Based on the research results of nonsingular problems, some scholars at home and abroad go on to explore singular nonlinear equations. Based on previous research results , three numerical methods for singular problems are studied in this paper. There are four parts in this paper, as following.1. Aiming at the nonlinear equation as F ( x ) = 0, both research status on singular problems and its necessity are introduced.2. Combining extrapolation and Newton-Moser iteration method, a new iteration scheme for solving singular nonlinear equation is constructed. Its convergence order is increased. Finally , its validity is tested by numerical example.3. For nonsingular equations, King-Werner iteration method was an effective method and its convergence order is 1 + 2. In this paper, the modified King-Werner iteration scheme for solving singular equations is given. The convergence for the modified iteration scheme is proved and its convergence rate greatly is improved.4. Aiming at singular problems of the nonlinear equation, the convergence of parallel secant method is proved and its convergence rate is obtained. A modified term is added to the iteration scheme of parallel secant method. The computational complexity of modified parallel secant method isn't almost increased, but its convergence rate greatly is improved.