New Acceleration Iterateive Methods For Singular Problems

Many equations arising in practical application are singular nonlinear equations, such as saddle points, bifurcation points and fold points etc. So studying the numerical method of solve singular problems has very important practical meaning. Most iterative methods are well studied for non-singular problems, so the research for solving singular problems is a complementary to the nonlinear theory. Decker, Kelley, H. B. Keller have studied Newton method, Chord method and Quasi-Newton method for solving singular nonlinear equations, the convergence theorem is proved and the asymptotic rate of convergence is obtained. Without additional calculations, multi-step iterative methods are constructed for solving singular nonlinear problems by space geometry properties in this paper, the better asymptotic convergence rates are obtained.Several numerical methods for singular nonlinear equations are studied in this paper, the main results are as follows:First, the original method is improved, new acceleration iterative scheme is constructed, its convergence theorem is proved and the estimation of the convergence rate is given.Second, for a class of nonlinear singular equations in R n, a row-column- update Quasi-Nweton method is given, the iterative sequence converges to x *, the method keeps sparsity and symmetry. The sufficient conditions of convergence are given and the convergence speed is estimated.Third, the extrapolation technique is applied widely in series calculation, circumference rate calculation, difference and finite element methods. The extrapolation technique and King-Werner method are combined to construct a new iterative method for solving singular problems in Hilbert space. Without additional calculation, the asymptotic linear convergence rate is improved greatly.