Multi-Step Iterative Method To Solve Singular Problems

Equations arising in many actual nonlinear problems are singular equations, such as saddle points, bifurcation points and fold points etc. Therefore, an interest in the computation of such solution points has provided much of the motivation for the more recent attention directed toward singular problems. As the convergence properties about several types of iterative methods are studied for solving non-singular problems, so the research for singular problems complements the non-linear theory. H. B. Keller, C. T. Kelley and D. W. Decker have studied Newton's method, Chord method and Quasi-Newton method for solving singular nonlinear equations. In this paper, singular problems are solved by multi-step iterative method, the convergence theorem is proved and the asymptotic rate of convergence is obtained. Without additional calculations, the combination iterate in virtue of space geometry character has the better asymptotic rate of convergence. The method is generalized to solve high order singular problems in this paper.There are four parts in this paper. In the first chapter, the main research results about solving singular problems have been given, the main contents of this paper, the background and the significance of the research have also been described in the preface. In the second chapter, having an extensive application in series calculation, circumference rate calculation, difference and finite element methods, the extrapolation technique is employed to solve singular problems in Hilbert space. Without additional calculation, the asymptotic rate of convergence is improved greatly. In the third chapter, a new iterative method, namely the Parallel Secant method, is constructed to solve singular problems. The convergence theorem is proved and the asymptotic rate of convergence is obtained. The geometry character of Hilbert Space is used to modify Parallel Secant method of which asymptotic rate of convergence is improved with the same calculation. In the fourth chapter, King-Werner's method is considered to solve high order singular problems.