Solving Singular Problems By Geometry Character

Many equations arising in practical application are singular nonlinear equations, such as bifurcation points, inflexion etc. Decker, Kelley and H. B. Keller have used Newton's method, chord method and Quasi-Newton method to solve singular nonlinear equations, and proved the theory of convergence and obtained the asymptotic rate of convergence. On the premise of few additional calculations, it is significant that how carries on the combination iteration using the geometry character to improve the asymptotic rate of convergence, which has been studied in this paper.The whole paper is divided into four parts. Firstly, the development of singular problems home and abroad is expatiated in the preface, and the main contents, background and significance of this paper are introduced. Secondly, the extrapolation technique has an extensive application in series calculation, circumference rate calculation, difference and finite element methods. The extrapolation technique and Broyden's method are combined to construct a new iterative method for solving singular problems in Hilbert space and L p Space. Without additional calculation, the asymptotic linear rate of convergence is improved greatly. Thirdly, Newton-Moser method of singular nonlinear equations converges linearly in normal Banach space and the asymptotic rate of convergence is the root of a cube equation. The character of Hilbert Space and L p Space is utilized to modify Newton-Moser method for solving singular problems. The modified Newton-Moser method improves the asymptotic linear rate of convergence and has the same calculation as that of Newton-Moser method. At last, Halley, Chebyshev and Super-Halley iterative scheme are also important methods for solving nonlinear equations. As they have many advantages, they are widely used. Chebyshev method is modified for solving nonlinear singular equations and the asymptotic rate of convergence is given in this paper.