The Convergence Of Several Iterate Schemes For Solving Singular Problems

The problem of solving nonlinear equations is involved in many practicalapplications. The improvement of the nonlinear theory is hot and difficult spot in theresearch of mathematics problems. And the research on the singular problems is animportant work of the improving of nonlinear theory. To solve the singular nonlinearequations, iterative methods usually be taken to obtain an approximate solution of acertain accuracy. Newton’s method is the most classical iterative method. Manyscholars have researched on the transformation of Newton’s method. Abundantaccomplishment have been gained in the convergence rate and the acceleration ofiteration format.The convergence of several iterative schemes for solving singular problems wasdiscussed in this paper. To solve the problems of slow convergence rate and largecalculation amount, the modified iterative scheme was introduced. The convergencerate was greatly increased and the calculation amount was rarely increased throughthe modifiction. The main content of this article were:1. The research background and the current situation of the development of bothat home and abroad were introduced briefly in this article.2. Broyden method was widely used for the low calculation cost and theconvenient usability. Broyden method and its convergence rate were both improvedin this article so that it will be more suitable for solving singular nonlinear equations.3. Many achievements have been made in non-singular nonlinear equations byusing Chebyshev method. But there was few research on singular problems. In thispaper, the application of Chebyshev method in solving singular nonlinear equationswas disscussed. At the same time, Chebyshev method was improved and a higherconvergence rate was obtained.4. Newton’s method is the most classical methods of solving nonlinear equations.The Newton’s method for solving high-order singular problems was modified in thissection. The convergence rate was improved on the basis of no increasing the calculation amount through the modifiction.