The Study Of Multipoint Iterative Methods For Solving Nonlinear Equations

The problem solving nonlinear equations holds a very important position in the theory and practice not only of applied mathematics, but also of many branches of computer science, engineering sciences, finance, physics, and so on. Research on this problem also promotes a blend and development of mathematics and computing science. This thesis mainly concerns itself with multipoint iterative methods for solving nonlinear equations. Multipoint iterative methods are defined as methods that require evaluation of function and its derivatives at a number of values of the independent variable. The major goal of constructing new multipoint methods is to achieve as high as possible computational efficiency; in other words, it is desirable to achieve as high as possible convergence order with a fixed number of functional evaluations per iteration. Based on this idea, some new multipoint iterative methods are presented, the convergence order and the computational efficiency of the presented methods are studied and the numerical experiments are used to demonstrate the performance of the presented methods. The main results and innovations are as follows:1. Some Newton type iterative methods without memory are presented for solving nonlinear equations, including optimal fourth-order two-point methods and optimal eighth-order three-point methods. Based on the presented methods without memory, some Newton type iterative methods with memory are obtained by using the accelerating parameter. The convergence order and computational efficiency of the two-point and three-point methods are increased without any additional functional evaluations. Numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the computational efficiency and the performance of the presented methods.2. A n-point Newton type iterative method without memory is presented by the method of Hermite's interpolation. The new method has the optimal order 2". Based on the presented method without memory, the n-point Newton type iterative method with memory is obtained by using one accelerating parameter. The accelerating parameter is calculated using information available from the current and previous iterations. The improvement of convergence order is achieved without any additional functional evaluations so that the proposed method with memory has a high computational efficiency. Numerical examples are included to confirm theoretical results and demonstrate convergence behavior of the proposed method.3. Some Steffensen type iterative methods without memory are presented for solving nonlinear equations, including optimal fourth-order three-point methods, seventh-order four-point method and optimal eighth-order four-point methods. Based on the presented methods without memory with optimal order, some Steffensen type iterative methods with memory are obtained by using the accelerating parameter. The convergence order and computational efficiency of the three-point and four-point Steffensen type iterative methods are increased without any additional functional evaluations. Numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the computational efficiency and the performance of the presented methods.4. A (w+1)-point Steffensen type iterative method without memory is presented by the method of Newton's interpolation. The new method has the optimal order 2" requiring n+1 functional evaluations per iteration. Based on the presented method without memory, the (n+1)-point Steffensen type iterative method with memory is obtained by using n+1 accelerating parameters in per full iteration. The maximal convergence order of the new Steffensen type iterative method with memory is(2n+1+1-?22(n+1)+1)/2, which is higher than the existing any iterative methods. Numerical examples are included to confirm theoretical results and demonstrate convergence behavior of the proposed methods.5. A new sixth-order Newton type iterative method is presented for solving nonli-near systems, Per iteration, the new method needs to compute the LU decomposition of Jacobian Matrix only once. A new computational efficiency index is used to compare the efficiencies of the new method and some know methods. It is proved that the new method is more efficient. Numerical experiments are performed, which support the theoretical results.6. Four Steffensen type iterative methods with various local order of convergence are presented for solving nonlinear systems. A development of an inverse first-order divided difference operator for multivariable function is applied to prove the local convergence order of the new methods. The computational efficiency is compared with some relevant methods. Numerical examples show that the new iterative methods can save computing time and improve computational efficiency.