Several Modified Iterations Of Solving Nonlinear Equations

When researching the social and natural phenomena or solving engineering problems with mathematical tools, many occurrences are transformed to the problems of solving non-linear equations f(x)=0. Therefore, both in theory and practice, solving nonlinear equations are very important issues. The iterative method is one of the most impor-tant method for finding roots of nonlinear equations. For the pros and cons of iterative method of solving nonlinear equations, the speed and quality of the results have signif-icant impact. So for reality, constructing efficient iterative method has much scientific value and practical significance. The iterative algorithm for solving nonlinear equations discussed in this paper is based on the Newton method, Steffensen method and Ostrowski method. We focus on two aspects for finding new-iterative methods, by increasing the it-eration steps, approximating the function values and adopting parameters. Respectively, we make a number of new variants, and present iterative methods of approximating single root within the real number, and the computational efficiency indexes of these methods reach maximum values (called optimal iterations), the effectiveness algorithms are verified by numerical tests. The paper is divided into three chapters, the contents are concluded as follows:In the first chapter, we summarize related definitions, background and current situ-ation about the iterative method for solving nonlinear equations, review results obtained by scholars in recent years in this research field.In the second chapter, we present the research background and current situation of Steffensen iterative algorithms, we construct five new two-step fourth order convergent Steffensen type iterative algorithms with efficiency index1.587, prove theoretically the correctness of conclusions by means of Taylor expansion, and give numerical tests to show these improved methods’efficiency. Five algorithms are introduced with parameters as the weights, on one hand, the iterative efficiency indexes achieve the best, such as the third and fifth theorems by choosing parameters, new iterative algorithms can be obtained. On the other hand, we generalize results in the existing literatures, such as the first, second and fourth theorems contain the first two steps in the eight order convergence in some three-step fourth-order convergent algorithms. In short, each step of each iteration only need to calculate three function values, thus avoid the complicated calculation of the derivative. This also gives a good foundation for constructing higher order convergent algorithms.In the third chapter, we give the background and current situation of Ostrowski iterative algorithms, construct a class of three-step eighth order convergent Ostrowski type iterative algorithms with efficiency index1.682, prove theoretically the correctness of the results by means of Taylor expansion, and give numerical tests to show these improved methods’efficiency. The main results of the promotion of such algorithms with two-variant weights function, and then to find a function meet certain properties to make the iterative efficiency indexes to achieve the best, it is much more extensive than with weight function of one-variant. Meanwhile, this works improve many corresponding results of the literatures.