Advances In Iterative Methods For Nonlinear Equations PDF Download

Many of the problems in experimental sciences and other disciplines can be expressed in the form of nonlinear equations. The solution of these equations is rarely obtained in closed form. With the development of computers, these problems can be addressed by numerical algorithms that approximate the solution. Specifically, fixed point iterative methods are used, which generate a convergent sequence presumably to the solution of the equation or system of equations. Since J.F. Traub (Iterative methods for the solution of equations, Prentice-Hall, N.J. 1964) initiated the qualitative as well the quantitative analysis of iterative methods in the 1960s, iterative methods for nonlinear systems has been a constantly interesting field of study for numerical analysts. Our contribution to this field is the analysis and construction of new iterative methods, by improving the order of convergence and computational efficiency either of these or other known methods. To study the new iterative methods that we have proposed, we reviewed analyzed and improved classic concepts of computational order of convergence, the error equation, and the computational cost of an iterative method for both an equation and a system of nonlinear equations. Specifically, we have worked on the following points: - We computed the local order of convergence for known two-step and new multi-step iterative methods by means of expansions in formal developments in power series of the function F, the Jacobian operator, the inverse Jacobian operator and the divided difference operator and its inverse operator. - We generated some measures that approximate the order of convergence. Four new variants for computing the computational order of convergence (COC) are given: one requires the value of the root, whilst the other three do not. - We constructed families of iterative schemes that are variants of Newton's method and Chebyshev's method and improve the order and the efficiency. - We studied several families of the modified Secant method (Secant, Kurchatov and Steffensen), evaluated variants of these methods and chose the most efficient. - We generalized the concepts of efficiency index and computational efficiency for nonlinear equations to systems of nonlinear equations. This has been termed the computational efficiency index (CEI). - We considered that in iterative process using variable precision, the accuracy will increase as the computation proceeds. The final result will be obtained as precisely as possible, depending on the computer and the software. - We expressed the cost of evaluating elementary functions in terms of products. This cost depends on the computer, the software and the arithmetic that we used. The above numerical calculations were performed in the algebraic system called MAPLE. - We presented a new way of comparing elapsed time for different iterative schemes. This consists of estimating the time required to achieve a correct decimal of the solution by the method selected. That is, we measured the relationship between the time to fulfill the stop criterion and the total number of correct decimals obtained by method. The five papers selected for this compendium were published in scientific journals in the area of applied mathematics. The impact factor of these journals is, in all cases, in the first third according to the classification of the Journal of Citation Reports. There are four preceding papers that no are part of this report by its publication date.