Study of Some Higher-Order Methods for Multiple Roots of Nonlinear Equations

Abstract

One of the most basic and earliest problem of numerical analysis concerns with finding effi- ciency and accurately the approximate solution of the nonlinear equation of the form: f(x) = 0, where f : I ⊆R→R is a nonlinear sufficiently differentiable function in an interval I. Analytic methods for obtaining exact solutions of such problems are almost non-existence. Therefore, one has to find the approximate solutions by relying on numerical methods which are based on iterative procedures. There are several one-point as well as multi-point iterative methods are available in the literature to solve these equations. Therefore, the construction of iterative methods for solving nonlinear equations is practically important and interesting task, which has attracted the attention of many researchers around the world. Therefore, the main goal and motivation in the development of new equally competitive methods is to achieve highest computational efficiency with a fixed number of function evaluations per iteration. In our thesis, we have proposed several new one-point as well as multi-point families of methods for obtaining multiple roots of nonlinear equations. Further, we also intend to construct fourth-order optimal families of Jarratt’s method. In majority of the tested examples, numerical results have shown that all the proposed methods can compete with their classical counterparts. For the comparison of iterative methods, one should take into account their convergence orders, the numerical stability, computational costs, asymptotic error constants, the dependence of convergence on the choice of initial guesses, the simple body structures, basins of attraction which provide their dynamic behavior and so on. The study of some of the mentioned features is often complicated so for better comparison, we have taken four section i.e. number of iterations, Computational order of convergence (COC), absolute error is calculated by using same total number of function evaluations (TNFE=12) and basins of attraction. Many computer algebra software systems are available such as Mathematica, Matlab and Mapple etc. We use computer algebra software namely, Wolfram Mathematica-9 in multi precision in the computation of nonlinear scalar equations throughout this work. We accept an approximate solution up to any specific degree rather than the exact root. Therefore, the following stopping criteria is used for computer program: (i)|xn+1−xn|< ε, (ii)|f(xn+1)|< ε. When the above stopping criterion is satisfied, xn+1 is consider as the required root.