Optimal Iterative Family for Solving Non-Linear Equations

Abstract

One of the most important and challenging problems in scientific and engineering applications is to find solutions of non-linear equations. In order to find out the approximate solutions of these equations, one has to adopt numerical techniques based on iteration procedures. Newton’s method is probably the well-known iterative method for solving these equations. In recent years, many modifications of Newton’s method has been proposed in literature, which have either equal or better performance than Newton’s method. The Chapter 1 is an introductory chapter and gives a brief survey of literature. Several real world problems have been considered for which the numerical solutions are require for solving scalar non-linear equations. The fundamental concepts and classification of iterative methods and their striking features are also stated. The research work on the iterative method carried out in solving non-linear equations. Chapter 2 presents quartically convergent families of ellipse method for the solution of scalar non-linear equation, permitting f′(x) = 0 near the root. Quartically convergent variant of ellipse method (QVEM) have been proposed, convergence analysis and numerical examples are also stated in this chapter. Chapter 3 presents the basin of attraction of iterative method for solving non-linear equations. Different methods generate different basin of attraction. The different shades of these colors indicate the speed of convergence to the respective roots (in terms of number of iteration required to get very close to the root).