Symbolic computation and computer graphics as tools for developing and studying new root-finding methods

Many very difficult problems in applied mathematics and other scientific disciplines cannot be solved without powerful computational systems, such as symbolic computation and computer graphics. In this paper we construct two new families of the fourth order iterative methods for finding a multiple real or complex zero of a given function. For developing these methods, a recurrent formula for generating iterative methods of higher order for solving nonlinear equations is applied and implemented by symbolic computation through several programs in computer algebra system Mathematica. Symbolic computation was the only tool for solving the considered complex problem since it provides handling and manipulating complex mathematical expressions and other mathematical objects. The properties of the proposed rapidly convergent methods are illustrated by several numerical examples. To examine the convergence behavior of the presented methods, we also give the dynamic study of these methods using basins of attraction. Such a methodology, besides a visualization of iterative processes, deliveries very important features on iterations including running CPU time and average number of iterations, as a function of starting points. The program for plotting basins of attraction in Mathematica is included.