Computer visualization and dynamic study of new families of root-solvers

Abstract

The considered topic, the dynamic study of new root solvers using computer tools, is actually a “bridge” between computer science and applied mathematics. The goal of this paper is the construction and dynamic study of two new one-parameter families for solving nonlinear equations using advanced computer tools such as symbolic computation, computer graphics and multi-precision arithmetic. First family, based on Popovski’s third order method (Popovski, 1980), is modified and adapted for finding multiple zeros of differentiable functions. Its dynamic study is performed using basins of attraction and associated quantitative data. In the second part this one-point family for a single zero serves for the derivation of a very efficient family of iterative methods with corrections for the simultaneous determination of all multiple zeros of algebraic polynomials. The convergence analysis, performed by the help of symbolic computation in computer algebra system Mathematica, has shown that the order of convergence of the proposed family is four, five and six, depending of the type of used corrective approximations. A very fast convergence rate is obtained without any additional evaluations of a given polynomial P and its derivatives P′ and P′′, which points to the high computational efficiency of new methods. Choosing different values of the involved parameter, the presented family generates a variety of simultaneous methods. Employing multi-precision arithmetic, it has been shown by numerical experiments that four particular methods from the family produce approximations of very high accuracy. Computer visualization, carried out by plotting trajectories of zero approximations, points to the stability and robustness of the proposed simultaneous methods and indicates a conjecture on their globally convergent properties, one of the most important features of simultaneous methods for solving polynomial equations.