Computer tools for the construction and analysis of some efficient root-finding simultaneous methods

Abstract

Using the tools provided by computer algebra system Mathematica, we consider two iterative methods of high efficiency for the simultaneous approximation of simple or multiple (real or complex) zeros of algebraic polynomials. The proposed methods are based on the fourth-order Schr?der-like methods of the first and second kind. We prove that the order of convergence of both basic total-step simultaneous methods is equal to five. Using corrective approximations produced by methods of order two, three and four for finding a single multiple zero, the convergence order is increased from five to six, seven, and eight, respectively. The increased convergence speed is attained with negligible number of additional arithmetic operations, which significantly increases the computational efficiency of the accelerated methods. Convergence properties of the proposed methods are demonstrated by numerical examples and graphics visualization by plotting trajectories of zero approximations. Flows of iterative processes, presented by these trajectories, point to the stability and robustness of the proposed methods.