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

Abstract

Using the tools provided by computer algebra system {\it Mathematica}, we consider two iterative methods of high efficiency for the
simultaneous approximation of simple or multiple (real or complex) zeros of
algebraic polynomials are considered. The proposed methods are based on the fourth-order Schr\"oder-like methods of the first and second kind. By means of symbolic computation 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. Besides, single-step versions of the proposed methods, based on the Gauss-Seidel
approach, are constructed. 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.