Fast Root-Finding of Nonlinear Equations in Geometric Computation

Abstract

Computing the roots of polynomials is an important issue in various geometric problems. In this paper, we introduce a new family of iterative methods with sixth and seventh order convergence for nonlinear equations (or polynomials). The new method is obtained by combining a different fourth-order iterative method with Newtons method and using the approximation based on the divided difference to replace the derivative. It can improve the order of convergence and reduce the required number of functional evaluations per step. Numerical comparisons demonstrate the performance of the presented methods.