An iteration formula for the simultaneous determination of the zeros of a polynomial

摘要

The need for efficient algorithms for determining zeros of given polynomials has been stressed in many applications. In this paper we give a new cubic iteration method for determining simultaneously all the zeros of a polynomial (assumed distinct) starting with ‘reasonably close’ initial approximations (also assumed distinct).The polynomial is expressed as an expansion in terms of the starting and their correction terms.A formula which gives cubic convergence without involving second derivatives is derived by retaining terms up to second order of the expansion in the correction terms.Numerical evidence is given to illustrate the cubic convergence of the process.