Point estimation of simultaneous methods for solving polynomial equations: a survey

One of the most important problems in solving nonlinear equations is the construction of such initial conditions which provide both the guaranteed and fast convergence of the considered numerical algorithm. Smaleʹs approach from 1981, known as “point estimation theory”, treats convergence conditions and the domain of convergence in solving an equation f(z)=0 using only the information of f at the initial point z0. A procedure of this type is applied in this paper to iterative methods for the simultaneous approximation of simple zeros of polynomial equations. We have stated new, refined initial conditions which ensure the guaranteed convergence of the most frequently used simultaneous methods for solving algebraic equations: the Durand–Kerner method, Börsch-Supan method, the Ehrlich–Aberth method and the square-root family of one parameter methods. The stated initial conditions are of significant practical importance since they are computationally verifiable; they depend only on the coefficients of a given polynomial, its degree n and initial approximations to polynomial zeros.