On the guaranteed convergence of the fourth order simultaneous method for polynomial zeros

摘要

One of the most important and challenging problems in solving nonlinear equations is the construction of computationally verifiable initial conditions that provide both the guaranteed and fast convergence of the considered numerical method. A suitable convergence procedure, based partially on Smale's “point estimation theory” from 1981, is applied in this paper to the fourth order iterative method for the simultaneous approximation of simple zeros of polynomials, proposed by Zheng and Sun in 1999. We have stated initial conditions which ensure the guaranteed convergence of this method. These conditions are of significant practical importance since they depend only on available data: the coefficients of a given polynomial, its degree n and initial approximations to polynomial zeros.