On the convergence of high-order Ehrlich-type iterative methods for approximating all zeros of a polynomial simultaneously

摘要

We study a family of high order Ehrlich-type methods for approximating allzeros of a polynomial simultaneously. Let us denote by $T^{(1)}$ the famousEhrlich method (1967). Starting from $T^{(1)}$, Kjurkchiev and Andreev (1987)have introduced recursively a sequence ${(T^{(N)})_{N=1}^\infty}$ of iterativemethods for simultaneous finding polynomial zeros. For given $N \ge 1$, theEhrlich-type method $T^{(N)}$ has the order of convergence ${2 N + 1}$. In thispaper, we establish two new local convergence theorems as well as a semilocalconvergence theorem (under computationally verifiable initial conditions andwith a posteriori error estimate) for the Ehrlich-type methods $T^{(N)}$. Ourfirst local convergence theorem generalizes a result of Proinov (2015) andimproves the result of Kjurkchiev and Andreev (1987). The second localconvergence theorem generalizes another recent result of Proinov (2015), butonly in the case of maximum-norm. Our semilocal convergence theorem is thefirst result in this direction.