On the convergence of high-order Ehrlich-type iterative methods for approximating all zeros of a polynomial simultaneously
For numerical analysts working on polynomial root-finding, this provides rigorous convergence guarantees for high-order simultaneous methods, though the results are incremental extensions of existing theory.
The paper establishes new local and semilocal convergence theorems for high-order Ehrlich-type iterative methods for simultaneously approximating all polynomial zeros, improving upon prior results and providing the first semilocal convergence result for these methods.
We study a family of high order Ehrlich-type methods for approximating all zeros of a polynomial simultaneously. Let us denote by $T^{(1)}$ the famous Ehrlich method (1967). Starting from $T^{(1)}$, Kjurkchiev and Andreev (1987) have introduced recursively a sequence ${(T^{(N)})_{N=1}^\infty}$ of iterative methods for simultaneous finding polynomial zeros. For given $N \ge 1$, the Ehrlich-type method $T^{(N)}$ has the order of convergence ${2 N + 1}$. In this paper, we establish two new local convergence theorems as well as a semilocal convergence theorem (under computationally verifiable initial conditions and with a posteriori error estimate) for the Ehrlich-type methods $T^{(N)}$. Our first local convergence theorem generalizes a result of Proinov (2015) and improves the result of Kjurkchiev and Andreev (1987). The second local convergence theorem generalizes another recent result of Proinov (2015), but only in the case of maximum-norm. Our semilocal convergence theorem is the first result in this direction.