On a family of Weierstrass-type root-finding methods with accelerated convergence
For numerical analysts, this provides practical convergence guarantees for higher-order simultaneous polynomial root-finding methods, though the work is incremental as it extends existing local results.
This paper extends the convergence analysis of a family of Weierstrass-type root-finding methods from local to semilocal, providing computationally verifiable initial conditions and a posteriori error estimates for methods of arbitrary order N+1.
Kyurkchiev and Andreev (1985) constructed an infinite sequence of Weierstrass-type iterative methods for approximating all zeros of a polynomial simultaneously. The first member of this sequence of iterative methods is the famous method of Weierstrass (1891) and the second one is the method of Nourein (1977). For a given integer $N \ge 1$, the $N$th method of this family has the order of convergence ${N+1}$. Currently in the literature, there are only local convergence results for these methods. The main purpose of this paper is to present semilocal convergence results for the Weierstrass-type methods under computationally verifiable initial conditions and with computationally verifiable a posteriori error estimates.