Convergence of a Family of Accelerated Schröder-like Methods for the Simultaneous Extraction of Simple and Multiple Polynomial Zeros

Abstract

In the present paper, we investigate the local and the semilocal convergence of a recently constructed family of accelerated simultaneous methods with two corrections applied to polynomials with simple and multiple zeros. A new local convergence theorem that provides explicit domains of initial conditions and error estimates ensuring the Q-convergence of the methods from the first iteration is established. Based on this result, we derive a semilocal convergence theorem, for simple zeros, with computable initial conditions and an error estimate, making it directly applicable in practice. Three numerical tests illustrate the applicability of the theoretical results.