A general semilocal convergence theorem for simultaneous methods for polynomial zeros and its applications to Ehrlich's and Dochev-Byrnev's methods
Provides improved theoretical convergence guarantees for practitioners using specific polynomial root-finding methods.
The paper establishes a general semilocal convergence theorem for simultaneous polynomial zero methods, yielding improved convergence results for Ehrlich's and Dochev-Byrnev's methods, and proves the identity of Dochev-Byrnev's and Prešić-Tanabe's methods.
In this paper, we establish a general semilocal convergence theorem (with computationally verifiable initial conditions and error estimates) for iterative methods for simultaneous approximation of polynomial zeros. As application of this theorem, we provide new semilocal convergence results for Ehrlich's and Dochev-Byrnev's root-finding methods. These results improve the results of Petković, Herceg and Ilić [Numer. Algorithms 17 (1998) 313--331] and Proinov [C.~R. Acad. Bulg. Sci. 59 (2006) 705--712]. We also prove that Dochev-Byrnev's method (1964) is identical to Pre{\v s}ić-Tanabe's method (1972).