Relationships between different types of initial conditions for simultaneous root finding methods
This work provides a theoretical framework for converting local convergence results into practical, verifiable conditions for simultaneous polynomial root-finding methods.
The paper establishes relationships between different types of initial conditions for simultaneous root-finding methods, showing that local convergence theorems can be converted into theorems with computationally verifiable initial conditions, which is of practical importance.
The construction of initial conditions of an iterative method is one of the most important problems in solving nonlinear equations. In this paper, we obtain relationships between different types of initial conditions that guarantee the convergence of iterative methods for simultaneous finding all zeros of a polynomial. In particular, we show that any local convergence theorem for a simultaneous method can be converted into a convergence theorem with computationally verifiable initial conditions which is of practical importance. Thus, we propose a new approach for obtaining semilocal convergence results for simultaneous methods via local convergence results.