On a new family of simultaneous methods with corrections for the inclusion of polynomial zeros

Abstract

A high-order one-parameter family of inclusion methods for the simultaneous inclusion of all simple complex zeros of a polynomial is presented. For specific values of the parameter, some known interval methods are obtained. The convergence rate of the basic fourth-order family is increased to 5 and 6 using Newton's and Halley's corrections, respectively. Using the concept of the R-order of convergence of mutually dependent sequences, we present a convergence analysis of the accelerated total-step and single-step methods with corrections. The suggested inclusion methods have great computational efficiency since an increase of the convergence rate is attained with only a few additional calculations. Two numerical examples are included to demonstrate the convergence properties of the proposed methods.

Keywords:

• Zeros of polynomials
• One-parameter family
• Iterative processes
• Simultaneous methods
• Convergence rate
• Acceleration of convergence
• Circular arithmetic

Acknowledgements

This research was supported by the Serbian Ministry of Science under grant number 144024.