The improved Farmer–Loizou method for finding polynomial zeros

Abstract

An improvement of the Farmer–Loizou method for the simultaneous determination of simple roots of algebraic polynomials is proposed. Using suitable corrections of Newton's type, the convergence of the basic method is increased from 4 to 5 without any additional calculations. In this manner, a higher computational efficiency of the improved method is achieved. We prove a local convergence of the presented method under initial conditions which depend on a geometry of zeros and their initial approximations. Numerical examples are given to demonstrate the convergence behaviour of the proposed method and related methods.

Keywords:

• polynomial zeros
• simultaneous methods
• iterative methods
• convergence order
• convergence conditions

2000 AMS Subject Classification :

• 65H05

Acknowledgements

This work was supported by the Serbian Ministry of Science under grant 174022.