A family of root-finding methods with accelerated convergence*

Dedicated to Professor Jürgen Herzberger on the occasion of his 65th birthday.
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Abstract

A parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered. The convergence of the basic method of the fourth order is accelerated using Newton's and Halley's corrections thus generating total-step methods of orders five and six. Further improvements are obtained by applying the Gauss-Seidel approach. Accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations. Detailed convergence analysis and two numerical examples are given.

Keywords

Determination of polynomial zeros
Simultaneous iterative methods
Convergence analysis
Accelerated convergence
R-order of convergence

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The authors would like to thank the anonymous referees who made valuable comments and suggestions that improved the presentation.