Computing multiple zeros using a class of quartically convergent methods

Abstract

Targeting a new multiple zero finder, in this paper, we suggest an efficient two-point class of methods, when the multiplicity of the root is known. The theoretical aspects are investigated and show that each member of the contributed class achieves fourth-order convergence by using three functional evaluations per full cycle. We also employ numerical examples to evaluate the accuracy of the proposed methods by comparison with other existing methods. For functions with finitely many real roots in an interval, relatively little literature is known, while in applications, the users wish to find all the real zeros at the same time. Hence, the second aim of this paper will be presented by designing a fourth-order algorithm, based on the developed methods, to find all the real solutions of a nonlinear equation in an interval using the programming package Mathematica 8.

Keywords

• Multiplicity
• Two-step methods
• Mathematica
• All the real solutions
• Finitely many zeros