An alternative analysis for the local convergence of iterative methods for multiple roots including when the multiplicity is unknown

ABSTRACT

In this paper we propose an alternative for the study of local convergence radius and the uniqueness radius for some third-order methods for multiple roots whose multiplicity is known. The main goal is to provide an alternative that avoids the use of sophisticated properties of divided differences that are used in already published papers about local convergence for multiple roots. We defined the local study by using a technique taking into consideration a bounding condition for the $(m+i)\mathrm{th}$ derivative of the function $f\left(x\right)$ with i=1,2. In the case that the method uses first and second derivative in its iterative expression and i=1 in case the method only uses first derivative. Furthermore we implement a numerical analysis in the following sense. Since the radius of local convergence for high-order methods decreases with the order, we must take into account the analysis of ITS behaviour when we introduce a new iterative method. Finally, we have used these iterative methods for multiple roots for the case where the multiplicity m is unknown, so we estimate this factor by different strategies comparing the behaviour of the corresponding estimations and how this fact affect to the original method.

KEYWORDS:

• Nonlinear equations
• iterative methods
• multiple roots
• convergence ball
• local convergence

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Nonlinear algebraic or transcendental equations

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Secretaría de Educación Superior, Ciencia, Tecnología e Innovación (Convocatoria Abierta 2015 fase II).