An Optimal Iterative Technique for Multiple Root Finder of Nonlinear Problems

Abstract

In this paper, an optimal higher-order iterative technique to approximate the multiple roots of a nonlinear equation has been presented. The proposed technique has special properties: a two-point method that does not involve any derivatives, has an optimal convergence of fourth-order, is cost-effective, is more stable, and has better numerical results. In addition to this, we adopt the weight function approach at both substeps (which provide us with a more general form of two-point methods). Firstly, the convergence order is studied for multiplicity m=2,3 by Taylor’s series expansion and then general convergence for m≥4 is proved. We have demonstrated the applicability of our methods to six numerical problems. Out of them: the first one is the well-known Van der Waals ideal gas problem, the second one is used to study the blood rheology model, the third one is chosen from the linear algebra (namely, eigenvalue), and the remaining three are academic problems. We concluded on the basis of obtained CPU timing, computational order of convergence, and absolute errors between two consecutive iterations for which our methods illustrate better results as compared to earlier studies.

Keywords

• Kung–Traub conjecture
• nonlinear equations
• Newton’s method
• multiple roots