Optimal eighth-order multiple root finding iterative methods using bivariate weight function

Abstract

In this contribution, a novel eighth-order scheme is presented for solving nonlinear equations with multiple roots. The proposed scheme comprises of three steps with the modified Newton method as its first step followed by two weighted Newton steps involving one univariate and one bivariate function respectively. Analysis of convergence confirms that the presented scheme obtains optimal computational order of convergence. The efficiency of presented scheme is compared numerically with recent eighth-order methods. Functions like population growth problem, Newton’s beam problem, etc., have been considered for numerical experimentation. For the comparative study in the complex plane, we employed the concept of basins of attraction.

Keywords

• Multiple roots
• Eighth-order convergence
• Optimal method
• Newton method
• Nonlinear equations