An optimal eighth order derivative free multiple root finding numerical method and applications to chemistry

Abstract

In this paper, we present an optimal eighth order derivative-free family of methods for multiple roots which is based on the first order divided difference and weight functions. This iterative method is a three step method with the first step as Traub–Steffensen iteration and the next two taken as Traub–Steffensen-like iteration with four functional evaluations per iteration. We compare our proposed method with the recent derivative-free methods using some chemical engineering problems modelled as nonlinear equations with simple and multiple roots. Stability of the presented family of methods is demonstrated by using the graphical tool known as basins of attraction.

Parametric Iterative Method for Addressing an Embedded-Steel Constitutive Model with Multiple Roots

“…In this section, Model (6) and different academic problems are solved using the classical Rall's scheme and the iterative class (4), with the weight function described in (8) as G (µ) = 0,…”

“…when the multiplicity is unknown. In recent years, some optimal iterative methods for solving nonlinear problems with multiple roots have appeared in the literature, such as [8][9][10][11], for cases with known multiplicity and [12], for unknown multiplicity. In most cases, the iterative expression is complicated, which increases the computational cost.…”

Stability Analysis of a New Fourth-Order Optimal Iterative Scheme for Nonlinear Equations

“…It is widely known that fixed-point iterative methods play a fundamental role in scientific disciplines such as Celestial Mechanics (see for example [1,2]), Electrical Power Systems [3], Chemistry [4], Hydraulic Engineering [5], and Civil Engineering [6]. These algorithms provide approximate solutions when exact solutions are challenging to obtain or when problems are ill-conditioned, while offering computational efficiency in terms of both time and computational resources.…”