Effective Root-Finding Methods for Nonlinear Equations Based on Multiplicative Calculi

Abstract

In recent studies, papers related to the multiplicative based numerical methods demonstrate applicability and efficiency of these methods. Numerical root-finding methods are essential for nonlinear equations and have a wide range of applications in science and engineering. Therefore, the idea of root-finding methods based on multiplicative and Volterra calculi is self-evident. Newton-Raphson, Halley, Broyden, and perturbed root-finding methods are used in numerical analysis for approximating the roots of nonlinear equations. In this paper, Newton-Raphson methods and consequently perturbed root-finding methods are developed in the frameworks of multiplicative and Volterra calculi. The efficiency of these proposed root-finding methods is exposed by examples, and the results are compared with some ordinary methods. One of the striking results of the proposed method is that the rate of convergence for many problems are considerably larger than the original methods.

An efficient technique to solve nonlinear equations using multiplicative calculus

“…We compare our new methods (16) (*VIM), (17) (*Halley), and (19) (*VIM2) with (*NM) [25], (18), and the following methods:…”

“…Method 2.2 is the multiplicative Newton method for solving the multiplicative nonlinear equation (6). We would like to mention that Method 2.2 was derived in [25]. 16), it lessens to the following method for g(x) = 1 :…”

“…In this section, we consider some numerical examples to demonstrate the efficiency and performance of newly developed methods; see Tables 1-3 and Figures 1-3. We compare our new methods (16) (*VIM), ( 17) (*Halley), and ( 19) (*VIM2) with (*NM) [25], (18), and the following methods:…”

Construction of a numerical method for finding the zeros of both smooth and nonsmooth functions

“…Paper [5] proposed the method of fourth order based on the Newton's method. In [6], authors built an efficient method for finding the roots based on the multiplicative computations. These methods require a derivative of the function, which, as was already mentioned, is quite difficult to calculate.…”

Nonlinear System Stability and Behavioral Analysis for Effective Implementation of Artificial Lower Limb

“…Through some specific features, the implementation of system identification regarding forward dynamics and model generation is possible. The determination of stability in the nonlinear domain [23,24] preferably nourishes the idea of the planning of artificial limb [25,26] design. Until now, it has not been reported in any open journal that lower limb system stability can be analyzed using a nonlinear stability analysis method [27,28] in the field of bio robotics.…”