One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations

Abstract

In this study, we construct the one parameter optimal derivative-free iterative family to find the multiple roots of an algebraic nonlinear function. Many researchers developed the higher order iterative techniques by the use of the new function evaluation or the first-order or second-order derivative of functions to evaluate the multiple roots of a nonlinear equation. However, the evaluation of the derivative at each iteration is a cumbersome task. With this motivation, we design the second-order family without the utilization of the derivative of a function and without the evaluation of the new function. The proposed family is optimal as it satisfies the convergence order of Kung and Traub’s conjecture. Here, we use one parameter a for the construction of the scheme, and for a=1, the modified Traub method is its a special case. The order of convergence is analyzed by Taylor’s series expansion. Further, the efficiency of the suggested family is explored with some numerical tests. The obtained results are found to be more efficient than earlier schemes. Moreover, the basin of attraction of the proposed and earlier schemes is also analyzed.

Derivative-Free Iterative Schemes for Multiple Roots of Nonlinear Functions

“…Finally, we also compare with a second order method proposed by Kansal et al [12], which is given by…”

“…Expression (12), is respectively denoted by (MM1), (MM2), (MM3), and (MM4) for b = 6 7 , 2 3 , 3 4 , and 5 6 . These values of parameter b are the best for the numerical results, as claimed by Kansal et al in [12].…”

Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

“…Example 4. Now, we considered the problem of a continuous stirred-tank reactor [26] shown in Figure 1. Here, components A and R are fed to the reactor at rates Q and q − Q, respectively.…”

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

“…It is noteworthy that most of the higher order optimal methods that were obtained during the past years require derivative evaluation of the involved function, as in [1], while higher order derivativefree methods exist rarely in literature because it is a difficult task to preserve the order of convergence when we replace derivatives by differences. Kansal et al [7] and Kumar et al [8] gave second order iterative schemes to find repeated roots of nonlinear equations. Sharma et al [17], Kumar et al [9,10], Behl et al [2] and Rani and Kansal [13] proposed a fourth-order root finding methods for multiple roots.…”