A 4th-Order Optimal Extension of Ostrowski’s Method for Multiple Zeros of Univariate Nonlinear Functions

Abstract

We present a new optimal class of Ostrowski's method for obtaining multiple zeros of univariate nonlinear functions. Several researchers tried to construct an optimal family of Ostrowski's method for multiple zeros, but they did not have success in this direction. The new strategy adopts a weight function approach. The design structure of new families of Ostrowski's technique is simpler than the existing classical families of the same order for multiple zeros. The classical Ostrowski's method of fourth-order can obtain a particular form for the simple root. Their efficiency is checked on a good number of relevant numerical examples. These results demonstrate the performance of our methods. We find that the new methods are just as competent as other existing robust techniques available in the literature.

Modified King’s Family for Multiple Zeros of Scalar Nonlinear Functions

“…For justifying the proposed scheme in Equation ( 6), we have compared our new schemes defined in Equation (26), and Equation (30) denoted by MM1, and MM2, and compared with the existing methods defined in Equations ( 2)-( 4), denoted by LM, SM, and ZM, respectively. Moreover, the results are compared with the expression in Equation ( 5), (for equation number RM (32) of article [22]).…”

“…Recently, Behl and Hamdan [22] focused on the extension of Ostrowski's methods for finding the multiple zeros, and given as:…”

“…In literature, various researchers analyzed the variants of King's family for solving the scalar nonlinear equation with multiplicity m = 1. Recently, Behl and Hamdan [22] extended the Ostrowski's method for multiple zero of a function. Whereas Sharma and Sharma [11] focused on the Jarratt's method and modified it for computing the multiple roots.…”

An optimal derivative-free King's family for multiple zeros and its dynamics

“…, 2015; Behl et al. , 2019; Behl and Al-Hamdan, 2019). These methods can be broadly classified into two categories: methods with derivatives and methods without derivatives.…”

“…The above method (2) converges quadratically for multiple roots and requires the computation of the derivative at each step. However, many higher-order scheme based on (2) have been presented and explored in the literature (see references Hansen and Patrick, 1976;Victory and Neta, 1983;Dong, 1987;Osada, 1994;Neta, 2008;Li et al, 2009Li et al, , 2010Sharma and Sharma, 2010;Zhou et al, 2011;Sharifi et al, 2012;Soleymani et al, 2013;Geum et al, 2015;Kansal et al, 2015;Behl and Al-Hamdan, 2019). These methods can be broadly classified into two categories: methods with derivatives and methods without derivatives.…”