Abstract
There is no doubt that there is plethora of optimal fourth-order iterative approaches available to estimate the simple zeros of nonlinear functions. We can extend these method/methods for multiple zeros but the main issue is to preserve the same convergence order. Therefore, numerous optimal and non-optimal modifications have been introduced in the literature to preserve the order of convergence. Such count of methods that can estimate the multiple zeros are limited in the scientific literature. With this point, a new optimal fourth-order scheme is presented for multiple zeros with known multiplicity. The proposed scheme is based on the weight function strategy involving functions in ratio. Moreover, the scheme is optimal as it satisfies the hypothesis of Kung–Traub conjecture. An exhaustive study of the convergence is shown to determine the fourth order of the methods under certain conditions. To demonstrate the validity and appropriateness for the proposed family, several numerical experiments have been performed. The numerical comparison highlights the effectiveness of scheme in terms of accuracy, stability, and CPU time.
Highlights
With the rapid growth of the numerical field, various physical and technical applications [1,2,3] are justifying the importance for solving the nonlinear equations
Such problems are arise in various fields of natural and physical sciences, including the heat and fluid flow problems, initial and boundary value problems, as well as problems associated with global positioning systems (GPS)
While discussing about the root finding of nonlinear equation of the form f ( x ) = 0, where f ( x ) is real function defined in a domain D ⊆ R, we pictured the classical Newton’s method and for multiple roots, the modified Newton method [4,5,6]
Summary
With the rapid growth of the numerical field, various physical and technical applications [1,2,3] are justifying the importance for solving the nonlinear equations. A board community of researchers suggested several optimal [7,8,9,10,11,12,13,14,15,16,17,18,19] and non-optimal [20,21] multipoint iterative methods for estimating the multiple zeros of a function on the basis of Kung–Traub conjecture. It is a challenging problem in the field of numerical analysis, to construct an optimal scheme of King’s family for approximating the multiple zero of a function Motivating from this idea, we have made an attempt to extend the King’s family [23] to optimal multipoint iterative method for obtaining the desire multiple zero of an input function.
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