Modified Optimal Class of Newton-Like Fourth-Order Methods for Multiple Roots

Munish Kansal,Ramandeep Behl,Mohammed Ali A Mahnashi,Fouad Othman Mallawi

doi:10.3390/sym11040526

Abstract

Here, we propose optimal fourth-order iterative methods for approximating multiple zeros of univariate functions. The proposed family is composed of two stages and requires 3 functional values at each iteration. We also suggest an extensive convergence analysis that demonstrated the establishment of fourth-order convergence of the developed methods. It is interesting to note that some existing schemes are found to be the special cases of our proposed scheme. Numerical experiments have been performed on a good number of problems arising from different disciplines such as the fractional conversion problem of a chemical reactor, continuous stirred tank reactor problem, and Planck’s radiation law problem. Computational results demonstrates that suggested methods are better and efficient than their existing counterparts.

Highlights

Importance of solving nonlinear problems is justified by numerous physical and technical applications over the past decades
We are interested in presenting a new optimal class of parametric-based iterative methods having fourth-order convergence which exploit weight function technique for computing multiple zeros
We proposed a wide general optimal class of iterative methods for approximating multiple zeros of nonlinear functions numerically

Summary

Introduction

Importance of solving nonlinear problems is justified by numerous physical and technical applications over the past decades. When we discuss about iterative solvers for obtaining multiple roots with known multiplicity m ≥ 1 of scalar equations of the type g( x ) = 0, where g : D ⊆ R → R, modified Newton’s technique [1,2] ( known as Rall’s method) is the most popular and classical iterative scheme, which is defined by x s +1 = x s − m g( xs ). In 2013, Zhou et al [13], presented a family of 4-order optimal iterative methods, defined as follows: g( x ). We are interested in presenting a new optimal class of parametric-based iterative methods having fourth-order convergence which exploit weight function technique for computing multiple zeros. It is interesting to note that the optimal fourth-order families (5) and (6) can be considered as special cases of our scheme for some particular values of free parameters. The new scheme can be treated as more general family for approximating multiple zeros of nonlinear functions.

Construction of the Family

Numerical Experiments

Methods n xs

Conclusions