Abstract
We suggest a derivative-free optimal method of second order which is a new version of a modification of Newton’s method for achieving the multiple zeros of nonlinear single variable functions. Iterative methods without derivatives for multiple zeros are not easy to obtain, and hence such methods are rare in literature. Inspired by this fact, we worked on a family of optimal second order derivative-free methods for multiple zeros that require only two function evaluations per iteration. The stability of the methods was validated through complex geometry by drawing basins of attraction. Moreover, applicability of the methods is demonstrated herein on different functions. The study of numerical results shows that the new derivative-free methods are good alternatives to the existing optimal second-order techniques that require derivative calculations.
Highlights
Finding the roots of nonlinear functions is a significant problem in numerical mathematics and has numerous applications in different parts of applied sciences [1,2]
There are many iterative methods of different order of convergence to approximate the multiple roots of g( x ) = 0
The motivation for developing higher order methods is closely related to the Kung–Traub conjecture [20] that establishes an upper bound for the order of convergence to be attained with a specific number of functional evaluations, ρc ≤ 2μ−1, where ρc is order of convergence and μ represents functions’ evaluations
Summary
Finding the roots of nonlinear functions is a significant problem in numerical mathematics and has numerous applications in different parts of applied sciences [1,2]. There are many iterative methods of different order of convergence to approximate the multiple roots of g( x ) = 0, (see, for example [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]). Such methods require the evaluations of derivatives of either first order or first and second order. The methods which follow the Kung–Traub conjecture are called optimal methods
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