An Efficient Derivative Free One-Point Method with Memory for Solving Nonlinear Equations

Janak Raj Sharma,Sunil Kumar,Clemente Cesarano

doi:10.3390/math7070604

Janak Raj Sharma, Sunil Kumar + Show 1 more

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https://doi.org/10.3390/math7070604

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Abstract

We propose a derivative free one-point method with memory of order 1.84 for solving nonlinear equations. The formula requires only one function evaluation and, therefore, the efficiency index is also 1.84. The methodology is carried out by approximating the derivative in Newton’s iteration using a rational linear function. Unlike the existing methods of a similar nature, the scheme of the new method is easy to remember and can also be implemented for systems of nonlinear equations. The applicability of the method is demonstrated on some practical as well as academic problems of a scalar and multi-dimensional nature. In addition, to check the efficacy of the new technique, a comparison of its performance with the existing techniques of the same order is also provided.

Highlights

In this study, we consider the problem of solving the nonlinear equations F ( x ) = 0; whereinF : D ⊂ Rm → Rm is a univariate function when m = 1 or multivariate function when m > 1 on an open domain D, by iterative methods
Newton’s method [1,2,3] is one of the basic one-point methods which has quadratic convergence and requires one function and one derivative evaluation per iteration but it may diverge if the derivative is very small or zero
Researchers have proposed some derivative free one-point methods, for example, the Secant method [2], the Traub method [2], the Muller method [4,5], the Jarratt and Nudds method [6] and the Sharma method [7]. These methods are classified as one-point methods with memory whereas Newton’s method is a one-point method without memory

Summary

Introduction

Newton’s method [1,2,3] is one of the basic one-point methods which has quadratic convergence and requires one function and one derivative evaluation per iteration but it may diverge if the derivative is very small or zero. To overcome this problem, researchers have proposed some derivative free one-point methods, for example, the Secant method [2], the Traub method [2], the Muller method [4,5], the Jarratt and Nudds method [6] and the Sharma method [7]. All the above mentioned one-point methods with memory require one function evaluation per iteration and possess order of convergence 1.84 except Secant which has order 1.62

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