scholarly journals Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 766 ◽  
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Basins Of Attraction ◽  
Higher Order ◽  
Order Derivative ◽  
Multiple Roots ◽  
Nonlinear Functions ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Derivative Free ◽  
Graphical Tool ◽  
Derivative Free Methods

A number of higher order iterative methods with derivative evaluations are developed in literature for computing multiple zeros. However, higher order methods without derivative for multiple zeros are difficult to obtain and hence such methods are rare in literature. Motivated by this fact, we present a family of eighth order derivative-free methods for computing multiple zeros. Per iteration the methods require only four function evaluations, therefore, these are optimal in the sense of Kung-Traub conjecture. Stability of the proposed class is demonstrated by means of using a graphical tool, namely, basins of attraction. Boundaries of the basins are fractal like shapes through which basins are symmetric. Applicability of the methods is demonstrated on different nonlinear functions which illustrates the efficient convergence behavior. Comparison of the numerical results shows that the new derivative-free methods are good competitors to the existing optimal eighth-order techniques which require derivative evaluations.

scholarly journals An Efficient Class of Traub–Steffensen-Type Methods for Computing Multiple Zeros

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 65 ◽  
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Clemente Cesarano
Keyword(s):  
Iterative Methods ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Order Derivative ◽  
Third Order ◽  
Higher Order Methods ◽  
Multiple Zeros ◽  
Derivative Free ◽  
Graphical Tool ◽  
Numerical Experimentation

Numerous higher-order methods with derivative evaluations are accessible in the literature for computing multiple zeros. However, higher-order methods without derivatives are very rare for multiple zeros. Encouraged by this fact, we present a family of third-order derivative-free iterative methods for multiple zeros that require only evaluations of three functions per iteration. Convergence of the proposed class is demonstrated by means of using a graphical tool, namely basins of attraction. Applicability of the methods is demonstrated through numerical experimentation on different functions that illustrates the efficient behavior. Comparison of numerical results shows that the presented iterative methods are good competitors to the existing techniques.

scholarly journals Optimal One-Point Iterative Function Free from Derivatives for Multiple Roots

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 709 ◽  
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Ioannis K. Argyros
Keyword(s):  
Complex Geometry ◽  
Basins Of Attraction ◽  
Second Order ◽  
Multiple Roots ◽  
Single Variable ◽  
Optimal Method ◽  
Multiple Zeros ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
The Stability

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.

An efficient class of fourth-order derivative-free method for multiple-roots

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sunil Kumar ◽  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Ioannis K. Argyros
Keyword(s):  
Multiple Root ◽  
Second Step ◽  
Optimal Order ◽  
Multiple Roots ◽  
Nonlinear Functions ◽  
New Methods ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Derivative Free Method ◽  
Good Number

Abstract Many optimal order multiple root techniques, which use derivatives in the algorithm, have been proposed in literature. Many researchers tried to construct an optimal family of derivative-free methods for multiple roots, but they did not get success in this direction. With this as a motivation factor, here, we present a new optimal class of derivative-free methods for obtaining multiple roots of nonlinear functions. This procedure involves Traub–Steffensen iteration in the first step and Traub–Steffensen-like iteration in the second step. Efficacy is checked on a good number of relevant numerical problems that verifies the efficient convergent nature of the new methods. Moreover, we find that the new derivative-free methods are just as competent as the other existing robust methods that use derivatives.

scholarly journals Generating Optimal Eighth Order Methods for Computing Multiple Roots

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1947
Author(s):  
Deepak Kumar ◽  
Sunil Kumar ◽  
Janak Raj Sharma ◽  
Matteo d’Amore
Keyword(s):  
Real Life ◽  
Nonlinear Function ◽  
Basins Of Attraction ◽  
Computational Time ◽  
Multiple Roots ◽  
Time Stability ◽  
Academic Problems ◽  
Eighth Order ◽  
Multiple Zeros ◽  
New Family

There are a few optimal eighth order methods in literature for computing multiple zeros of a nonlinear function. Therefore, in this work our main focus is on developing a new family of optimal eighth order iterative methods for multiple zeros. The applicability of proposed methods is demonstrated on some real life and academic problems that illustrate the efficient convergence behavior. It is shown that the newly developed schemes are able to compete with other methods in terms of numerical error, convergence and computational time. Stability is also demonstrated by means of a pictorial tool, namely, basins of attraction that have the fractal-like shapes along the borders through which basins are symmetric.

scholarly journals New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Rajinder Thukral
Keyword(s):  
Nonlinear Equations ◽  
Convergence Order ◽  
Order Derivative ◽  
Optimal Order ◽  
Eighth Order ◽  
Optimal Order Of Convergence ◽  
New Family ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Iteration Methods

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based onnevaluations could achieve optimal convergence order of . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented.

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