New seventh and eighth order derivative free methods for solving nonlinear equations

2017 ◽  
Vol 10 (4) ◽  
pp. 103-115 ◽  
Author(s):  
Bhavna Panday ◽  
J. P. Jaiswal
Keyword(s):  
Nonlinear Equations ◽  
Order Derivative ◽  
Eighth Order ◽  
Derivative Free ◽  
Derivative Free Methods

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.

scholarly journals A New Derivative-Free Method to Solve Nonlinear Equations

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 583
Author(s):  
Beny Neta
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Equations ◽  
High Order ◽  
Order Derivative ◽  
Eighth Order ◽  
High Order Derivative ◽  
Derivative Free ◽  
Derivative Free Method

A new high-order derivative-free method for the solution of a nonlinear equation is developed. The novelty is the use of Traub’s method as a first step. The order is proven and demonstrated. It is also shown that the method has much fewer divergent points and runs faster than an optimal eighth-order derivative-free method.

scholarly journals Eighth-Order Iterative Methods without Derivatives for Solving Nonlinear Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
R. Thukral
Keyword(s):  
Nonlinear Equations ◽  
Convergence Order ◽  
Optimal Order ◽  
Eighth Order ◽  
Optimal Order Of Convergence ◽  
New Methods ◽  
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 on evaluations, could achieve optimal convergence order . 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.

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.

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