scholarly journals A Family of Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Alicia Cordero ◽  
José L. Hueso ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa
Keyword(s):  
Fourth Order ◽  
Order Of Convergence ◽  
Nonsmooth Equations ◽  
Order Convergence ◽  
Linear Combinations ◽  
Numerical Examples ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Seventh Order ◽  
The Given

A family of derivative-free methods of seventh-order convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to 1.6266. Also, numerical examples are used to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other derivative-free methods, including some optimal fourth-order ones, in the sense of Kung-Traub’s conjecture.

scholarly journals Solving Nonsmooth Equations Using Derivative-Free Methods

2012 ◽  
Vol 3 ◽  
pp. 37-45
Author(s):  
M.S.M. Bahgat ◽  
M.A. Hafiz
Keyword(s):  
Error Analysis ◽  
Nonlinear Equations ◽  
Divided Differences ◽  
Nonsmooth Equations ◽  
Linear Combinations ◽  
Cubic Convergence ◽  
Numerical Examples ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
The Given

In this paper, a family of derivative-free methods of cubic convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to 1.442. The convergence and error analysis are given. Also, numerical examples are used to show the performance of the presented methods and to compare with other derivative-free methods. And, were applied these methods on smooth and nonsmooth equations.

scholarly journals Two Bi-Accelerator Improved with Memory Schemes for Solving Nonlinear Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
J. P. Jaiswal
Keyword(s):  
Theoretical Result ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Order Convergence ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Memory Schemes ◽  
With Memory ◽  
Interpolatory Polynomial

The present paper is devoted to the improvement of theR-order convergence of with memory derivative free methods presented by Lotfi et al. (2014) without doing any new evaluation. To achieve this aim one more self-accelerating parameter is inserted, which is calculated with the help of Newton’s interpolatory polynomial. First theoretically it is proved that theR-order of convergence of the proposed schemes is increased from 6 to 7 and 12 to 14, respectively, without adding any extra evaluation. Smooth as well as nonsmooth examples are discussed to confirm theoretical result and superiority of the proposed schemes.

scholarly journals On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1091 ◽  
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Lorentz Jäntschi
Keyword(s):  
Fourth Order ◽  
Multiple Root ◽  
Second Step ◽  
Optimal Order ◽  
Multiple Roots ◽  
Order Convergence ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Derivative Information

A number of optimal order multiple root techniques that require derivative evaluations in the formulas have been proposed in literature. However, derivative-free optimal techniques for multiple roots are seldom obtained. By considering this factor as motivational, here we present a class of optimal fourth order methods for computing multiple roots without using derivatives in the iteration. The iterative formula consists of two steps in which the first step is a well-known Traub–Steffensen scheme whereas second step is a Traub–Steffensen-like scheme. The Methodology is based on two steps of which the first is Traub–Steffensen iteration and the second is Traub–Steffensen-like iteration. Effectiveness is validated on different problems that shows the robust convergent behavior of the proposed methods. It has been proven that the new derivative-free methods are good competitors to their existing counterparts that need derivative information.

scholarly journals Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1242
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Eulalia Martínez ◽  
Majed Aali Alsulami
Keyword(s):  
Iterative Methods ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Real Life ◽  
Multiple Roots ◽  
Computational Results ◽  
Nonlinear Functions ◽  
First Order ◽  
Life Problems ◽  
Derivative Free

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

An Accelerated Iterative Method with Third-Order Convergence

2012 ◽  
Vol 220-223 ◽  
pp. 2658-2661
Author(s):  
Zhong Yong Hu ◽  
Liang Fang ◽  
Lian Zhong Li
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
New Method ◽  
Order Convergence ◽  
Third Order ◽  
Numerical Examples ◽  
Modified Newton’S Method ◽  
Jarratt Method

We present a new modified Newton's method with third-order convergence and compare it with the Jarratt method, which is of fourth-order. Based on this new method, we obtain a family of Newton-type methods, which converge cubically. Numerical examples show that the presented method can compete with Newton's method and other known third-order modifications of Newton's method.

scholarly journals Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Gustavo Fernández-Torres ◽  
Juan Vásquez-Aquino
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Classical Method ◽  
Multiple Roots ◽  
Order Convergence ◽  
Numerical Examples ◽  
Optimal Methods

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.

Semilocal Convergence of a Seventh-Order Method in Banach Spaces Under Hölder Continuity Condition

2020 ◽  
Vol 87 (1-2) ◽  
pp. 56
Author(s):  
Neha Gupta ◽  
J. P. Jaiswal
Keyword(s):  
Banach Spaces ◽  
Error Bound ◽  
Order Of Convergence ◽  
Nonlinear Operator ◽  
Lipschitz Continuity ◽  
Semilocal Convergence ◽  
Continuity Condition ◽  
Theoretical Development ◽  
Numerical Examples ◽  
Seventh Order

The motive of this article is to analyze the semilocal convergence of a well existing iterative method in the Banach spaces to get the solution of nonlinear equations. The condition, we assume that the nonlinear operator fulfills the Hölder continuity condition which is softer than the Lipschitz continuity and works on the problems in which either second order Frèchet derivative of the nonlinear operator is challenging to calculate or does not hold the Lipschitz condition. In the convergence theorem, the existence of the solution x<sup>*</sup> and its uniqueness along with prior error bound are established. Also, the <em>R</em>-order of convergence for this method is proved to be at least 4+3q. Two numerical examples are discussed to justify the included theoretical development followed by an error bound expression.

Efficient Halley-like Methods for the Inclusion of Multiple Zeros of Polynomials

2012 ◽  
Vol 12 (3) ◽  
pp. 351-366 ◽  
Author(s):  
Miodrag S. Petković ◽  
Mimica R. Milošević
Keyword(s):  
Fixed Point ◽  
Convergence Rate ◽  
Fourth Order ◽  
Interval Arithmetic ◽  
Order Of Convergence ◽  
Single Step ◽  
Zeros Of Polynomials ◽  
Numerical Examples ◽  
Multiple Zeros ◽  
Inclusion Methods

AbstractStarting from suitable zero-relation, we derive higher-order iterative methods for the simultaneous inclusion of polynomial multiple zeros in circular complex interval arithmetic. The convergence rate is increased using a family of two-point methods of the fourth order for solving nonlinear equations as a predictor. The methods are more efficient compared to existing inclusion methods for multiple zeros, based on fixed point relations. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis of the total-step and the single-step methods. The proposed self-validated methods possess a great computational efficiency since the acceleration of the convergence rate from four to seven is achieved only by a few additional calculations. To demonstrate convergence behavior of the presented methods, two numerical examples are given.

scholarly journals On the fourth order zero-finding methods for polynomials

Filomat ◽  
2003 ◽  
pp. 35-46
Author(s):  
Snezana Ilic ◽  
Lidija Rancic
Keyword(s):  
Fourth Order ◽  
Order Of Convergence ◽  
Simultaneous Approximation ◽  
New Method ◽  
Main Attention ◽  
Simple Complex ◽  
Order Zero ◽  
Numerical Examples ◽  
Convergence Behavior ◽  
Complex Zeros

The fourth order methods for the simultaneous approximation of simple complex zeros of a polynomial are considered. The main attention is devoted to a new method that may be regarded as a modification of the well known cubically convergent Ehrlich-Aberth method. It is proved that this method has the order of convergence equals four. Two numerical examples are given to demonstrate the convergence behavior of the studied methods.

scholarly journals Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
F. Soleymani
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
General Class ◽  
Analytical Study ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Numerical Examples ◽  
First Order

This paper contributes a very general class of two-point iterative methods without memory for solving nonlinear equations. The class of methods is developed using weight function approach. Per iteration, each method of the class includes two evaluations of the function and one of its first-order derivative. The analytical study of the main theorem is presented in detail to show the fourth order of convergence. Furthermore, it is discussed that many of the existing fourth-order methods without memory are members from this developed class. Finally, numerical examples are taken into account to manifest the accuracy of the derived methods.

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