scholarly journals New Modification of Behl's Method Free from Second Derivative with an Optimal Order of Convergence

2019 ◽  
Vol 1 (2) ◽  
Author(s):  
Wartono Wartono ◽  
Revia Agustiwari ◽  
Rahmawati Rahmawati
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Efficiency Index ◽  
Optimal Order ◽  
Second Derivative ◽  
Third Order ◽  
Optimal Order Of Convergence ◽  
Steffensen’S Method ◽  
Evaluation Functions ◽  
Taylor’S Series

AbstractBehl’s method is one of the iterative methods to solve a nonlinear equation that converges cubically. In this paper, we modified the iterative method with real parameter β using second Taylor’s series expansion and reduce the second derivative of the proposed method using the equality of Chun-Kim and Newton Steffensen. The result showed that the proposed method has a fourth-order convergence for b = 0 and involves three evaluation functions per iteration with the efficiency index equal to 41/3 = 1.5874. Numerical simulation is presented for several functions to demonstrate the performance of the new method. The final results show that the proposed method has better performance as compared to some other iterative methods.Keywords: efficiency index; third-order iterative method; Chun-Kim’s method; Newton-Steffensen’s method; nonlinear equation. AbstrakMetode Behl adalah salah satu metode iterasi yang digunakan untuk menyelesaikan persamaan nonlinear dengan orde konvergensi tiga. Pada artikel ini, modifikasi terhadap metode iterasi menggunakan ekspansi deret Taylor orde dua dengan parameter β  dan turunan kedua dihilangkan menggunakan penyetaraan dari metode Chun-Kim dan Newton-Steffensen. Hasil kajian menunjukkan bahwa metode iterasi yang diusulkan memiliki orde konvergensi empat untuk b = 0 dan melibatkan tiga evaluasi fungsi setiap iterasinya dengan indeks efisiensi sebesar 41/3 = 1,5874. Simulasi numerik dilakukan terhadap beberapa fungsi untuk menunjukkan performa modifikasi metode iterasi yang diusulkan. Hasil akhir menunjukkan bahwa metode iterasi tersebut mempunyai performa lebih baik dibandingkan dengan beberapa metode iterasi lainnya.Kata kunci: indeks efisiensi; metode iterasi orde tiga; metode Chun-Kim; metode Newton- Steffensen; persamaan nonlinear.

scholarly journals Modifikasi Metode Householder Tiga Parameter yang Bebas Turunan Kedua dengan Orde Konvergensi Optimal

2021 ◽  
Vol 18 (1) ◽  
pp. 62-74
Author(s):  
Wartono ◽  
M Zulianti ◽  
Rahmawati
Keyword(s):  
Numerical Simulation ◽  
Iterative Method ◽  
Iterative Methods ◽  
Efficiency Index ◽  
New Method ◽  
Third Order ◽  
The Third ◽  
New Iterative Method ◽  
Better Than ◽  
Taylor’S Series

The Householder’s method is one of the iterative methods with a third-order convergence that used to solve a nonlinear equation. In this paper, the authors modified the iterative method using the expansion of second order Taylor’s series and approximated its second derivative using equality of two the third-order iterative methods. Based on the results of the study, it was found that the new iterative method has a fourth-order of convergence and requires three evaluations of function with an efficiency index of 1,587401. Numerical simulation is given by using several functions to compare the performance between the new method with other iterative methods. The results of numerical simulation show that the performance of the new method is better than other iterative methods.

Three-Step Method with Fifth-Order Convergence for Nonlinear Equation

2012 ◽  
Vol 524-527 ◽  
pp. 3824-3827 ◽  
Author(s):  
Li Sun ◽  
Liang Fang ◽  
Yun Wang
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Efficiency Index ◽  
Order Method ◽  
Order Convergence ◽  
Step Method ◽  
Third Order ◽  
Second Derivatives ◽  
Fifth Order ◽  
Better Than

We present a fifth-order iterative method for the solution of nonlinear equation. The new method is based on the Noor's third-order method, which is a modified Householder method without second derivatives. Its efficiency index is 1.4953 which is better than that of Newton's method and Noor's method. Numerical results show the efficiency of the proposed method.

scholarly journals A General Three-Step Class of Optimal Iterations for Nonlinear Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Fazlollah Soleymani ◽  
Solat Karimi Vanani ◽  
Abtin Afghani
Keyword(s):  
Nonlinear Equation ◽  
General Class ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Numerical Examples ◽  
First Order ◽  
Engineering Problems ◽  
Optimal Efficiency

Many of the engineering problems are reduced to solve a nonlinear equation numerically, and as a result, an especial attention to suggest efficient and accurate root solvers is given in literature. Inspired and motivated by the research going on in this area, this paper establishes an efficient general class of root solvers, where per computing step, three evaluations of the function and one evaluation of the first-order derivative are used to achieve the optimal order of convergence eight. The without-memory methods from the developed class possess the optimal efficiency index 1.682. In order to show the applicability and validity of the class, some numerical examples are discussed.

Three-Step Method with Fifth-Order Convergence for Nonlinear Equation

2013 ◽  
Vol 846-847 ◽  
pp. 1274-1277
Author(s):  
Ying Peng Zhang ◽  
Li Sun
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Efficiency Index ◽  
Newton Methods ◽  
Order Convergence ◽  
Step Method ◽  
Second Derivatives ◽  
Fifth Order ◽  
Modified Newton Methods ◽  
Better Than

We present a fifth-order iterative method for the solution of nonlinear equation. The new method is based on two ordinary methods, which are modified Newton methods without second derivatives. Its efficiency index is 1.37973 which is better than that of Newton's method. Numerical results show the efficiency of the proposed method.

scholarly journals Four-Point Optimal Sixteenth-Order Iterative Method for Solving Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Malik Zaka Ullah ◽  
A. S. Al-Fhaid ◽  
Fayyaz Ahmad
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Iterative Scheme ◽  
Optimal Order ◽  
Optimal Order Of Convergence

We present an iterative method for solving nonlinear equations. The proposed iterative method has optimal order of convergence sixteen in the sense of Kung-Traub conjecture (Kung and Traub, 1974); it means that the iterative scheme uses five functional evaluations to achieve 16(=25-1) order of convergence. The proposed iterative method utilizes one derivative and four function evaluations. Numerical experiments are made to demonstrate the convergence and validation of the iterative method.

scholarly journals Further Acceleration of the Simpson Method for Solving Nonlinear Equations

2018 ◽  
Vol 14 (2) ◽  
pp. 7631-7639
Author(s):  
Rajinder Thukral
Keyword(s):  
Nonlinear Equation ◽  
Fourth Order ◽  
Efficiency Index ◽  
New Method ◽  
Order Method ◽  
Third Order ◽  
Type Method ◽  
New Formula ◽  
Order Four ◽  
Better Than

There are two aims of this paper, firstly, we present an improvement of the classical Simpson third-order method for finding zeros a nonlinear equation and secondly, we introduce a new formula for approximating second-order derivative. The new Simpson-type method is shown to converge of the order four.  Per iteration the new method requires same amount of evaluations of the function and therefore the new method has an efficiency index better than the classical Simpson method.  We examine the effectiveness of the new fourth-order Simpson-type method by approximating the simple root of a given nonlinear equation. Numerical comparisons is made with classical Simpson method to show the performance of the presented method.

scholarly journals Optimal Derivative-Free Root Finding Methods Based on Inverse Interpolation

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 164
Author(s):  
Moin-ud-Din Junjua ◽  
Fiza Zafar ◽  
Nusrat Yasmin
Keyword(s):  
Nonlinear Equation ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Science And Engineering ◽  
Optimal Order Of Convergence ◽  
Van Der Waals Equation ◽  
First Order ◽  
Root Finding ◽  
Derivative Free ◽  
Inverse Interpolation

Finding a simple root for a nonlinear equation f ( x ) = 0 , f : I ⊆ R → R has always been of much interest due to its wide applications in many fields of science and engineering. Newton’s method is usually applied to solve this kind of problems. In this paper, for such problems, we present a family of optimal derivative-free root finding methods of arbitrary high order based on inverse interpolation and modify it by using a transformation of first order derivative. Convergence analysis of the modified methods confirms that the optimal order of convergence is preserved according to the Kung-Traub conjecture. To examine the effectiveness and significance of the newly developed methods numerically, several nonlinear equations including the van der Waals equation are tested.

scholarly journals A New Sixth-Order Steffensen-Type Iterative Method for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Nonlinear Equations ◽  
Efficiency Index ◽  
Sixth Order ◽  
New Method ◽  
Derivative Free ◽  
Better Than

Based on iterative method proposed by Basto et al. (2006), we present a new derivative-free iterative method for solving nonlinear equations. The aim of this paper is to develop a new method to find the approximation of the root α of the nonlinear equation f(x)=0. This method has the efficiency index which equals 61/4=1.5651. The benefit of this method is that this method does not need to calculate any derivative. Several examples illustrate that the efficiency of the new method is better than that of previous methods.

scholarly journals Some Third Order Methods for Solving Nonlinear Equations

2018 ◽  
Vol 13 (1) ◽  
pp. 169-174
Author(s):  
Jivandhar Jnawali
Keyword(s):  
Numerical Methods ◽  
Nonlinear Equation ◽  
Numerical Experiment ◽  
Nonlinear Equations ◽  
Efficiency Index ◽  
Subject Classification ◽  
Numerical Comparison ◽  
Single Variable ◽  
Third Order ◽  
Matlab Software

 There are several third order numerical methods having same efficiency index appeared in literature for solving nonlinear equations of a single variable. Practically, if we apply these methods in different nonlinear equations, we can observe that all methods are not performed equally for given nonlinear equations. The main objective of this paper is to show through numerical experiment that the performance of some third order methods having the same efficiency index does not perform equally for particular nonlinear equation. For the numerical comparison, we use Matlab software.2010 AMS Subject Classification: 65H05 Journal of the Institute of Engineering, 2017, 13(1): 169-174

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