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.

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.

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.

Design and Analysis of a New Class of Derivative-Free Optimal Order Methods for Nonlinear Equations

2018 ◽  
Vol 15 (03) ◽  
pp. 1850010 ◽  
Author(s):  
Janak Raj Sharma ◽  
Ioannis K. Argyros ◽  
Deepak Kumar
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Numerical Examples ◽  
New Class ◽  
New Methods ◽  
Derivative Free ◽  
Local Convergence Analysis ◽  
Theoretical Results

We develop a general class of derivative free iterative methods with optimal order of convergence in the sense of Kung–Traub hypothesis for solving nonlinear equations. The methods possess very simple design, which makes them easy to remember and hence easy to implement. The Methodology is based on quadratically convergent Traub–Steffensen scheme and further developed by using Padé approximation. Local convergence analysis is provided to show that the iterations are locally well defined and convergent. Numerical examples are provided to confirm the theoretical results and to show the good performance of new methods.

scholarly journals A General Class of Derivative Free Optimal Root Finding Methods Based on Rational Interpolation

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Fiza Zafar ◽  
Nusrat Yasmin ◽  
Saima Akram ◽  
Moin-ud-Din Junjua
Keyword(s):  
General Class ◽  
Order Of Convergence ◽  
Rational Interpolation ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Iterative Schemes ◽  
Root Finding ◽  
New Methods ◽  
Derivative Free ◽  
Special Cases

We construct a new general class of derivative freen-point iterative methods of optimal order of convergence2n-1using rational interpolant. The special cases of this class are obtained. These methods do not need Newton’s iterate in the…first step of their iterative schemes. Numerical computations are presented to show that the new methods are efficient and can be seen as better alternates.

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 A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

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 The Convergence Ball and Error Analysis of the Relaxed Secant Method

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Rongfei Lin ◽  
Qingbiao Wu ◽  
Minhong Chen ◽  
Lu Liu
Keyword(s):  
Nonlinear Equation ◽  
Error Analysis ◽  
Error Estimate ◽  
Secant Method ◽  
Speed Of Convergence ◽  
Convergence Ball ◽  
Numerical Examples ◽  
Lipschitz Continuous ◽  
First Order ◽  
Nonlinear Equation Systems

A relaxed secant method is proposed. Radius estimate of the convergence ball of the relaxed secant method is attained for the nonlinear equation systems with Lipschitz continuous divided differences of first order. The error estimate is also established with matched convergence order. From the radius and error estimate, the relation between the radius and the speed of convergence is discussed with parameter. At last, some numerical examples are given.

scholarly journals On the Acceleration of the Multi-Level Monte Carlo Method

2015 ◽  
Vol 52 (2) ◽  
pp. 307-322 ◽  
Author(s):  
Kristian Debrabant ◽  
Andreas Röβler
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Weak Approximation ◽  
Mean Square ◽  
Optimal Order Of Convergence ◽  
Computational Costs ◽  
The Mean ◽  
Multi Level

The multi-level Monte Carlo method proposed by Giles (2008) approximates the expectation of some functionals applied to a stochastic process with optimal order of convergence for the mean-square error. In this paper a modified multi-level Monte Carlo estimator is proposed with significantly reduced computational costs. As the main result, it is proved that the modified estimator reduces the computational costs asymptotically by a factor (p / α)2 if weak approximation methods of orders α and p are applied in the case of computational costs growing with the same order as the variances decay.

Sign in / Sign up

Export Citation Format

Share Document