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 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 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.

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 On Dynamics of Iterative Techniques for Nonlinear Equation with Applications in Engineering

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Mudassir Shams ◽  
Nazir Ahmad Mir ◽  
Naila Rafiq ◽  
A. Othman Almatroud ◽  
Saima Akram
Keyword(s):  
Nonlinear Equation ◽  
Numerical Test ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Optimal Order ◽  
Multiple Roots ◽  
Optimal Order Of Convergence ◽  
Single Root ◽  
Root Finding

In this article, we construct an optimal family of iterative methods for finding the single root and then extend this family for determining all the distinct as well as multiple roots of single-variable nonlinear equations simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in the case of single root finding methods and 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The computational cost, basins of attraction, efficiency, log of residual, and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in the literature.

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.

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 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 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.

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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