scholarly journals A Class of Three-Step Derivative-Free Root Solvers with Optimal Convergence Order

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
F. Soleymani ◽  
S. Karimi Vanani ◽  
M. Jamali Paghaleh
Keyword(s):  
Efficiency Index ◽  
The Novel ◽  
Step Cycle ◽  
Eighth Order ◽  
Derivative Free ◽  
Order Root ◽  
Multipoint Iterative Methods ◽  
Optimal Efficiency ◽  
Scalar Equations ◽  
Better Than

A class of three-step eighth-order root solvers is constructed in this study. Our aim is fulfilled by using an interpolatory rational function in the third step of a three-step cycle. Each method of the class reaches the optimal efficiency index according to the Kung-Traub conjecture concerning multipoint iterative methods without memory. Moreover, the class is free from derivative calculation per full iteration, which is important in engineering problems. One method of the class is established analytically. To test the derived methods from the class, we apply them to a lot of nonlinear scalar equations. Numerical examples suggest that the novel class of derivative-free methods is better than the existing methods of the same type in the literature.

scholarly journals Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
F. Soleymani ◽  
S. Shateyi
Keyword(s):  
Optimization Problems ◽  
Efficiency Index ◽  
Fixed Number ◽  
Computational Science ◽  
Optimal Order ◽  
Objective Functions ◽  
Eighth Order ◽  
Numerical Tests ◽  
Derivative Free ◽  
Optimal Efficiency

Optimization problems defined by (objective) functions for which derivatives are unavailable or available at an expensive cost are emerging in computational science. Due to this, the main aim of this paper is to attain as high as possible of local convergence order by using fixed number of (functional) evaluations to find efficient solvers for one-variable nonlinear equations, while the procedure to achieve this goal is totally free from derivative. To this end, we consider the fourth-order uniparametric family of Kung and Traub to suggest and demonstrate two classes of three-step derivative-free methods using only four pieces of information per full iteration to reach the optimal order eight and the optimal efficiency index 1.682. Moreover, a large number of numerical tests are considered to confirm the applicability and efficiency of the produced methods from the new classes.

scholarly journals Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Alicia Cordero ◽  
Moin-ud-Din Junjua ◽  
Juan R. Torregrosa ◽  
Nusrat Yasmin ◽  
Fiza Zafar
Keyword(s):  
Iterative Methods ◽  
High Efficiency ◽  
Nonlinear Function ◽  
Efficiency Index ◽  
Simple Zero ◽  
Eighth Order ◽  
New Family ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results

We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear function. Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-memory class. The extension of new family to the with-memory one is also presented which attains the convergence order 15.5156 and a very high efficiency index 15.51561/4≈1.9847. Some particular schemes of the with-memory family are also described. Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.

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 Solving Nondifferentiable Nonlinear Equations by New Steffensen-Type Iterative Methods with Memory

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
J. P. Jaiswal
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
The Self ◽  
Function Evaluation ◽  
Eighth Order ◽  
Derivative Free ◽  
With Memory

It is attempted to present two derivative-free Steffensen-type methods with memory for solving nonlinear equations. By making use of a suitable self-accelerator parameter in the existing optimal fourth- and eighth-order without memory methods, the order of convergence has been increased without any extra function evaluation. Therefore, its efficiency index is also increased, which is the main contribution of this paper. The self-accelerator parameters are estimated using Newton’s interpolation. To show applicability of the proposed methods, some numerical illustrations are presented.

A Variant of Newton Method with Eighth-Order Convergence for Solving Nonlinear Equations

2012 ◽  
Vol 490-495 ◽  
pp. 51-55
Author(s):  
Liang Fang
Keyword(s):  
Nonlinear Equations ◽  
Newton Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Eighth Order ◽  
Second Derivatives ◽  
Better Than

In this paper, we present a variant of Newton method with order of convergence eight for solving nonlinear equations. The method is free from second derivatives. It requires three evaluations of the functions and two evaluations of derivatives in each step. Therefore the efficiency index of the presented method is 1.5157 which is better than that of classical Newton’s method 1.4142. Some numerical experiments illustrate that the proposed method is more efficient and performs better than classical Newton's method.

An Efficient Sixth-Order Convergent Newton-Type Iterative Method for Nonlinear Equations

2012 ◽  
Vol 220-223 ◽  
pp. 2585-2588
Author(s):  
Zhong Yong Hu ◽  
Fang Liang ◽  
Lian Zhong Li ◽  
Rui Chen
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Efficiency Index ◽  
Sixth Order ◽  
Type Method ◽  
Second Derivatives ◽  
Better Than

In this paper, we present a modified sixth order convergent Newton-type method for solving nonlinear equations. It is free from second derivatives, and requires three evaluations of the functions and two evaluations of derivatives per iteration. Hence the efficiency index of the presented method is 1.43097 which is better than that of classical Newton’s method 1.41421. Several results are given to illustrate the advantage and efficiency the algorithm.

A Set-theoretic Approach to Reasoning Services for the Description Logic 𝒟 ℒ D 4,×

2020 ◽  
Vol 176 (3-4) ◽  
pp. 349-384
Author(s):  
Domenico Cantone ◽  
Marianna Nicolosi-Asmundo ◽  
Daniele Francesco Santamaria
Keyword(s):  
Description Logic ◽  
Description Logics ◽  
Theoretic Approach ◽  
The Novel ◽  
Data Types ◽  
Left Hand ◽  
Rule Languages ◽  
Tableau System ◽  
High Scalability ◽  
Better Than

In this paper we consider the most common TBox and ABox reasoning services for the description logic 𝒟ℒ〈4LQSR,x〉(D) ( 𝒟 ℒ D 4,× , for short) and prove their decidability via a reduction to the satisfiability problem for the set-theoretic fragment 4LQSR. 𝒟 ℒ D 4,× is a very expressive description logic. It combines the high scalability and efficiency of rule languages such as the SemanticWeb Rule Language (SWRL) with the expressivity of description logics. In fact, among other features, it supports Boolean operations on concepts and roles, role constructs such as the product of concepts and role chains on the left-hand side of inclusion axioms, role properties such as transitivity, symmetry, reflexivity, and irreflexivity, and data types. We further provide a KE-tableau-based procedure that allows one to reason on the main TBox and ABox reasoning tasks for the description logic 𝒟 ℒ D 4,× . Our algorithm is based on a variant of the KE-tableau system for sets of universally quantified clauses, where the KE-elimination rule is generalized in such a way as to incorporate the γ-rule. The novel system, called KEγ-tableau, turns out to be an improvement of the system introduced in [1] and of standard first-order KE-tableaux [2]. Suitable benchmark test sets executed on C++ implementations of the three mentioned systems show that in several cases the performances of the KEγ-tableau-based reasoner are up to about 400% better than the ones of the other two systems.

scholarly journals Design of a Novel Six-Axis Wrist Force Sensor

Sensors ◽  
2018 ◽  
Vol 18 (9) ◽  
pp. 3120 ◽  
Author(s):  
Shanshan Hu ◽  
Huaiyang Wang ◽  
Yong Wang ◽  
Zhengshi Liu
Keyword(s):  
Simulation Analysis ◽  
Dynamic Performance ◽  
Force Sensor ◽  
The Novel ◽  
Element Analysis ◽  
Body Design ◽  
Design Idea ◽  
Bridge Circuits ◽  
Design Ideas ◽  
Better Than

A novel elastic body design idea of six-axis wrist force sensor with a floating beam was raised based on the analysis of the robot six-axis wrist force sensor with a floating beam. The design ideas improve the sensor’s dynamic performance significantly, while not reducing its sensitivity. First, the design ideas were described in detail, which were analyzed by mechanical modeling and were verified by finite element analysis. Second, the static simulation analysis of the novel elastomer of sensor was carried out. According to the strain distribution performance, the position of the strain gauges pasted and the connection mode of the full-bridge circuits were decided, which can achieve theoretical decoupling. Finally, the comparison between the static and dynamic performance of the novel sensor and the original sensor with floating beams was done. The results show that the static and dynamic performance of the novel six-axis wrist sensor are all better than the original sensor.

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