scholarly journals A biparametric family of four-step sixteenth-order root-finding methods with the optimal efficiency index

2011 ◽  
Vol 24 (8) ◽  
pp. 1336-1342 ◽  
Author(s):  
Young Hee Geum ◽  
Young Ik Kim
Keyword(s):  
Efficiency Index ◽  
Root Finding ◽  
Order Root ◽  
Optimal Efficiency

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 A New Nonlinear Ninth-Order Root-Finding Method with Error Analysis and Basins of Attraction

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1996
Author(s):  
Sania Qureshi ◽  
Higinio Ramos ◽  
Abdul Karim Soomro
Keyword(s):  
Nonlinear Models ◽  
Efficiency Index ◽  
Basins Of Attraction ◽  
Nonlinear Phenomena ◽  
Numerical Schemes ◽  
Engineering Research ◽  
Root Finding ◽  
Order Root ◽  
The Lorenz System ◽  
Root Finding Method

Nonlinear phenomena occur in various fields of science, business, and engineering. Research in the area of computational science is constantly growing, with the development of new numerical schemes or with the modification of existing ones. However, such numerical schemes, objectively need to be computationally inexpensive with a higher order of convergence. Taking into account these demanding features, this article attempted to develop a new three-step numerical scheme to solve nonlinear scalar and vector equations. The scheme was shown to have ninth order convergence and requires six function evaluations per iteration. The efficiency index is approximately 1.4422, which is higher than the Newton’s scheme and several other known optimal schemes. Its dependence on the initial estimates was studied by using real multidimensional dynamical schemes, showing its stable behavior when tested upon some nonlinear models. Based on absolute errors, the number of iterations, the number of function evaluations, preassigned tolerance, convergence speed, and CPU time (sec), comparisons with well-known optimal schemes available in the literature showed a better performance of the proposed scheme. Practical models under consideration include open-channel flow in civil engineering, Planck’s radiation law in physics, the van der Waals equation in chemistry, and the steady-state of the Lorenz system in meteorology.

scholarly journals Computational geometry as a tool for studying root-finding methods

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1019-1027 ◽  
Author(s):  
Ivan Petkovic ◽  
Lidija Rancic
Keyword(s):  
Computational Geometry ◽  
Basins Of Attraction ◽  
Dynamic Study ◽  
Third Order ◽  
Root Finding ◽  
Mathematical Visualization ◽  
Order Root ◽  
Insight Into ◽  
Basins Of Attractions

We present an efficient method from Computational geometry, a branch of computer science devoted to the study of algorithms, for mathematical visualization of a third order root solver. For many decades the quality of iterative methods for solving nonlinear equations were analyzed only by using numerical experiments. The disadvantage of this approach is the inconvenient fact that convergence behavior strictly depends on the choice of initial approximations and the structure of functions whose zeros are sought, which often makes the convergence analysis very hard and incomplete. For this reason in this paper we apply dynamic study of iterative processes relied on basins of attraction, a new and powerful methodology developed at the beginning of the 21th century. This approach provides graphic visualization of the behavior of convergent sequences and, consequently, offers considerably better insight into the quality of applied root solvers, especially into the domain of convergence. For demonstration, we present dynamic study of one parameter family of Halley?s type introduced in the first part of the paper. Characteristics of this family are discussed by basins of attractions for various values of the involved parameter. Special attention is devoted to clusters of polynomial roots, one of the most difficult problems in the topic. The analysis of the methods and presentation of basins of attractions are performed by the computer algebra system Mathematica.

scholarly journals Higher-Order Root-Finding Algorithms and Their Basins of Attraction

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Amir Naseem ◽  
M. A. Rehman ◽  
Thabet Abdeljawad
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Complex Polynomials ◽  
Convergence Criteria ◽  
Root Finding ◽  
Variational Iteration ◽  
Order Root ◽  
Iteration Technique

In this paper, we proposed and analyzed three new root-finding algorithms for solving nonlinear equations in one variable. We derive these algorithms with the help of variational iteration technique. We discuss the convergence criteria of these newly developed algorithms. The dominance of the proposed algorithms is illustrated by solving several test examples and comparing them with other well-known existing iterative methods in the literature. In the end, we present the basins of attraction using some complex polynomials of different degrees to observe the fractal behavior and dynamical aspects of the proposed algorithms.

scholarly journals An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Rajni Sharma ◽  
Janak Raj Sharma
Keyword(s):  
Computational Cost ◽  
Efficiency Index ◽  
Optimal Order ◽  
Computational Results ◽  
Order Convergence ◽  
Eighth Order ◽  
Root Finding ◽  
First Derivative ◽  
The Family ◽  
Iteration Methods

We derive a family of eighth-order multipoint methods for the solution of nonlinear equations. In terms of computational cost, the family requires evaluations of only three functions and one first derivative per iteration. This implies that the efficiency index of the present methods is 1.682. Kung and Traub (1974) conjectured that multipoint iteration methods without memory based on n evaluations have optimal order . Thus, the family agrees with Kung-Traub conjecture for the case . Computational results demonstrate that the developed methods are efficient and robust as compared with many well-known methods.

scholarly journals A New Three-Step Root-Finding Numerical Method and Its Fractal Global Behavior

2021 ◽  
Vol 5 (4) ◽  
pp. 204
Author(s):  
Asifa Tassaddiq ◽  
Sania Qureshi ◽  
Amanullah Soomro ◽  
Evren Hincal ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Nonlinear Models ◽  
Approximate Solutions ◽  
Efficiency Index ◽  
Basins Of Attraction ◽  
Systems Of Nonlinear Equations ◽  
Step Method ◽  
Root Finding ◽  
Complex Functions ◽  
Minimum Number ◽  
Increasing Demand

There is an increasing demand for numerical methods to obtain accurate approximate solutions for nonlinear models based upon polynomials and transcendental equations under both single and multivariate variables. Keeping in mind the high demand within the scientific literature, we attempt to devise a new nonlinear three-step method with tenth-order convergence while using six functional evaluations (three functions and three first-order derivatives) per iteration. The method has an efficiency index of about 1.4678, which is higher than most optimal methods. Convergence analysis for single and systems of nonlinear equations is also carried out. The same is verified with the approximated computational order of convergence in the absence of an exact solution. To observe the global fractal behavior of the proposed method, different types of complex functions are considered under basins of attraction. When compared with various well-known methods, it is observed that the proposed method achieves prespecified tolerance in the minimum number of iterations while assuming different initial guesses. Nonlinear models include those employed in science and engineering, including chemical, electrical, biochemical, geometrical, and meteorological models.

Some eighth-order root-finding three-step methods

2010 ◽  
Vol 15 (3) ◽  
pp. 536-544 ◽  
Author(s):  
Jisheng Kou ◽  
Xiuhua Wang ◽  
Yitian Li
Keyword(s):  
Eighth Order ◽  
Root Finding ◽  
Order Root
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