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 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.

scholarly journals A Newton method for harmonic mappings in the plane

2019 ◽  
Vol 40 (4) ◽  
pp. 2777-2801
Author(s):  
Olivier Sète ◽  
Jan Zur
Keyword(s):  
Basins Of Attraction ◽  
Harmonic Mappings ◽  
Iterative Root ◽  
Number Of Solutions ◽  
Critical Set ◽  
Complex Formulation ◽  
Root Finding ◽  
Matlab Implementation ◽  
Root Finding Method ◽  
Finding Method

Abstract We present an iterative root finding method for harmonic mappings in the complex plane, which is a generalization of Newton’s method for analytic functions. The complex formulation of the method allows an analysis in a complex variables spirit. For zeros close to poles of $f = h + \overline{g}$ we construct initial points for which the harmonic Newton iteration is guaranteed to converge. Moreover, we study the number of solutions of $f(z) = \eta $ close to the critical set of $f$ for certain $\eta \in \mathbb{C}$. We provide a MATLAB implementation of the method, and illustrate our results with several examples and numerical experiments, including phase plots and plots of the basins of attraction.

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 On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

2021 ◽  
Author(s):  
Krzysztof Gdawiec ◽  
Wiesław Kotarski ◽  
Agnieszka Lisowska
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Basins Of Attraction ◽  
Picard Iteration ◽  
Mann Iteration ◽  
Iteration Parameter ◽  
Root Finding ◽  
Base Method ◽  
Fabric Design ◽  
Root Finding Method

AbstractThere are two main aims of this paper. The first one is to show some improvement of the robust Newton’s method (RNM) introduced recently by Kalantari. The RNM is a generalisation of the well-known Newton’s root finding method. Since the base method is undefined at critical points, the RNM allows working also at such points. In this paper, we improve the RNM method by applying the Mann iteration instead of the standard Picard iteration. This leads to an essential decrease in the number of root finding steps without visible destroying the sharp boundaries among the basins of attractions presented in polynomiographs. Furthermore, we investigate visually the dynamics of the RNM with the Mann iteration together with the basins of attraction for varying Mann’s iteration parameter with the help of polynomiographs for several polynomials. The second aim of this paper is to present the intriguing polynomiographs obtained from the dynamics of the RNM with the Mann iteration under various sequences used in this iteration. The obtained polynomiographs differ considerably from the ones obtained with the RNM and are interesting from the artistic perspective. Moreover, they can easily find applications in wallpaper or fabric design.

scholarly journals Influence of Elastomeric Bearings in Tension on the Seismic Performance of Base-Isolated r.c. Buildings

2020 ◽  
Vol 11 (1) ◽  
pp. 82
Author(s):  
Fabio Mazza ◽  
Mirko Mazza
Keyword(s):  
Base Isolation ◽  
Nonlinear Models ◽  
Vertical Component ◽  
Computer Code ◽  
Nonlinear Phenomena ◽  
Isolation System ◽  
Elastomeric Bearings ◽  
Near Fault ◽  
Base Isolation System ◽  
Near Fault Earthquakes

Elastomeric bearings are commonly used in base-isolation systems to protect the structures from earthquake damages. Their design is usually developed by using nonlinear models where only the effects of shear and compressive loads are considered, but uncertainties still remain about consequences of the tensile loads produced by severe earthquakes like the near-fault ones. The present work aims to highlight the relapses of tension on the response of bearings and superstructure. To this end, three-, seven- and ten-storey r.c. framed buildings are designed in line with the current Italian seismic code, with a base-isolation system constituted of High-Damping-Rubber Bearings (HDRBs) designed for three values of the ratio between the vertical and horizontal stiffnesses. Experimental and analytical results available in literature are used to propose a unified nonlinear model of the HDRBs, including cavitation and post-cavitation of the elastomer. Nonlinear incremental dynamic analyses of the test structures are carried out using a homemade computer code, where other models of HDRBs considering only some nonlinear phenomena are implemented. Near-fault earthquakes with comparable horizontal and vertical components, prevailing horizontal component and prevailing vertical component are considered as seismic input. Numerical results highlight that a precautionary estimation of response parameters of the HDRBs is attained referring to the proposed model, while its effects on the nonlinear response of the superstructure are less conservative.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

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