scholarly journals One-Point Optimal Family of Multiple Root Solvers of Second-Order

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 655 ◽  
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Clemente Cesarano
Keyword(s):  
Weight Function ◽  
Basin Of Attraction ◽  
Multiple Root ◽  
Second Order ◽  
Weight Functions ◽  
Modified Newton Method ◽  
The Family ◽  
Dynamical Nature ◽  
Iterative Functions ◽  
The Basin Of Attraction

This manuscript contains the development of a one-point family of iterative functions. The family has optimal convergence of a second-order according to the Kung-Traub conjecture. This family is used to approximate the multiple zeros of nonlinear equations, and is based on the procedure of weight functions. The convergence behavior is discussed by showing some essential conditions of the weight function. The well-known modified Newton method is a member of the proposed family for particular choices of the weight function. The dynamical nature of different members is presented by using a technique called the “basin of attraction”. Several practical problems are given to compare different methods of the presented family.

scholarly journals A Family of Multiple-Root Finding Iterative Methods Based on Weight Functions

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2194
Author(s):  
Francisco I. Chicharro ◽  
Rafael A. Contreras ◽  
Neus Garrido
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Numerical Test ◽  
Multiple Root ◽  
Initial Guess ◽  
Dynamical Analysis ◽  
Weight Functions ◽  
Root Finding ◽  
Wide Range ◽  
The Family

A straightforward family of one-point multiple-root iterative methods is introduced. The family is generated using the technique of weight functions. The order of convergence of the family is determined in its convergence analysis, which shows the constraints that the weight function must satisfy to achieve order three. In this sense, a family of iterative methods can be obtained with a suitable design of the weight function. That is, an iterative algorithm that depends on one or more parameters is designed. This family of iterative methods, starting with proper initial estimations, generates a sequence of approximations to the solution of a problem. A dynamical analysis is also included in the manuscript to study the long-term behavior of the family depending on the parameter value and the initial guess considered. This analysis reveals the good properties of the family for a wide range of values of the parameter. In addition, a numerical test on academic and engineering multiple-root functions is performed.

ON THE PROBLEM OF RESTORING A FUNCTION IN THREE DIMENSIONAL SPACE BY THE FAMILY OF CONES

2021 ◽  
Vol 10 (11) ◽  
pp. 3505-3513
Author(s):  
Z.Kh. Ochilov ◽  
M.I. Muminov
Keyword(s):  
Weight Function ◽  
Exact Solutions ◽  
Special Form ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Weight Functions ◽  
The Family ◽  
The Given ◽  
Three Dimensional Space

In this paper, we consider the problem of recovering a function in three-dimensional space from a family of cones with a weight function of a special form. Exact solutions of the problem are obtained for the given weight functions. A class of parameters for the problem that has no solution is constructed.

scholarly journals Capture Zones of the Family of Functions λzmexp(z)

2003 ◽  
Vol 13 (09) ◽  
pp. 2623-2640 ◽  
Author(s):  
Núria Fagella ◽  
Antonio Garijo
Keyword(s):  
Fixed Point ◽  
Critical Point ◽  
Basin Of Attraction ◽  
Capture Zone ◽  
Connected Component ◽  
Dynamical Plane ◽  
Capture Zones ◽  
The Family ◽  
Parameter Values ◽  
The Basin Of Attraction

We consider the family of entire transcendental maps given by Fλ,m(z)=λzm exp (z) where m≥2. All functions Fλ,m have a superattracting fixed point at z=0, and a critical point at z = -m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behavior, i.e. λ values such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then nonlocally connected. However, there are parameter values for which the boundary of the immediate basin of z=0 is a quasicircle.

scholarly journals One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2223
Author(s):  
Munish Kansal ◽  
Ali Saleh Alshomrani ◽  
Sonia Bhalla ◽  
Ramandeep Behl ◽  
Mehdi Salimi
Keyword(s):  
Basin Of Attraction ◽  
Nonlinear Function ◽  
Second Order ◽  
Function Evaluation ◽  
Multiple Roots ◽  
Derivative Free ◽  
The One ◽  
Special Case ◽  
The Basin Of Attraction ◽  
Taylor’S Series

In this study, we construct the one parameter optimal derivative-free iterative family to find the multiple roots of an algebraic nonlinear function. Many researchers developed the higher order iterative techniques by the use of the new function evaluation or the first-order or second-order derivative of functions to evaluate the multiple roots of a nonlinear equation. However, the evaluation of the derivative at each iteration is a cumbersome task. With this motivation, we design the second-order family without the utilization of the derivative of a function and without the evaluation of the new function. The proposed family is optimal as it satisfies the convergence order of Kung and Traub’s conjecture. Here, we use one parameter a for the construction of the scheme, and for a=1, the modified Traub method is its a special case. The order of convergence is analyzed by Taylor’s series expansion. Further, the efficiency of the suggested family is explored with some numerical tests. The obtained results are found to be more efficient than earlier schemes. Moreover, the basin of attraction of the proposed and earlier schemes is also analyzed.

scholarly journals Strongly self-inverse weighted graphs

2020 ◽  
Vol 36 (36) ◽  
pp. 80-89
Author(s):  
Abraham Berman ◽  
Naomi Shaked-Monderer ◽  
Swarup Kumar Panda
Keyword(s):  
Weight Function ◽  
Bipartite Graph ◽  
Weighted Graph ◽  
Perfect Matchings ◽  
Weight Functions ◽  
Weighted Graphs ◽  
Positive Weight ◽  
The Family

Let G be a connected, bipartite graph. Let Gw denote the weighted graph obtained from G by assigning weights to its edges using the positive weight function w : E(G) ! (0;1). In this article we consider a class Hnmc of bipartite graphswith unique perfect matchings and the family WG of weight functions with weight 1 on the matching edges, and characterize all pairs G in Hnmc and w in WG such that Gw is strongly self-inverse.

scholarly journals Optimization and Stability of Heat Engines: The Role of Entropy Evolution

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 865 ◽  
Author(s):  
Julian Gonzalez-Ayala ◽  
Moises Santillán ◽  
Maria Santos ◽  
Antonio Calvo Hernández ◽  
José Mateos Roco
Keyword(s):  
Local Stability ◽  
Production Efficiency ◽  
Basin Of Attraction ◽  
Heat Engine ◽  
Time Constraints ◽  
Thermal Gradients ◽  
Heat Engines ◽  
Low Dissipation ◽  
The Basin Of Attraction

Local stability of maximum power and maximum compromise (Omega) operation regimes dynamic evolution for a low-dissipation heat engine is analyzed. The thermodynamic behavior of trajectories to the stationary state, after perturbing the operation regime, display a trade-off between stability, entropy production, efficiency and power output. This allows considering stability and optimization as connected pieces of a single phenomenon. Trajectories inside the basin of attraction display the smallest entropy drops. Additionally, it was found that time constraints, related with irreversible and endoreversible behaviors, influence the thermodynamic evolution of relaxation trajectories. The behavior of the evolution in terms of the symmetries of the model and the applied thermal gradients was analyzed.

scholarly journals Time Delay and Feedback Control of an Inverted Pendulum With Stick Slip Friction

2007 ◽  
Author(s):  
Sue Ann Campbell ◽  
Stephanie Crawford ◽  
Kirsten Morris
Keyword(s):  
Time Delay ◽  
Basin Of Attraction ◽  
Critical Time ◽  
Viscous Friction ◽  
Experimental System ◽  
Friction Model ◽  
Stable Equilibrium Point ◽  
Feedback Controllers ◽  
Stick Slip ◽  
The Basin Of Attraction

We consider an experimental system consisting of a pendulum, which is free to rotate 360 degrees, attached to a cart which can move in one dimension. There is stick slip friction between the cart and the track on which it moves. Using two different models for this friction we design feedback controllers to stabilize the pendulum in the upright position. We show that controllers based on either friction model give better performance than one based on a simple viscous friction model. We then study the effect of time delay in this controller, by calculating the critical time delay where the system loses stability and comparing the calculated value with experimental data. Both models lead to controllers with similar robustness with respect to delay. Using numerical simulations, we show that the effective critical time delay of the experiment is much less than the calculated theoretical value because the basin of attraction of the stable equilibrium point is very small.

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