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.

scholarly journals Fixed Point Root-Finding Methods of Fourth-Order of Convergence

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 769 ◽  
Author(s):  
Alicia Cordero ◽  
Lucía Guasp ◽  
Juan R. Torregrosa
Keyword(s):  
Weight Function ◽  
Dynamical Behavior ◽  
Nonlinear Problems ◽  
Dynamical Analysis ◽  
Function Technique ◽  
Weight Functions ◽  
Good Tool ◽  
Quadratic Polynomials ◽  
Root Finding ◽  
The Stability

In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub’s conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties.

scholarly journals An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 672 ◽  
Author(s):  
Saima Akram ◽  
Fiza Zafar ◽  
Nusrat Yasmin
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Multiple Root ◽  
Multiple Roots ◽  
Order Convergence ◽  
Eighth Order ◽  
Root Finding ◽  
New Family ◽  
Two Parameters ◽  
Theoretical Results

In this paper, we introduce a new family of efficient and optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity ( m ≥ 1 ) . We use the weight function approach involving one and two parameters to develop the new family. A comprehensive convergence analysis is studied to demonstrate the optimal eighth-order convergence of the suggested scheme. Finally, numerical and dynamical tests are presented, which validates the theoretical results formulated in this paper and illustrates that the suggested family is efficient among the domain of multiple root finding methods.

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.

Numerical solutions and analysis of diffusion for new generalized fractional Burgers equation

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Yufeng Xu ◽  
Om Agrawal
Keyword(s):  
Finite Difference ◽  
Diffusion Process ◽  
Weight Function ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Weight Functions ◽  
Linear Recurrence ◽  
Scale Function ◽  
Special Cases ◽  
Wide Range

AbstractIn this paper, numerical solutions of Burgers equation defined by using a new Generalized Time-Fractional Derivative (GTFD) are discussed. The numerical scheme uses a finite difference method. The new GTFD is defined using a scale function and a weight function. Many existing fractional derivatives are the special cases of it. A linear recurrence relationship for the numerical solutions of the resulting system of linear equations is found via finite difference approach. Burgers equations with different fractional orders and coefficients are computed which show that this numerical method is simple and effective, and is capable of solving the Burgers equation accurately for a wide range of viscosity values. Furthermore, we study the influence of the scale and the weight functions on the diffusion process of Burgers equation. Numerical simulations illustrate that a scale function can stretch or contract the diffusion on the time domain, while a weight function can change the decay velocity of the diffusion process.

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 On iterative techniques for estimating all roots of nonlinear equation and its system with application in differential equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mudassir Shams ◽  
Naila Rafiq ◽  
Nasreen Kausar ◽  
Praveen Agarwal ◽  
Choonkil Park ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basin Of Attraction ◽  
Numerical Test ◽  
Computational Cost ◽  
System Of Nonlinear Equations ◽  
Single Root ◽  
Root Finding

AbstractIn this article, we construct a family of iterative methods for finding a single root of nonlinear equation and then generalize this family of iterative methods for determining all roots of nonlinear equations simultaneously. Further we extend this family of root estimating methods for solving a system of nonlinear equations. Convergence analysis shows that the order of convergence is 3 in case of the single root finding method as well as for the system of nonlinear equations and is 5 for simultaneous determination of all distinct and multiple roots of a nonlinear equation. The computational cost, basin 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 literature.

scholarly journals ON JARRATT'S FAMILY OF OPTIMAL FOURTH-ORDER ITERATIVE METHODS AND THEIR DYNAMICS

Fractals ◽  
2014 ◽  
Vol 22 (04) ◽  
pp. 1450013 ◽  
Author(s):  
Changbum Chun ◽  
Beny Neta ◽  
Sujin Kim
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Fourth Order ◽  
Optimal Methods ◽  
The Family

P. Jarratt has developed a family of fourth-order optimal methods. He suggested two members of the family. The dynamics of one of those was discussed previously. Here we show that the family can be written using a weight function and analyze all members of the family to find the best performer.

scholarly journals Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
Naila Rafiq ◽  
Saima Akram ◽  
Nazir Ahmad Mir ◽  
Mudassir Shams
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Dynamical Behavior ◽  
Numerical Test ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Single Root ◽  
Root Finding ◽  
The Stability

In this article, we first construct a family of optimal 2-step iterative methods for finding a single root of the nonlinear equation using the procedure of weight function. We then extend these methods for determining all roots simultaneously. Convergence analysis is presented for both cases to show that the order of convergence is 4 in case of the single-root finding method and is 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The dynamical behavior is presented to analyze the stability of fixed and critical points of the rational operator of one-point iterative methods. The computational cost, basins of attraction, efficiency, log of the residual, and numerical test examples show that the newly constructed methods are more efficient as compared with the existing methods in the literature.

scholarly journals DYNAMICAL ANALYSIS TO EXPLAIN THE NUMERICAL ANOMALIES IN THE FAMILY OF ERMAKOV-KALITLIN TYPE METHODS

2019 ◽  
Vol 24 (3) ◽  
pp. 335-350
Author(s):  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
Pura Vindel
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Dynamical Analysis ◽  
Test Functions ◽  
Iterative Schemes ◽  
New Family ◽  
The Family ◽  
The Stability ◽  
Appl Math ◽  
Parametric Class

In this paper, we study the dynamics of an iterative method based on the Ermakov-Kalitkin class of iterative schemes for solving nonlinear equations. As it was proven in ”A new family of iterative methods widening areas of convergence, Appl. Math. Comput.”, this family has the property of getting good estimations of the solution when Newton’s method fails. Moreover, the set of converging starting points for several non-polynomial test functions was plotted and they showed to be wider in the case of proposed methods than in Newton’s case, for small values of the parameter. Now, we make a complex dynamical analysis of this parametric class in order to justify the stability properties of this family.

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.

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