scholarly journals Geometric Construction of Eighth-Order Optimal Families of Ostrowski’s Method

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Ramandeep Behl ◽  
S. S. Motsa
Keyword(s):  
Fourth Order ◽  
Weighted Average ◽  
Efficiency Index ◽  
Basins Of Attraction ◽  
Geometric Construction ◽  
The Other ◽  
Initial Value ◽  
Eighth Order ◽  
Numerical Examples ◽  
Ostrowski’S Method

Based on well-known fourth-order Ostrowski’s method, we proposed many new interesting optimal families of eighth-order multipoint methods without memory for obtaining simple roots. Its geometric construction consists in approximatingfn′atznin such a way that its average with the known tangent slopesfn′atxnandynis the same as the known weighted average of secant slopes and then we apply weight function approach. The adaptation of this strategy increases the convergence order of Ostrowski's method from four to eight and its efficiency index from 1.587 to 1.682. Finally, a number of numerical examples are also proposed to illustrate their accuracy by comparing them with the new existing optimal eighth-order methods available in the literature. It is found that they are very useful in high precision computations. Further, it is also noted that larger basins of attraction belong to our methods although the other methods are slow and have darker basins while some of the methods are too sensitive upon the choice of the initial value.

scholarly journals Design and Complex Dynamics of Potra–Pták-Type Optimal Methods for Solving Nonlinear Equations and Its Applications

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 942 ◽  
Author(s):  
Prem B. Chand ◽  
Francisco I. Chicharro ◽  
Neus Garrido ◽  
Pankaj Jain
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Complex Dynamics ◽  
Basins Of Attraction ◽  
Weight Functions ◽  
Order Method ◽  
Eighth Order ◽  
Numerical Examples ◽  
Fourth Order Method ◽  
Cubic Polynomials

In this paper, using the idea of weight functions on the Potra–Pták method, an optimal fourth order method, a non optimal sixth order method, and a family of optimal eighth order methods are proposed. These methods are tested on some numerical examples, and the results are compared with some known methods of the corresponding order. It is proved that the results obtained from the proposed methods are compatible with other methods. The proposed methods are tested on some problems related to engineering and science. Furthermore, applying these methods on quadratic and cubic polynomials, their stability is analyzed by means of their basins of attraction.

scholarly journals Some New Higher Order Weighted Newton Methods for Solving Nonlinear Equation with Applications

2019 ◽  
Vol 24 (2) ◽  
pp. 59
Author(s):  
Parimala Sivakumar ◽  
Jayakumar Jayaraman
Keyword(s):  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Efficiency Index ◽  
Higher Order ◽  
Function Technique ◽  
Function Evaluation ◽  
Step Method ◽  
Eighth Order ◽  
Additional Function

Three new iterative methods for solving scalar nonlinear equations using weight function technique are presented. The first one is a two-step fifth order method with four function evaluations which is improved from a two-step Newton’s method having same number of function evaluations. By this, the efficiency index of the new method is improved from 1.414 to 1.495. The second one is a three step method with one additional function evaluation producing eighth order accuracy with efficiency index 1.516. The last one is a new fourth order optimal two-step method with efficiency index 1.587. All these three methods are better than Newton’s method and many other equivalent higher order methods. Convergence analyses are established so that these methods have fifth, eighth and fourth order respectively. Numerical examples ascertain that the proposed methods are efficient and demonstrate better performance when compared to some equivalent and optimal methods. Seven application problems are solved to illustrate the efficiency and performance of the proposed methods.

scholarly journals An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Rajni Sharma ◽  
Ashu Bahl
Keyword(s):  
Fourth Order ◽  
Dynamical Behavior ◽  
Computational Cost ◽  
Efficiency Index ◽  
Basins Of Attraction ◽  
Optimal Order ◽  
Extraneous Fixed Points ◽  
Fourth Order Method ◽  
Jarratt Method ◽  
Better Than

We present a new fourth order method for finding simple roots of a nonlinear equation f(x)=0. In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its first derivative. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The conjugacy maps and extraneous fixed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.

scholarly journals Optimal Methods for Finding Simple and Multiple Roots of Nonlinear Equations and their Basins of Attraction

2020 ◽  
Vol 37 (1-2) ◽  
pp. 14-29
Author(s):  
Prem Bahadur Chand
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Multiple Root ◽  
Basins Of Attraction ◽  
Weight Functions ◽  
Multiple Roots ◽  
Order Method ◽  
Numerical Examples ◽  
Fourth Order Method ◽  
Simple And Multiple Roots

In this paper, using the variant of Frontini-Sormani method, some higher order methods for finding the roots (simple and multiple) of nonlinear equations are proposed. In particular, we have constructed an optimal fourth order method and a family of sixth order method for finding a simple root. Further, an optimal fourth order method for finding a multiple root of a nonlinear equation is also proposed. We have used different weight functions to a cubically convergent For ntini-Sormani method for the construction of these methods. The proposed methods are tested on numerical examples and compare the results with some existing methods. Further, we have presented the basins of attraction of these methods to understand their dynamics visually.

scholarly journals On generalized Halley-like methods for solving nonlinear equations

2019 ◽  
Vol 13 (2) ◽  
pp. 399-422
Author(s):  
Miodrag Petkovic ◽  
Ljiljana Petkovic ◽  
Beny Neta
Keyword(s):  
Computer Graphics ◽  
Nonlinear Equation ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Multiple Root ◽  
Basins Of Attraction ◽  
Third Order ◽  
Numerical Examples ◽  
Theoretical Results

Generalized Halley-like one-parameter families of order three and four for finding multiple root of a nonlinear equation are constructed and studied. This presentation is, actually, a mixture of theoretical results, algorithmic aspects, numerical experiments, and computer graphics. Starting from the proposed class of third order methods and using an accelerating procedure, we construct a new fourth order family of Halley's type. To analyze convergence behavior of two presented families, we have used two methodologies: (i) testing by numerical examples and (ii) dynamic study using basins of attraction.

scholarly journals Exponential spline approach for the solution of fourth order obstacle boundary value problems

Filomat ◽  
2012 ◽  
Vol 26 (5) ◽  
pp. 993-1004 ◽  
Author(s):  
Arshad Khan ◽  
Shilpi Bisht
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Boundary Value ◽  
Contact Problems ◽  
The Other ◽  
Numerical Illustration ◽  
Quintic Spline ◽  
Numerical Examples ◽  
Exponential Spline ◽  
Present Technique

An exponential quintic spline technique at mid knots is developed for approximating the solution of system of fourth order boundary value problems associated with obstacle, unilateral and contact problems. The present technique gives a class of methods of order two, four and six. Two numerical examples are considered for the numerical illustration of the proposed method. It is shown that the method developed in this paper is more efficient than the other finite difference, collocation and spline methods.

scholarly journals Modification of Newton-Househölder Method for Determining Multiple Roots of Unknown Multiplicity of Nonlinear Equations

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1020
Author(s):  
Syahmi Afandi Sariman ◽  
Ishak Hashim ◽  
Faieza Samat ◽  
Mohammed Alshbool
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Accuracy ◽  
Basins Of Attraction ◽  
Convergence Order ◽  
Multiple Roots ◽  
Eighth Order ◽  
Numerical Examples

In this study, we propose an extension of the modified Newton-Househölder methods to find multiple roots with unknown multiplicity of nonlinear equations. With four functional evaluations per iteration, the proposed method achieves an optimal eighth order of convergence. The higher the convergence order, the quicker we get to the root with a high accuracy. The numerical examples have shown that this scheme can compete with the existing methods. This scheme is also stable across all of the functions tested based on the graphical basins of attraction.

A HIGHLY ACCURATE AND EFFICIENT TRIGONOMETRICALLY-FITTED P-STABLE THREE-STEP METHOD FOR PERIODIC INITIAL-VALUE PROBLEMS

2006 ◽  
Vol 17 (04) ◽  
pp. 545-560
Author(s):  
ZHONGCHENG WANG ◽  
YONGMING DAI ◽  
DONGMEI WU
Keyword(s):  
Initial Value Problems ◽  
The Other ◽  
New Method ◽  
Phase Lag ◽  
Initial Value ◽  
Step Method ◽  
Eighth Order ◽  
Numerical Illustration ◽  
Even Order ◽  
Periodic Initial Value Problems

In this paper we present a new three-step method, which is a trigonometrically-fitted P-stable Obrechkoff method with phase-lag (frequency distortion) infinity. In this new method, we make use of higher-even-order derivatives including the eighth-order to increase the accuracy. On the other hand, we adopt a special structure to reduce the computational complexity of high-derivatives. The numerical illustration demonstrate that the new method has advantage in accuracy, periodic stability and efficiency.

scholarly journals Composition Methods for Dynamical Systems Separable into Three Parts

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 533 ◽  
Author(s):  
Fernando Casas ◽  
Alejandro Escorihuela-Tomàs
Keyword(s):  
Dynamical Systems ◽  
Fourth Order ◽  
Initial Value Problems ◽  
Splitting Methods ◽  
Initial Value ◽  
Optimization Criteria ◽  
Numerical Examples ◽  
Composition Methods ◽  
Improved Performance ◽  
Continuous Problem

New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a way that each part is explicitly solvable. The methods are obtained by applying different optimization criteria and preserve geometric properties of the continuous problem by construction. Different numerical examples exhibit their improved performance with respect to previous splitting methods in the literature.

Sign in / Sign up

Export Citation Format

Share Document