scholarly journals HIGHER-ORDER FAMILIES OF MULTIPLE ROOT FINDING METHODS SUITABLE FOR NON-CONVERGENT CASES AND THEIR DYNAMICS

2019 ◽  
Vol 24 (3) ◽  
pp. 422-444 ◽  
Author(s):  
Ramandeep Behl ◽  
Vinay Kanwar ◽  
Young Ik Kim
Keyword(s):  
Multiple Root ◽  
Multiple Roots ◽  
Robust Methods ◽  
Order Convergence ◽  
Dynamical Study ◽  
Root Finding ◽  
Modified Newton’S Method ◽  
Stable Manner ◽  
Schröder’S Method ◽  
First Time

In this paper, we present many new one-parameter families of classical Rall’s method (modified Newton’s method), Schröder’s method, Halley’s method and super-Halley method for the first time which will converge even though the guess is far away from the desired root or the derivative is small in the vicinity of the root and have the same error equations as those of their original methods respectively, for multiple roots. Further, we also propose an optimal family of iterative methods of fourth-order convergence and converging to a required root in a stable manner without divergence, oscillation or jumping problems. All the methods considered here are found to be more effective than the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods.

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 An Efficient Family of Optimal Eighth-Order Multiple Root Finders

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 310 ◽  
Author(s):  
Fiza Zafar ◽  
Alicia Cordero ◽  
Juan Torregrosa
Keyword(s):  
Life Science ◽  
Real Life ◽  
Multiple Root ◽  
Dynamical Analysis ◽  
Multiple Roots ◽  
Eighth Order ◽  
Science And Engineering ◽  
New Family ◽  
Modified Newton’S Method ◽  
Numerical Performance

Finding a repeated zero for a nonlinear equation f ( x ) = 0 , f : I ⊆ R → R has always been of much interest and attention due to its wide applications in many fields of science and engineering. Modified Newton’s method is usually applied to solve this kind of problems. Keeping in view that very few optimal higher-order convergent methods exist for multiple roots, we present a new family of optimal eighth-order convergent iterative methods for multiple roots with known multiplicity involving a multivariate weight function. The numerical performance of the proposed methods is analyzed extensively along with the basins of attractions. Real life models from life science, engineering, and physics are considered for the sake of comparison. The numerical experiments and dynamical analysis show that our proposed methods are efficient for determining multiple roots of nonlinear equations.

scholarly journals Efficient Three-Step Class of Eighth-Order Multiple Root Solvers and Their Dynamics

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 837
Author(s):  
R. A. Alharbey ◽  
Munish Kansal ◽  
Ramandeep Behl ◽  
J. A. Tenreiro Machado
Keyword(s):  
General Class ◽  
Real Life ◽  
Multiple Root ◽  
Order Convergence ◽  
Eighth Order ◽  
Dynamical Study ◽  
New Methods ◽  
Life Problems ◽  
Special Cases ◽  
New Strategy

This article proposes a wide general class of optimal eighth-order techniques for approximating multiple zeros of scalar nonlinear equations. The new strategy adopts a weight function with an approach involving the function-to-function ratio. An extensive convergence analysis is performed for the eighth-order convergence of the algorithm. It is verified that some of the existing techniques are special cases of the new scheme. The algorithms are tested in several real-life problems to check their accuracy and applicability. The results of the dynamical study confirm that the new methods are more stable and accurate than the existing schemes.

scholarly journals New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permittingf′(xn)=0

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Ramandeep Behl ◽  
V. Kanwar
Keyword(s):  
Higher Order ◽  
The Other ◽  
Robust Methods ◽  
Challenging Problem ◽  
Order Convergence ◽  
Dynamical Study ◽  
Globally Convergent ◽  
Special Cases ◽  
Globally Convergent Methods ◽  
Ostrowski’S Method

Construction of higher-order optimal and globally convergent methods for computing simple roots of nonlinear equations is an earliest and challenging problem in numerical analysis. Therefore, the aim of this paper is to present optimal and globally convergent families of King's method and Ostrowski's method having biquadratic and eight-order convergence, respectively, permittingf′(x)=0in the vicinity of the required root. Fourth-order King's family and Ostrowski's method can be seen as special cases of our proposed scheme. All the methods considered here are found to be more effective to the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods.

scholarly journals On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1091 ◽  
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Lorentz Jäntschi
Keyword(s):  
Fourth Order ◽  
Multiple Root ◽  
Second Step ◽  
Optimal Order ◽  
Multiple Roots ◽  
Order Convergence ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Derivative Information

A number of optimal order multiple root techniques that require derivative evaluations in the formulas have been proposed in literature. However, derivative-free optimal techniques for multiple roots are seldom obtained. By considering this factor as motivational, here we present a class of optimal fourth order methods for computing multiple roots without using derivatives in the iteration. The iterative formula consists of two steps in which the first step is a well-known Traub–Steffensen scheme whereas second step is a Traub–Steffensen-like scheme. The Methodology is based on two steps of which the first is Traub–Steffensen iteration and the second is Traub–Steffensen-like iteration. Effectiveness is validated on different problems that shows the robust convergent behavior of the proposed methods. It has been proven that the new derivative-free methods are good competitors to their existing counterparts that need derivative information.

An efficient class of fourth-order derivative-free method for multiple-roots

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sunil Kumar ◽  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Ioannis K. Argyros
Keyword(s):  
Multiple Root ◽  
Second Step ◽  
Optimal Order ◽  
Multiple Roots ◽  
Nonlinear Functions ◽  
New Methods ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Derivative Free Method ◽  
Good Number

Abstract Many optimal order multiple root techniques, which use derivatives in the algorithm, have been proposed in literature. Many researchers tried to construct an optimal family of derivative-free methods for multiple roots, but they did not get success in this direction. With this as a motivation factor, here, we present a new optimal class of derivative-free methods for obtaining multiple roots of nonlinear functions. This procedure involves Traub–Steffensen iteration in the first step and Traub–Steffensen-like iteration in the second step. Efficacy is checked on a good number of relevant numerical problems that verifies the efficient convergent nature of the new methods. Moreover, we find that the new derivative-free methods are just as competent as the other existing robust methods that use derivatives.

An Accelerated Iterative Method with Third-Order Convergence

2012 ◽  
Vol 220-223 ◽  
pp. 2658-2661
Author(s):  
Zhong Yong Hu ◽  
Liang Fang ◽  
Lian Zhong Li
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
New Method ◽  
Order Convergence ◽  
Third Order ◽  
Numerical Examples ◽  
Modified Newton’S Method ◽  
Jarratt Method

We present a new modified Newton's method with third-order convergence and compare it with the Jarratt method, which is of fourth-order. Based on this new method, we obtain a family of Newton-type methods, which converge cubically. Numerical examples show that the presented method can compete with Newton's method and other known third-order modifications of Newton's method.

scholarly journals Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Gustavo Fernández-Torres ◽  
Juan Vásquez-Aquino
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Classical Method ◽  
Multiple Roots ◽  
Order Convergence ◽  
Numerical Examples ◽  
Optimal Methods

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.

Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equations

2011 ◽  
Vol 60 (2) ◽  
pp. 145-159 ◽  
Author(s):  
Marcin Ligas ◽  
Piotr Banasik
Keyword(s):  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
System Of Nonlinear Equations ◽  
Order Convergence ◽  
Ellipsoidal Height ◽  
Third Order ◽  
The Third ◽  
Geodetic Coordinates ◽  
Modified Newton’S Method

Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equationsA new method to transform from Cartesian to geodetic coordinates is presented. It is based on the solution of a system of nonlinear equations with respect to the coordinates of the point projected onto the ellipsoid along the normal. Newton's method and a modification of Newton's method were applied to give third-order convergence. The method developed was compared to some well known iterative techniques. All methods were tested on three ellipsoidal height ranges: namely, (-10 - 10 km) (terrestrial), (20 - 1000 km), and (1000 - 36000 km) (satellite). One iteration of the presented method, implemented with the third-order convergence modified Newton's method, is necessary to obtain a satisfactory level of accuracy for the geodetic latitude (σφ < 0.0004") and height (σh< 10-6km, i.e. less than a millimetre) for all the heights tested. The method is slightly slower than the method of Fukushima (2006) and Fukushima's (1999) fast implementation of Bowring's (1976) method.

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