Halley's Method Research Articles

Application of the Inflection Point in the Evaluation of the Halley and Newton-Raphson Techniques for Finding the Root of Non-Linear Equations

Numeric Method is one of the methods used to solve nonlinear equation roots. Many methods can be used, both open methods and closed methods. In this case, the method used is closed, namely the Newton-Raphson Method and Halley Method. The research aims to find out the comparison result between the Newton-Raphson Method and the Halley Method. The research used a literature method from a book, journal, and any other literature, where it connected with the topic. The steps used are formulation problem, finding and collecting information, describing and explaining the information, analysis, and conclusion of the result. The conclusion can be explained with a table of data and explanations Based on data analysis, it can be stated that the Halley Method is faster toward convergence compared to the Newton-Raphson Method based on the first case or second case.

From Halley to Secant: Redefining root finding with memory‐based methods including convergence and stability

Root‐finding methods solve equations and identify unknowns in physics, engineering, and computer science. Memory‐based root‐seeking algorithms may look back to expedite convergence and enhance computational efficiency. Real‐time systems, complicated simulations, and high‐performance computing demand frequent, large‐scale calculations. This article proposes two unique root‐finding methods that increase the convergence order of the classical Newton–Raphson (NR) approach without increasing evaluation time. Taylor's expansion uses the classical Halley method to create two memory‐based methods with an order of 2.4142 and an efficiency index of 1.5538. We designed a two‐step memory‐based method with the help of Secant and NR algorithms using a backward difference quotient. We demonstrate memory‐based approaches' robustness and stability using visual analysis via polynomiography. Local and semilocal convergence are thoroughly examined. Finally, proposed memory‐based approaches outperform several existing memory‐based methods when applied to models including a thermistor, path traversed by an electron, sheet‐pile wall, adiabatic flame temperature, and blood rheology nonlinear equation.

ON NEW HYBRID ROOT-FINDING ALGORITHMS FOR SOLVING TRANSCENDENTAL EQUATIONS USING EXPONENTIAL AND HALLEY'S METHODS

The objective of this paper is to propose two new hybrid root finding algorithms for solving transcendental equations. The proposed algorithms are based on the well-known root finding methods namely the Halley's method, regula-falsi method and exponential method. We show using numerical examples that the proposed algorithms converge faster than other related methods. The first hybrid algorithm consists of regula-falsi method and exponential method (RF-EXP). In the second hybrid algorithm, we use regula falsi method and Halley's method (RF-Halley). Several numerical examples are presented to illustrate the proposed algorithms, and comparison of these algorithms with other existing methods are presented to show the efficiency and accuracy. The implementation of the proposed algorithms is presented in Microsoft Excel (MS Excel) and the mathematical software tool Maple.

An Accelerated Regularized Chebyshev–Halley Method for Unconstrained Optimization

In machine learning, most models can be transformed into unconstrained optimization problems, so how to solve the unconstrained optimization problem for different objective functions is always a hot issue. In this paper, a class of unconstrained optimization where objection function has [Formula: see text]th-order derivative and Lipschitz continuous simultaneously is studied. To handle such problems, we propose an accelerated regularized Chebyshev–Halley method based on the Accelerated Hybrid Proximal Extragradient (A-HPE) framework. It proves that convergence complexity of the proposed method is [Formula: see text], which is consistent with the lower iteration complexity bound for third-order tensor methods. Numerical experiments on functions in machine learning demonstrate the promising performance of the proposed method.

Newton-Like Components in the Chebyshev–Halley Family of Degree n Polynomials

We study the Chebyshev–Halley methods applied to the family of polynomials f_{n,c}(z)=z^n+c, for nge 2 and cin mathbb {C}^{*}. We prove the existence of parameters such that the immediate basins of attraction corresponding to the roots of unity are infinitely connected. We also prove that, for n ge 2, the corresponding dynamical plane contains a connected component of the Julia set, which is a quasiconformal deformation of the Julia set of the map obtained by applying Newton’s method to f_{n,-1}.

A general approach to the study of the convergence of Picard iteration with an application to Halley's method for multiple zeros of analytic functions

In this paper, we define a new wide class of iteration functions and then we use it to prove a general convergence theorem that provides exact domain of initial approximations to guarantee the high Q-order of convergence of Picard iteration generated by this class of functions. As an application of this theorem, we prove some local convergence theorems about the famous Halley's method for simple and multiple zeros of analytic functions. All obtained results are supplied with a priori and a posteriori error estimates as well as with assessments of the asymptotic error constants.

Disconnected Julia set of Halley's method for exponential maps

We investigate the Halley method of exponential maps. Our main result is that, unlike Newton's method, the Julia set of Halley's method may be disconnected when applied to entire maps of form where p and q are polynomials and q is non-constant. We also describe the nature of the fixed points and classify rational Halley's maps of entire functions.

A comparison between Fukushima-Halley algorithm and Trilateration algorithm for geodetic conversion

The geodetic transformation of Cartesian coordinates into their elliptical equivalent is a fundamental problem in geodesy. The Fukushima algorithm accelerated by Halley method (Fukushima-Halley) is considered the standard in this conversion. The Trilateration algorithm is a recent algorithm solving the conversion problem through a computational geometry approach. This study compared the Trilateration algorithm to the Fukushima-Halley algorithm in aspects of accuracy of results, time efficiency, and space efficiency. Also, the parallel version of both algorithms was established using the Master-Slave technique and compared. The Trilateration Algorithm showed a slightly higher accuracy compared to Fukushima-Halley algorithm, which allocated less space in memory, and was 2.6 faster in sequential version compared to 1.9 in the parallel version. The study introduced a benchmark for arithmetic operation on the testing machine to be used in time efficiency comparison.

Some Approximate Schemes for Solving Nonlinear Equations

Some iterative algorithms for solving nonlinear equation $f(x) = 0$ are suggested and investigated using Taylor series and homotopy perturbation technique. These algorithms can be viewed as extensions and generalization of well known methods such as Householder and Halley methods with cubic convergence. Convergence of the proposed methods has been discussed and analyzed. Several numerical examples are given to illustrate the efficiency of the suggested algorithms for solving nonlinear equations. Comparison with other iterative schemes is carried out to show the validity and performance of these algorithms.

Unified Convergence Analysis of Chebyshev–Halley Methods for Multiple Polynomial Zeros

In this paper, we establish two local convergence theorems that provide initial conditions and error estimates to guarantee the Q-convergence of an extended version of Chebyshev–Halley family of iterative methods for multiple polynomial zeros due to Osada (J. Comput. Appl. Math. 2008, 216, 585–599). Our results unify and complement earlier local convergence results about Halley, Chebyshev and Super–Halley methods for multiple polynomial zeros. To the best of our knowledge, the results about the Osada’s method for multiple polynomial zeros are the first of their kind in the literature. Moreover, our unified approach allows us to compare the convergence domains and error estimates of the mentioned famous methods and several new randomly generated methods.

Анализ точности и формальное обоснование численного решения нелинейных уравнений методами Ньютона и Хэйли при использовании арифметики с плавающей запятой

The paper describes particularities of using floating-point arithmetics for finding solutions of nonlinear equations by the means of numerical approximation. Analysis of scientific literature shows scarcity of works studying these methods of numerical solvers in presence of limitations and particularities imposed by algebras of floating-point numbers despite well-known significance of these aspects when it comes to overall accuracy, predictability and usability of numerical solvers. Therefore, the paper describes some interesting results of theoretical and experimental study of these wellknown Newton's and Halley's methods from the point of view of their implementability and problems that arise when floating-point arithmetic is used. The analysis is conducted both theoretically and experimentally using floating-point machines. On one hand, the experiments demonstrate correspondence between the predicted efficiency factors and ones that are measured, but on the other hand these measurements contradict intuitively predicted behavior of solvers if no floating-point specifics are taken into account.

A note on machine method for root extraction

The present investigation deals with the critical study of the works of Lancaster and Traub, who have developed $n$th root extraction methods of a real number. It is found that their developed methods are equivalent and the particular cases of Halley's and Householder's methods. Again the methods presented by them are easily obtained from simple modifications of Newton's method, which is the extension of Heron's square root iteration formula. Further, the rate of convergency of their reported methods are studied.

Application of Numerical Analysis for Solving Nonlinear Equation

For the last years, the various of Newton's formula have become common iterative numerical techniques to realize approximate solutions to the zeros of nonlinear equations of PV cell (single diode) model foe electronic applications. These techniques do not necessitate the computation of second derivative of the functions but it need only first derivative of it's functions. In this paper, we introduce a new proposed method Dekker's Formula with five evaluations per iterations based on Accelerated Predictor-Corrector Halley's method. Numerical experiments produce that the new algorithm can determine with the standard Newton's algorithm.

Why Does Child Mortality Decrease With Age? Modeling the Age-Associated Decrease in Mortality Rate Using WHO Metadata From 25 Countries.

Background: Our previous study analyzed the age trajectory of mortality (ATM) in 14 European countries, while this study aimed at investigating ATM in other continents and in countries with a higher level of mortality. Data from 11 Non-European countries were used.Methods: The number of deaths was extracted from the WHO mortality database. The Halley method was used to calculate the mortality rates in all possible calendar years and all countries combined. This method enables us to combine more countries and more calendar years in one hypothetical population.Results: The age trajectory of total mortality (ATTM) and also ATM due to specific groups of diseases were very similar in the 11 non-European countries and in the 14 European countries. The level of mortality did not affect the main results found in European countries. The inverse proportion was valid for ATTM in non-European countries with two exceptions.Slower or no mortality decrease with age was detected in the first year of life, while the inverse proportion model was valid for the age range (1, 10) years in most of the main chapters of ICD10.Conclusions: The decrease in child mortality with age may be explained as the result of the depletion of individuals with congenital impairment. The majority of deaths up to the age of 10 years were related to congenital impairments, and the decrease in child mortality rate with age was a demonstration of population heterogeneity. The congenital impairments were latent and may cause death even if no congenital impairment was detected.

Extending the convergence domain of deformed Halley method under $$\omega$$ condition in Banach spaces

In this study, we present an improved local analysis using $$\omega$$ condition to enlarge the convergence domain of deformed Halley method. This analysis avoids the use of the extra assumption on the boundedness of the first derivative of the nonlinear operator. Finally, numerical tests confirm that our analysis provides a larger convergence domain, in comparison to the earlier study, without using additional conditions.

Semilocal Convergence of Modified Chebyshev–Halley Method for Nonlinear Operators in Case of Unbounded Third Derivative

Semilocal Convergence of Modified Chebyshev–Halley Method for Nonlinear Operators in Case of Unbounded Third Derivative

AN IMPROVED LOCAL ANALYSIS OF DEFORMED HALLEY METHOD IN BANACH SPACES

AN IMPROVED LOCAL ANALYSIS OF DEFORMED HALLEY METHOD IN BANACH SPACES

Edmond Halley's Contributions to Hydrogeology.

Edmond Halley's Contributions to Hydrogeology.

A Comprehensive Physics-Based Current–Voltage SPICE Compact Model for 2-D-Material-Based Top-Contact Bottom-Gated Schottky-Barrier FETs

In this article, we report the development of a novel physics-based analytical model for explaining the current-voltage relationship in Schottky barrier (SB) 2-D-material field effect transistors (FETs). The model has at its core the calculation of the surface-potential (SP) which is accomplished by invoking 2-D density of states in conjunction with Fermi-Dirac (FD) distribution for electron and hole statistics. The explicit computation for the SP, carried out using the Lambert-W function together with Halley's method, is used to construct the SP-based band-diagram for realizing the transparency of the SBs. Thereafter, the ambipolar current is derived in terms of the electron and hole injection phenomena-the thermionic emission and Fowler-Nordheim tunneling mechanisms-at the SB contacts. Furthermore, drift-diffusion current is derived in terms of the SP and incorporated in the model to account for the scattering in the intrinsic 2-D channel. Finally, the Verilog-A model is validated against experimental I-V data reported in the literature for two different 2-D-material systems. This is the first demonstration of an explicit SP-based SPICE model for ambipolar SB-2-D-FETs that is simultaneously built on tunneling-emission and drift-diffusion formalisms.

Why Does Child Mortality Decrease With Age? Modeling the Age-Associated Decrease in Mortality Rate Using WHO Metadata From 14 European Countries.

Background: Mortality rate rapidly decreases with age after birth, and, simultaneously, the spectrum of death causes show remarkable changes with age. This study analyzed age-associated decreases in mortality rate from diseases of all main chapters of the 10th revision of the International Classification of Diseases.Methods: The number of deaths was extracted from the mortality database of the World Health Organization. As zero cases could be ascertained for a specific age category, the Halley method was used to calculate the mortality rates in all possible calendar years and in all countries combined.Results: All causes mortality from the 1st day of life to the age of 10 years can be represented by an inverse proportion model with a single parameter. High coefficients of determination were observed for total mortality in all populations (arithmetic mean = 0.9942 and standard deviation = 0.0039).Slower or no mortality decrease with age was detected in the 1st year of life, while the inverse proportion method was valid for the age range [1, 10) years in most of all main chapters with three exceptions. The decrease was faster for the chapter “Certain conditions originating in the perinatal period” (XVI).The inverse proportion was valid already from the 1st day for the chapter “Congenital malformations, deformations and chromosomal abnormalities” (XVII).The shape of the mortality decrease was very different for the chapter “Neoplasms” (II) and the rates of mortality from neoplasms were age-independent in the age range [1, 10) years in all populations.Conclusion: The theory of congenital individual risks of death is presented and can explain the results. If it is valid, latent congenital impairments may be present among all cases of death that are not related to congenital impairments. All results are based on published data, and the data are presented as a supplement.