The Householder's method is a root-find algorithm which is a natural extension of both the Newton's method and the Halley's method. The current paper focuses on approximating the square root of a positive real number based on these methods. The resulting algorithms can be expressed using Chebyshev polynomials. An extension to the $n$th root is also proposed.

ABSTRACT This paper proposed a new multistage hybrid algorithm MH‐IK to solve the inverse kinematics (IK) problem of redundant manipulators in terms of position, orientation, and collision errors. An improved cyclic coordinate descent method is proposed to satisfy the iteration of prismatic joints and avoid singularities. Halley's method is integrated to achieve fast convergence. On the basis of geometric analysis and the Beta distribution, a global search method is developed to overcome the local minimum problem. Four strategies are designed to further improve the adaptability during iterative loops. The proposed algorithm is applied to a seven‐degrees‐of‐freedom mill relining manipulator. Unlike conventional arms, the IK solution of this special manipulator has more requirements to be satisfied, including solvability, stability, real‐time, joint limits, and collision avoidance in a confined inner workspace. The superiority of MH‐IK is demonstrated through convergence analysis, ablation and comparison experiments against nine nonlinear algorithms and three millisecond‐level solvers. Furthermore, the hardware experiments are carried out in an established digital twin system to further verify the effectiveness of the proposed method. Although MH‐IK lacks solution diversity, it significantly outperforms other algorithms in almost all metrics, such as 5.27 ms solving time, 99.5% success rate, and 0.51% collision rate.

High-order methods are particularly crucial for achieving highly accurate solutions or satisfying high-order optimality conditions. However, most existing high-order methods require solving complex high-order Taylor polynomial models, which pose significant computational challenges. In this paper, we propose a Chebyshev–Halley method with gradient regularization, which retains the convergence advantages of high-order methods while effectively addressing computational challenges in polynomial model solving. The proposed method incorporates a quadratic regularization term with an adaptive parameter proportional to a certain power of the gradient norm, thereby ensuring a closed-form solution at each iteration. In theory, the method achieves a global convergence rate of O(k−3) or even O(k−5), attaining the optimal rate of third-order methods without requiring additional acceleration techniques. Moreover, it maintains local superlinear convergence for strongly convex functions. Numerical experiments demonstrate that the proposed method compares favorably with similar methods in terms of efficiency and applicability.

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.

The paper discusses two algorithms for accurately determining solutions to the transcendental thermionic emission equation, which is the cornerstone of forward electrical behavior in Schottky diodes. The numerical techniques are developed based on the Newton-Raphson and Halley methods. Both approaches use distinct forms for the thermionic emission expression, emphasizing robustness against numerical overflows. Parameter initialization, complexity and applicability are discussed for each technique. A comparison is carried out between forward characteristics simulated with the two methods, which are then also used for characterizing real SiC-Schottky diodes. Results evince complete compatibility and highly accurate approximations of experimental measurements (R2≅99.9%) on devices with different contact compositions.

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.

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.

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.

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

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.

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 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.

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.

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.

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.

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.

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

AN IMPROVED LOCAL ANALYSIS OF DEFORMED HALLEY METHOD IN BANACH SPACES