Halley's Method Research Articles

Some variants of Chebyshev–Halley methods free from second derivative

In this paper, we present some new modifications of Chebyshev–Halley methods free from second derivative. The convergence analysis shows that the new methods are third-order convergent and require one function and two first derivative evaluations per iteration. Several numerical examples are given to illustrate the performance of the presented methods.

Sixth-order variants of Chebyshev–Halley methods for solving non-linear equations

In this paper, we present a family of new variants of Chebyshev–Halley methods with sixth-order convergence. Compared with Chebyshev–Halley methods, the new methods require one additional evaluation of the function. The numerical results presented show that the new methods compete with Chebyshev–Halley methods.

A new modified Halley method without second derivatives for nonlinear equation

In a recent paper, Noor and Noor [K. Inayat Noor, M. Aslam Noor, Predictor–corrector Halley method for nonlinear equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.11.023] have suggested and analyzed a predictor–corrector method Halley method for solving nonlinear equations. In this paper, we modified this method by using the finite difference scheme, which has a quintic convergence. We have compared this modified Halley method with some other iterative of fifth-orders convergence methods, which shows that this new method is a robust one. Several examples are given to illustrate the efficiency and the performance of this new method.

On Chebyshev–Halley methods with sixth-order convergence for solving non-linear equations

In this paper, we present a family of new variants of Chebyshev–Halley methods. The new methods have sixth-order convergence although they only add one evaluation of the function at the point iterated by Chebyshev–Halley methods. The numerical results presented show that the new methods work better not only in order but in efficiency.

Some new sixth-order methods for solving non-linear equations

In this paper, we present some new sixth-order methods based on the Chebyshev–Halley methods. Per iteration the new methods require an additional evaluation of the function compared with Chebyshev–Halley methods. Analysis of efficiency shows that the new methods can compete with Newton’s method and the classical third-order methods, as we show in numerical results.

Certain improvements of Chebyshev–Halley methods with accelerated fourth-order convergence

In this paper, we present a new one-parameter family of methods which improves the classical Chebyshev–Halley methods with cubic convergence. The convergence analysis shows that each method of the family is quartically convergent and per iteration it requires one evaluation of the given function and two of its first derivative. The well-known Jarratt fourth-order method is shown to be part of the family. Several numerical examples are given to illustrate the efficiency and performance of the presented methods.

Some variants of Chebyshev–Halley methods with fifth-order convergence

In this paper, we present some new variants of Chebyshev–Halley methods. The local order of convergence is increased from 3 to 5. The new methods add one evaluation of the function at the point iterated by Chebyshev–Halley methods. The numerical results show that the new methods work better not only in order but in efficiency.

Predictor–corrector Halley method for nonlinear equations

In this paper, we suggest and analyze a new two-step predictor–corrector type iterative methods for solving nonlinear equations of the type f( x) = 0 by using the technique of updating the solution. This method can be viewed as a predictor–corrector iterative Halley’s method. We also consider the convergence analysis of the proposed method. To illustrate the efficiency of this new method, we give several examples and a comparison with the method of Grau and Diaz-Barrero [M. Grau, J.L. Diaz-Barrero, An improvement to Ostrowski root-finding method, Appl. Math. Comput. 173 (2006) 450–456] is given. This method can be considered as a refinement and improvement of the Halley method.

Modified Chebyshev–Halley methods with sixth-order convergence

In this paper, we present a family of modified Chebyshev–Halley methods. The new methods add one evaluation of the function at the point iterated by Chebyshev–Halley methods. Analysis of convergence shows that these methods have sixth-order convergence. Numerical results show that the new methods can be of practical interest.

The improvements of Chebyshev–Halley methods with fifth-order convergence

In this paper, we present a family of new fifth-order methods, which improves the classical Chebyshev–Halley methods with cubic convergence. The new methods add one evaluation of the function at the point iterated by Chebyshev–Halley methods. Analysis of efficiency shows that the new methods can be of practical interest, which is also corroborated by numerical tests.

Recurrence relations for a Newton-like method in Banach spaces

The convergence of iterative methods for solving nonlinear operator equations in Banach spaces is established from the convergence of majorizing sequences. An alternative approach is developed to establish this convergence by using recurrence relations. For example, the recurrence relations are used in establishing the convergence of Newton's method [L.B. Rall, Computational Solution of Nonlinear Operator Equations, Robert E. Krieger, New York, 1979] and the third order methods such as Halley's, Chebyshev's and super Halley's [V. Candela, A. Marquina, Recurrence relations for rational cubic methods I: the Halley method, Computing 44 (1990) 169–184; V. Candela, A. Marquina, Recurrence relations for rational cubic methods II: the Halley method, Computing 45 (1990) 355–367; J.A. Ezquerro, M.A. Hernández, Recurrence relations for Chebyshev-type methods, Appl. Math. Optim. 41 (2000) 227–236; J.M. Gutiérrez, M.A. Hernández, Third-order iterative methods for operators with bounded second derivative, J. Comput. Appl. Math. 82 (1997) 171–183; J.M. Gutiérrez, M.A. Hernández, Recurrence relations for the Super–Halley method, Comput. Math. Appl. 7(36) (1998) 1–8; M.A. Hernández, Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433–445 [10]]. In this paper, an attempt is made to use recurrence relations to establish the convergence of a third order Newton-like method used for solving a nonlinear operator equation F ( x ) = 0 , where F : Ω ⊆ X → Y be a nonlinear operator on an open convex subset Ω of a Banach space X with values in a Banach space Y . Here, first we derive the recurrence relations based on two constants which depend on the operator F. Then, based on this recurrence relations a priori error bounds are obtained for the said iterative method. Finally, some numerical examples are worked out for demonstrating our approach.

An analysis of the properties of the variants of Newton’s method with third order convergence

For the last five years, the variants of the Newton’s method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non-linear equations. These methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives. In this paper, we investigate about the relationship between these methods which are in fact based on the approximation of the second order derivative present in the third order limited Taylor expansion. We also prove that they are different forms of the Halley method and are all contractive iterative methods in a common neighbourhood. We extend some of these variants to multivariate cases and prove their respective local cubic convergence from their corresponding linear models.

Fewer Convergence Conditions for the Halley Method

We present a new semilocal convergence result of Newton-Kantorovich type for Halley's method, where fewer convergence conditions are required than all the existing ones until now.

Computing zeros of analytic functions in the complex plane without using derivatives

We present a package in Fortran 90 which solves f ( z ) = 0 , where z ∈ W ⊂ C without requiring the evaluation of derivatives, f ′ ( z ) . W is bounded by a simple closed curve and f ( z ) must be holomorphic within W . We have developed and tested the package to support our work in the modeling of high frequency and optical wave guiding and resonant structures. The respective eigenvalue problems are particularly challenging because they require the high precision computation of all multiple complex roots of f ( z ) confined to the specified finite domain. Generally f ( z ) , despite being holomorphic, does not have explicit analytical form thereby inhibiting evaluation of its derivatives. Program summary Title of program:EZERO Catalogue identifier:ADXY_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADXY_v1_0 Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland Computer:IBM compatible desktop PC Operating system:Fedora Core 2 Linux (with 2.6.5 kernel) Programming languages used:Fortran 90 No. of bits in a word:32 No. of processors used:one Has the code been vectorized:no No. of lines in distributed program, including test data, etc.:21045 Number of bytes in distributed program including test data, etc.:223 756 Distribution format:tar.gz Peripherals used:none Method of solution:Our package uses the principle of the argument to count the number of zeros encompassed by a contour and then computes estimates for the zeros. Refined results for each zero are obtained by application of the derivative-free Halley method with or without Aitken acceleration, as the user wishes.

Halley's method for operators with unbounded second derivative

A new semilocal convergence result of Newton–Kantorovich type for the Halley method is presented, where a new technique is provided to analyze the semilocal convergence. The usual convergence conditions are relaxed, since the second derivative F ″ of a nonlinear operator F satisfies ‖ F ″ ( x 0 ) ‖ ⩽ α instead of ‖ F ″ ( x ) ‖ ⩽ M , for all x in a subset of the domain of F, where α and M are positive real constants.

A construction of attracting periodic orbits for some classical third-order iterative methods

We use a family of root-finding iterative methods for finding roots of nonlinear equations. We present a procedure for constructing polynomials so that superattracting periodic orbits of any prescribed period occur when these methods are applied. This family includes Chebyshev's method, Halley's method, the super-Halley method, and the c-methods, as particular cases.

New Kantorovich-Type Conditions for Halley's Method

LF (x )= F � (x) −1 F �� (x)F � (x) −1 F (x), provided that F � (xn) −1 and I − 1 LF (xn) −1 exist at each step. The operators F � (x) and F �� (x) denote the first and the second Frs of the operator F. Under certain conditions, the Halley method produces a sequence which converges to a solution x ∗ of F (x )= 0. Basic results concerning the convergence of the iteration have been published under assumptions of Newton- Kantorovich type. In (2) and (10), an abundant list of references can be found, where several techniques for finding sufficient conditions for the convergence of Halley's process appear. Initially, Safiev (7) presented a convergence theorem for the Halley iteration under the following conditions

On the R-order of the Halley method

An R-order bound for the Halley method is obtained in this work, where an analysis of the convergence of the method is also presented under mild differentiability conditions. To do this, a new technique is developed, where the involved operator must satisfy some recurrence relations.

Halley?s Method for Calculating the Earth-Sun Distance

Edmund Halley [1] published in 1717 his paper on a method for calculating the Earth-Sun distance, using observations of the transit of Venus. His paper is in Latin. A full translation with comments was provided in a book by Ferguson [2] in 1761. Ferguson [3] published the first edition of his book on astronomy in 1756. In the third edition of this astronomy book in 1764, and in all subsequent editions, he added the whole of his 1761 book. A translation is therefore available in any of these editions. In 1809 the editors [4] of the Abridged Transactions of the Royal Society included a translation of Halley's paper. This translation omitted some small parts not considered relevant to the main presentation, but they importantly retained his diagram and symbology. Ferguson's comments provide a guide to Halley's method but they do not provide an accurate picture. Ferguson illustrates the method with a completely graphical approach whereas Halley used a diagram to explain and illustrate his approach but his values of the observer's parallaxes were obtained by calculation. Also Ferguson does not reproduce Halley's diagram or symbols exactly and in one instance used a wrong dimension. The abridged version does not reproduce Halley's diagram completely. Halley's method is often referred to as his "method of differences". As well it is called a "projection method". He said it used "the doctrine of parallaxes". A critical review of Halley's paper and method is presented here using the diagram from the original paper. It is shown that Halley's calculation method gives excellent results for the Earth-Sun distance. The Abridged Transactions version of Halley's paper is used and extracts are provided where necessary. It is assumed that many of the concepts associated with a transit of Venus are well known.

On bilateral convergence of Halley's method

On bilateral convergence of Halley's method