super-Halley Method Research Articles

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.

General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros.

Super-Halley method under majorant conditions in Banach spaces

In this paper, we have studied local convergence of Super-Halley method in Banach spaces under the assumption of second order majorant conditions. This approach allows us to obtain generalization of earlier convergence analysis under majorizing sequences. Two important special cases of the convergence analysis based on the premises of Kantorovich and Smale type conditions have also been concluded. To show efficacy of our approach we have given three numerical examples.

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

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.

Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order

A large part of the numerical procedures for obtaining the equilibrium path or load-displacement curve of structural problems with nonlinear behavior is based on the Newton-Raphson iterative scheme, to which is coupled the path-following methods. This paper presents new algorithms based on Potra-Ptak, Chebyshev and super-Halley methods combined with the Linear Arc-Length path-following method. The main motivation for using these methods is the cubic order convergence. To elucidate the potential of our approach, we present an analysis of space and plane trusses problems with geometric nonlinearity found in the literature. In this direction, we will make use of the Positional Finite Element Method, which considers the nodal coordinates as variables of the nonlinear system instead of displacements. The numerical results of the simulations show the capacity of the computational algorithm developed to obtain the equilibrium path with force and displacement limits points. The implemented iterative methods exhibit better efficiency as the number of time steps and necessary accumulated iterations until convergence and processing time, in comparison with classic methods of Newton-Raphson and Modified Newton-Raphson.

Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order

A large part of the numerical procedures for obtaining the equilibrium path or load-displacement curve of structural problems with nonlinear behavior is based on the Newton-Raphson iterative scheme, to which is coupled the path-following methods. This paper presents new algorithms based on Potra-Pták, Chebyshev and super-Halley methods combined with the Linear Arc-Length path-following method. The main motivation for using these methods is the cubic order convergence. To elucidate the potential of our approach, we present an analysis of space and plane trusses problems with geometric nonlinearity found in the literature. In this direction, we will make use of the Positional Finite Element Method, which considers the nodal coordinates as variables of the nonlinear system instead of displacements. The numerical results of the simulations show the capacity of the computational algorithm developed to obtain the equilibrium path with force and displacement limits points. The implemented iterative methods exhibit better efficiency as the number of time steps and necessary accumulated iterations until convergence and processing time, in comparison with classic methods of Newton-Raphson and Modified Newton-Raphson.

The convergence theorem for fourth-order super-Halley method in weaker conditions

In this paper, we establish the Newton-Kantorovich convergence theorem of a fourth-order super-Halley method under weaker conditions in Banach space, which is used to solve the nonlinear equations. Finally, some examples are provided to show the application of our theorem.

Ball convergence of some fourth and sixth-order iterative methods

We present a local convergence analysis for some families of fourth and sixth-order methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Earlier studies [V. Candela and A. Marquina, Recurrence relations for rational cubic methods II: The Chebyshev method, Computing 45 (1990) 355–367; C. Chun, P. Stanica and B. Neta, Third order family of methods in Banach spaces, Comput. Math. Appl. 61 (2011) 1665–1675; J. M. Gutiérrez and M. A. Hernández, Recurrence relations for the super-Halley method, Comput. Math. Appl. 36 (1998) 1–8; M. A. Hernández and M. A. Salanova, Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method, J. Comput. Appl. Math. 126 (2000) 131–143; M. A. Hernández, Chebyshev’s approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433–455; M. A. Hernández, Second-derivative-free variant of the Chebyshev method for nonlinear equations, J. Optim. Theory Appl. 104(3) (2000) 501–515; J. L. Hueso, E. Martinez and C. Teruel, Convergence, efficiency and dynamics of new fourth and sixth-order families of iterative methods for nonlinear systems, J. Comput. Appl. Math. 275 (2015) 412–420; Á. A. Magre nán, Estudio de la dinámica del método de Newton amortiguado, Ph.D. thesis, Servicio de Publicaciones, Universidad de La Rioja (2013), http://dialnet.unirioja.es/servlet/tesis?codigo=38821 ; J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970); M. S. Petkovic, B. Neta, L. Petkovic and J. Džunič, Multi-Point Methods for Solving Nonlinear Equations (Elsevier, 2013); J. F. Traub, Iterative Methods for the Solution of Equations, Automatic Computation (Prentice-Hall, Englewood Cliffs, NJ, 1964); X. Wang and J. Kou, Semilocal convergence and [Formula: see text]-order for modified Chebyshev–Halley methods, Numer. Algorithms 64(1) (2013) 105–126] have used hypotheses on the fourth Fréchet derivative of the operator involved. We use hypotheses only on the first Fréchet derivative in our local convergence analysis. This way, the applicability of these methods is extended. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples illustrating the theoretical results are also presented in this study.

Semilocal convergence analysis on the modifications for Chebyshev–Halley methods under generalized condition

In this paper, we consider the semilocal convergence for modifications of Chebyshev–Halley methods in Banach space. Compared with the results on super-Halley method studied in reference Gutiérrez and Hernández (1998)these modified methods need less computation of inversion, the R-order is improved, and the Lipschitz continuity of second derivative is also relaxed. We prove a theorem to show existence-uniqueness of solution. The R-order for these modified methods is analyzed under generalized condition.

Analysis of semilocal convergence for ameliorated super-Halley methods with less computation for inversion

In this paper, the semilocal convergence for ameliorated super-Halley methods in Banach spaces is considered. Different from the results in [J. M. Gutiérrez and M. A. Hernández, Comput. Math. Appl. 36 (1998) 1–8], these ameliorated methods do not need to compute a second derivative, the computation for inversion is reduced and the $R$-order is also heightened. Under a weaker condition, an existence–uniqueness theorem for the solution is proved.

Efficient families of Newton’s method and its variants suitable for non-convergent cases

In this paper, we derive new one-parameter family of Newton’s method, Schroder’s method, super-Halley method and Halley’s method respectively for finding simple zeros of nonlinear functions, permitting \(f^{\prime }(x_n)=0\) at some points in the vicinity of required root. Using the newly derived family of super-Halley method, we further obtain new interesting families of famous quartically convergent Traub–Ostrowski’s and Jarratt’s methods respectively. Further, the approach has been extended to solve a system of nonlinear equations. It is found by way of illustration that the proposed methods are very useful in high-precision computing environment and non-convergent cases.

Semilocal convergence of multi-point improved super-Halley-type methods without the second derivative under generalized weak condition

In this paper, we consider the semilocal convergence of multi-point improved super-Halley-type methods in Banach space. Different from the results of super-Halley method studied in reference Gutierrez, J.M. and Hernandez, M.A. (Comput. Math. Appl. 36,1---8, 1998) these methods do not require second derivative of an operator, the R-order is improved and the convergence condition is also relaxed. We prove a convergence theorem to show existence and uniqueness of the solution.

Local convergence for a multi-point family of super-Halley methods in a Banach space under weak conditions

Local convergence for a multi-point family of super-Halley methods in a Banach space under weak conditions

On a two-parameter Chebyshev-Halley-like family of optimal two-point fourth order methods free from second derivatives

The one-parameter Chebyshev-Halley family is an important family of one-point third order iterative methods which requires one function, one first and one second derivative evaluations. The famous Chebyshev, Halley and Super-Halley methods are its members. Nedzhibov-Hasanov-Petkov (Numer. Alg. 42:127–136, 2006) approximated the second derivative present in the Chebyshev-Halley family to obtain a two-parameter Chebyshev-Halley-like family of two-point iterative methods free from second derivative. Only one member of this family known as the famous Jarratt method is fourth order and satisfies the Kung-Traub Conjecture. Other members are third order. Recent advancement in this field of numerical analysis have made it possible to develop fourth order methods from third order ones with the same number of function evaluations using weight functions. In this work, we develop a two-parameter Chebyshev-Halley-like family of two-point fourth order methods using weight functions. We compare the special members of the new family with the old one through an numerical example to illustrate the efficiency of the new family.

Convergence of a parametric continuation method

The aim of this paper is to establish the semilocal convergence of a parameter based continuation method combining the Chebyshev's and the Super-Halley's methods for solving nonlinear equations in Banach spaces. The parameter α $in$ [0,1] be such that for α = 0 it reduces to the Chebyshev's method and for α = 1 to the Super-Halley's method. This convergence is established using recurrence relations under the assumption that the second order Frechet derivative satisfies the ω-continuity condition. This condition is milder than the Lipschitz and the Holder continuity conditions used for this purpose. A numerical example is given to show that the second order Frechet derivative satisfies the ω-continuity condition even when it fails to satisfy the Lipschitz and the Holder continuity conditions. A number of recurrence relations are derived based on two parameters. The existence and uniqueness regions along with a closed form of the error bounds in terms of a real parameter α $in$ [0, 1] for the solution x* is given. Two numerical examples are worked out to demonstrate the efficacy of the method. It is observed that our method gives better existence and uniqueness regions of the solution for both the examples when compared with the results obtained in [4] for both the Chebyshev's method (α = 0) and the Super-Halley's method (α = 1).

A continuation method for solving non-linear equations in ℝ

The aim of this paper is to describe two variants of a parameter-based continuation method connecting two third order iterative methods namely, the Chebyshev's method and the Super-Halley's method, for solving non-linear equations in R without using the second derivative. The convergence analysis is carried out to show the third order convergence of the methods. A number of numerical examples are worked out to illustrate the efficiency and performance of the methods. On comparison of the number of iterations taken by our methods with those obtained by Newton's method, the Chebyshev's method and the Super-Halley's method, it is observed that our methods take less number of iterations. It is further observed that for two different values of the parameter, our methods reduce to the Chebyshev's method and the Super-Halley's method free from second derivative.

Analysis of Convergence for Improved Chebyshev–Halley Methods Under Different Conditions

In this paper, we study a class of improved Chebyshev---Halley methods in Banach spaces and prove the semilocal convergence for these methods. Compared with the super-Halley method, these methods need one less inversion of an operator, and the R-order of these methods is also higher than the one of super-Halley method under the same conditions. Using recurrence relations, we analyze the semilocal convergence for these methods under two different convergence conditions. The convergence theorems are proved to show the existence and uniqueness of a solution. We also give some numerical results to show our approach.

Convergence analysis for a family of improved super-Halley methods under general convergence condition

In this paper, we focus on the semilocal convergence for a family of improved super-Halley methods for solving non-linear equations in Banach spaces. Different from the results in Wang et al. (J Optim Theory Appl 153:779---793, 2012), the condition of Holder continuity of third-order Frechet derivative is replaced by its general continuity condition, and the latter is weaker than former. Moreover, the R-order of the methods is also improved. By using the recurrence relations, we prove a convergence theorem to show the existence-uniqueness of the solution. The R-order of these methods is analyzed with the third-order Frechet derivative of the operator satisfies general continuity condition and Holder continuity condition.

Recurrence relations for Super-Halley's method with Hölder continuous second derivative in Banach spaces

The aim of this paper is to study the semilocal convergence of the Super-Halley's method used for solving nonlinear equations in Banach spaces by using the recurrence relations. This convergence is established under the assumption that the second Frëchet derivative of the involved operator satisfies the Hölder continuity condition which is milder than the Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. An existence-uniqueness theorem and a proori error estimates are provided for the solution x*. The R-order of the method equals to (2 + p) for p $in$ (0,1] is also established. Three numerical examples are worked out to demonstrate the efficacy of our approach. On comparison with the results obtained for the Super-Halley's method in [3] by using majorizing sequence, we observed improved existence and uniqueness regions for the solution x* by our approach.

Fourth-order convergence theorem by using majorizing functions for super-Halley method in Banach spaces

In this paper, we study the semilocal convergence of a multipoint fourth-order super-Halley method for solving nonlinear equations in Banach spaces. We establish the Newton–Kantorovich-type convergence theorem for the method by using majorizing functions. We also get the error estimate. In comparison with the results obtained in Wang et al. [X.H. Wang, C.Q. Gu, and J.S. Kou, Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces, Numer. Algorithms 56 (2011), pp. 497–516], we can provide a larger convergence radius. Finally, we report some numerical applications to demonstrate our approach.