Modified families of multi-point iterative methods for solving nonlinear equations

We further present some semi-discrete modifications to the cubically convergent iterative methods derived by Kanwar and Tomar (Modified families of Newton, Halley and Chebyshev methods, Appl. Math. Comput. http://dx.doi.org/10.1016/j.amc.2007.02.119) and derived a number of interesting new classes of third-order multi-point iterative methods free from second derivatives. Furthermore, several functions have been tested and all the methods considered are found to be effective and compared to the well-known existing third and fourth-order multi-point iterative methods.      

Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations

In this paper, we derive one-parameter families of Newton, Halley, Chebyshev, Chebyshev-Halley type methods, super-Halley, C-methods, osculating circle and ellipse methods respectively for finding simple zeros of nonlinear equations, permitting f ′ (x) = 0 at some points in the vicinity of the required root. Halley, Chebyshev, super-Halley methods and, as an exceptional case, Newton method are seen as the special cases of the family. All the methods of the family and various others are cubically convergent to simple roots except Newton’s or a family of Newton’s method.      

Reduced Recurrence Relations for the Chebyshev Method

In this paper, we give sufficient conditions ensuring the convergence of the Chebyshev method in Banach spaces. We use a new system of recurrence relations which simplifies those given by Kantorovich for the Newton method or those given by Candela and Marquina for the Chebyshev and Halley methods.     

Chebyshev–Halley methods for analytic functions

Two modifications of the family of Chebyshev–Halley methods are given. The first is to improve the rate of convergence to a multiple zero of an analytic function. The second is to find simultaneously all distinct zeros of a polynomial.     

A simply constructed third-order modifications of Newton's method

In this paper, we present a simple and easily applicable approach to construct some third-order modifications of Newton's method for solving nonlinear equations. It is shown by way of illustration that existing third-order methods can be employed to construct new third-order iterative methods. The proposed approach is applied to the classical Chebyshev–Halley methods to derive their second-derivative-free variants. Numerical examples are given to support that the methods thus obtained can compete with known third-order methods.     

一类修正的Halley迭代法的统一收敛性定理

In this paper a class of modified Halley iteration methods for simultaneously finding polynomial zeros is discussed. A unified convergence theorem is proposed and the efficiency analysis is given.     

On Halley-type iterations with free second derivative

In this paper, we relax the convergence conditions required in Ezquerro and Hernández (Int. J. Pure Appl. Math. 6(1) (2003) 103) for a multipoint third-order iteration of Halley type, where the conditions provided are the known ones for methods of order three.     

On a new family of high‐order iterative methods for the matrix pth root

The main goal of this paper is to approximate the principal pth root of a matrix by using a family of high‐order iterative methods. We analyse the semi‐local convergence and the speed of convergence of these methods. Concerning stability, it is well known that even the simplified Newton method is unstable. Despite it, we present stable versions of our family of algorithms. We test numerically the methods: we check the numerical robustness and stability by considering matrices that are close to be singular and badly conditioned. We find algorithms of the family with better numerical behavior than the Newton and the Halley methods. These two algorithms are basically the iterative methods proposed in the literature to solve this problem. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

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]].     

The Hirota’s bilinear method and the tanh–coth method for multiple-soliton solutions of the Sawada–Kotera–Kadomtsev–Petviashvili equation

In this work we derive a new completely integrable dispersive equation. The equation is obtained by combining the Sawada–Kotera (SK) equation with the sense of the Kadomtsev–Petviashvili (KP) equation. The newly derived Sawada–Kotera–Kadomtsev–Petviashvili (SK–KP) equation is studied by using the tanh–coth method, to obtain single-soliton solution, and by the Hirota bilinear method, to determine the N-soliton solutions. The study highlights the significant features of the employed methods and its capability of handling completely integrable equations.     

求解方程的一类迭代公式

Using the forms of Newton iterative function, the iterative function of Newton's method to handle the problem of multiple roots and the Halley iterative function, we give a class of iterative formulae for solving equations in one variable in this paper and show that their convergence order is at least quadratic. At last we employ our methods to solve some non-linear equations and compare them with Newton's method and Halley's method. Numerical results show that our iteration schemes are convergent if we choose two suitable parametric functions λ(x) and μ(x). Therefore, our iteration schemes are feasible and effective.     

Modified Chebyshev-Halley type method and its variants for computing multiple roots

We present two families of third order methods for finding multiple roots of nonlinear equations. One family is based on the Chebyshev-Halley scheme (for simple roots) and includes Halley, Chebyshev and Chun-Neta methods as particular cases for multiple roots. The second family is based on the variant of Chebyshev-Halley scheme and includes the methods of Dong, Homeier, Neta and Li et al. as particular cases. The efficacy is tested on a number of relevant numerical problems. It is observed that the new methods of the families are equally competitive with the well known special cases of the families.     

On the semilocal convergence behavior for Halley’s method

The present paper is concerned with the semilocal convergence problems of Halley’s method for solving nonlinear operator equation in Banach space. Under some so-called majorant conditions, a new semilocal convergence analysis for Halley’s method is presented. This analysis enables us to drop out the assumption of existence of a second root for the majorizing function, but still guarantee Q-cubic convergence rate. Moreover, a new error estimate based on a directional derivative of the twice derivative of the majorizing function is also obtained. This analysis also allows us to obtain two important special cases about the convergence results based on the premises of Kantorovich and Smale types.     

Optimized mean based second derivative-free families of Chebyshev–Halley type methods

In this paper, we present new interesting fourth-order optimal families of Chebyshev–Halley type methods free from second-order derivative. In terms of computational cost, eachmember of the families requires two functions and one first-order derivative evaluation per iteration, so that their efficiency indices are 1.587. It is found by way of illustration that the proposed methods are useful in high-precision computing environment. Moreover, it is also observed that larger basins of attraction belong to ourmethods although the othersmethods are slow and have darker basins while some of the methods are too sensitive upon the choice of the initial guess.     

并行Halley迭代法的修正及其效率分析

In this paper a modification of the parallel Halley iteration method for simultaneously finding polynomial zeros is discussed. The convergence and the convergence rate with high order are obtained and the efficiency analysis is given.     

Partitioned Runge–Kutta Methods in Lie-Group Setting

We introduce partitioned Runge–Kutta (PRK) methods as geometric integrators in the Runge–Kutta–Munthe-Kaas (RKMK) method hierarchy. This is done by first noticing that tangent and cotangent bundles are the natural domains for the differential equations to be solved. Next, we equip the (co)tangent bundle of a Lie group with a group structure and treat it as a Lie group. The structure of the differential equations on the (co)tangent-bundle Lie group is such that partitioned versions of the RKMK methods are naturally introduced. Numerical examples are included to illustrate the new methods.     

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.     

On Halley iteration

This paper proposes some modified Halley iterations for finding the zeros of polynomials. We investigate the non-overshoot properties of the modified Halley iterations and other important properties that play key roles in solving symmetric eigenproblems. We also extend Halley iteration to systems of polynomial equations in several variables. Received March 20, 1996 / Revised version received December 5, 1997     

Some new additive Runge–Kutta methods and their applications

We propose some new additive Runge–Kutta methods of orders ranging from 2 to 4 that may be used for solving some nonlinear system of ODEs, especially for the temporal discretization of some nonlinear systems of PDEs with constraints. Only linear ODEs or PDEs need to be solved at each time step with these new methods.