A new two-point scheme for multiple roots of nonlinear equations

In most of the earlier research for multiple zeros, in order to obtain a new iteration function from the existing scheme, the usual practice is to make no change at the first substep. In this paper, we explore the idea that what are the advantages if the flexibility of choice is also given at the first substep. Therefore, we present a new two-point sixth-order scheme for multiple roots (m>1). The main advantages of our scheme over the existing schemes are flexibility at both substeps, simple body structure, smaller residual error, smaller error difference between two consecutive iterations, and smaller asymptotic error constant. The development of the scheme is based on midpoint formula and weight functions of two variables. We compare our methods with the existing methods of the same order with real-life applications as well as standard test problems. From the numerical results, we find that our methods can be considered as better alternates for the existing methods of the same order. Finally, dynamical study of the proposed schemes is presented that confirms the theoretical results.     

New third and fourth order nonlinear solvers for computing multiple roots

We present a new third order method for finding multiple roots of nonlinear equations based on the scheme for simple roots developed by Kou et al. [J. Kou, Y. Li, X. Wang, A family of fourth-order methods for solving non-linear equations, Appl. Math. Comput. 188 (2007) 1031-1036]. Further investigation gives rise to new third and fourth order families of methods which do not require second derivative. The fourth order family has optimal order, since it requires three evaluations per step, namely one evaluation of function and two evaluations of first derivative. The efficacy is tested on a number of relevant numerical problems. Computational results ascertain that the present methods are competitive with other similar robust methods.     

Fifth-order iterative method for finding multiple roots of nonlinear equations

This paper presents a fifth-order iterative method as a new modification of Newton’s method for finding multiple roots of nonlinear equations with unknown multiplicity m. Its convergence order is analyzed and proved. Moreover, several numerical examples demonstrate that the proposed iterative method is superior to the existing methods.     

Two new classes of optimal Jarratt-type fourth-order methods

In this paper, we investigate the construction of some two-step without memory iterative classes of methods for finding simple roots of nonlinear scalar equations. The classes are built through the approach of weight functions and these obtained classes reach the optimal order four using one function and two first derivative evaluations per full cycle. This shows that our classes can be considered as Jarratt-type schemes. The accuracy of the classes is tested on a number of numerical examples. And eventually, it is observed that our contributions take less number of iterations than the compared existing methods of the same type to find more accurate approximate solutions of the nonlinear equations.     

Some second-derivative-free variants of Halley’s method for multiple roots

In this paper, we present two new families of third-order methods for finding multiple roots of nonlinear equations. Each of them is based on a variant of the Halley’s method (for simple roots) free from second derivative. One of the families requires one evaluation of the function and two of its first derivative per iteration, and the other family requires two evaluations of the function and one of its first derivative. Several numerical examples are given to illustrate the performance of the presented methods.     

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.     

A third-order modification of Newton’s method for multiple roots

In this paper, we present a new third-order modification of Newton’s method for multiple roots, which is based on existing third-order multiple root-finding methods. Numerical examples show that the new method is competitive to other methods for multiple roots.     

Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs

A general class of multi-step iterative methods for finding approximate real or complex solutions of nonlinear systems is presented. The well-known technique of undetermined coefficients is used to construct the first method of the class while the higher order schemes will be attained by a frozen Jacobian. The point of attraction theory will be taken into account to prove the convergence behavior of the main proposed iterative method. Then, it will be observed that an m-step method converges with 2m-order. A discussion of the computational efficiency index alongside numerical comparisons with the existing methods will be given. Finally, we illustrate the application of the new schemes in solving nonlinear partial differential equations.     

Derivative free multipoint iterative methods for simple and multiple roots

A one-parameter family of derivative free multipoint iterative methods of orders three and four are derived for finding the simple and multiple roots off(x)=0. For simple roots, the third order methods require three function evaluations while the fourth order methods require four function evaluations. For multiple roots, the third order methods require six function evaluations while the fourth order methods require eight function evaluations. Numerical results show the robustness of these methods.     

A new family of Schröder's method and its variants based on power means for multiple roots of nonlinear equations

In this article, we derive one-parameter family of Schröder's method based on Gupta et al.'s (K.C. Gupta, V. Kanwar, and S. Kumar, A family of ellipse methods for solving non-linear equations, Int. J. Math. Educ. Sci. Technol. 40 (2009), pp. 571–575) family of ellipse methods for the solution of nonlinear equations. Further, we introduce new families of Schröder-type methods for multiple roots with cubic convergence. Proposed families are derived from modified Newton's method for multiple roots and one-parameter family of Schröder's method. Numerical examples are also provided to show that these new methods are competitive to other known methods for multiple roots.     

一类新的求解非线性方程的七阶方法

利用权函数法给出了一类求解非线性方程单根的七阶收敛的方法.每步迭代需要计算三个函数值和一个导数值,因此方法的效率指数为1.627.数值试验给出了该方法与牛顿法及同类方法的比较,显示了该方法的优越性.最后指出Kou等人给出的七阶方法是方法的特例.     

New modified optimal families of King’s and Traub-Ostrowski’s methods

Based on quadratically convergent Schröder’s method, we derive many new interesting families of fourth-order multipoint iterative methods without memory for obtaining simple roots of nonlinear equations by using the weight function approach. The classical King’s family of fourth-order methods and Traub-Ostrowski’s method are obtained as special cases. According to the Kung-Traub conjecture, these methods have the maximal efficiency index because only three functional values are needed per step. Therefore, the fourth-order family of King’s family and Traub-Ostrowski’smethod are the main findings of the present work. The performance of proposed multipoint methods is compared with their closest competitors, namely, King’s family, Traub-Ostrowski’s method, and Jarratt’s method in a series of numerical experiments. All the methods considered here are found to be effective and comparable to the similar robust methods available in the literature.     

Modified Jarratt method for computing multiple roots

In this paper, we present a fourth order method for computing multiple roots of nonlinear equations. The method is based on Jarratt scheme for simple roots [P. Jarratt, Some efficient fourth order multipoint methods for solving equations, BIT 9 (1969) 119-124]. The method is optimal, since it requires three evaluations per step, namely one evaluation of function and two evaluations of first derivative. The efficacy is tested on a number of relevant numerical problems. It is observed that the present scheme is competitive with other similar robust methods.     

A new fourth-order iterative method for finding multiple roots of nonlinear equations

In this paper, we present a new fourth-order method for finding multiple roots of nonlinear equations. It requires one evaluation of the function and two of its first derivative per iteration. Finally, some numerical examples are given to show the performance of the presented method compared with some known third-order methods.     

Variants of Steffensen-secant method and applications

In this paper, a parametric variant of Steffensen-secant method and three fast variants of Steffensen-secant method for solving nonlinear equations are suggested. They achieve cubic convergence or super cubic convergence for finding simple roots by only using three evaluations of the function per step. Their error equations and asymptotic convergence constants are deduced. Modified Steffensen’s method and modified parametric variant of Steffensen-secant method for finding multiple roots are also discussed. In the numerical examples, the suggested methods are supported by the solution of nonlinear equations and systems of nonlinear equations, and the application in the multiple shooting method.     

Efficient derivative-free variants of Hansen-Patrick’s family with memory for solving nonlinear equations

In this paper, we present a new tri-parametric derivative-free family of Hansen-Patrick type methods for solving nonlinear equations numerically. The proposed family requires only three functional evaluations to achieve optimal fourth order of convergence. In addition, acceleration of convergence speed is attained by suitable variation of free parameters in each iterative step. The self-accelerating parameters are estimated from the current and previous iteration. These self-accelerating parameters are calculated using Newton’s interpolation polynomials of third and fourth degrees. Consequently, the R-order of convergence is increased from 4 to 7, without any additional functional evaluation. Furthermore, the most striking feature of this contribution is that the proposed schemes can also determine the complex zeros without having to start from a complex initial guess as would be necessary with other methods. Numerical experiments and the comparison of the existing robust methods are included to confirm the theoretical results and high computational efficiency.     

A class of multi‐step difference schemes by using Padé approximant

In this paper, we present a method for the construction of a class of multi‐step finite differences schemes for solving arbitrary order linear two‐point boundary value problems. The construction technique is based on Padé approximant. It is easy to derive multi‐step difference schemes, and it includes many existing schemes as its special cases. Numerical experiments show that the proposed schemes are flexible and convergent. Copyright © 2013 John Wiley & Sons, Ltd.     

On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods

A novel method of locating all real roots of systems of nonlinear equations is presented here. The root finding problem is transformed to optimization problem, enabling the application of global optimization methods. Among many methods that exist in global optimization literature, Multistart and Minfinder are applied here because of their ability to locate not only the global minimum but also all local minima of the objective function. This procedure enables to locate all the possible roots of the system. Various test cases have been examined in order to validate the proposed procedure. This methodology does not make use of a priori knowledge of the number of the existing roots in the same manner as the corresponding global optimization methodology which does not make use of a priori knowledge of the existed number of local minima. Application of the new methodology resulted in finding all the roots in all test cases. The proposed methodology is general enough to be applied in any root finding problem.     

A family of modified Ostrowski’s methods with optimal eighth order of convergence

In this paper, we derive a new family of eighth-order methods for obtaining simple roots of nonlinear equations by using the weight function method. Each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative, so that their efficiency indices are 1.682, which are optimal according to the Kung and Traub’s conjecture (1974) [2]. Numerical comparisons are made to show the performance of the derived method, as is shown in the numerical section.     

New eighth-order iterative methods for solving nonlinear equations

In this paper, three new families of eighth-order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods. Per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative. This implies that the efficiency index of the developed methods is 1.682, which is optimal according to Kung and Traub’s conjecture [7] for four function evaluations per iteration. Notice that Bi et al.’s method in [2] and [3] are special cases of the developed families of methods. In this study, several new examples of eighth-order methods with efficiency index 1.682 are provided after the development of each family of methods. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.