A Steffensen-like method and its higher-order variants

For solving nonlinear equations, we suggest a second-order parametric Steffensen-like method, which is derivative free and only uses two evaluations of the function in one step. We also suggest a variant of the Steffensen-like method which is still derivative free and uses four evaluations of the function to achieve cubic convergence. Moreover, a fast Steffensen-like method with super quadratic convergence and a fast variant of the Steffensen-like method with super cubic convergence are proposed by using a parameter estimation. The error equations and asymptotic convergence constants are obtained for the discussed methods. The numerical results and the basins of attraction support the proposed methods.     

A variant of Steffensen’s method of fourth-order convergence and its applications

In this paper, a variant of Steffensen’s method of fourth-order convergence for solving nonlinear equations is suggested. Its error equation and asymptotic convergence constant are proven theoretically and demonstrated numerically. The derivative-free method only uses three evaluations of the function per iteration to achieve fourth-order convergence. Its applications on systems of nonlinear equations and boundary-value problems of nonlinear ODEs are showed as well in the numerical examples.     

A family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs

In this paper, a family of fourth-order Steffensen-type two-step methods is constructed to make progress in including Ren-Wu-Bi’s methods [H. Ren, Q. Wu, W. Bi, A class of two-step Steffensen type methods with fourth-order convergence, Appl. Math. Comput. 209 (2009) 206-210] and Liu-Zheng-Zhao’s method [Z. Liu, Q. Zheng, P. Zhao, A variant of Steffensens method of fourth-order convergence and its applications, Appl. Math. Comput. 216 (2010) 1978-1983] as its special cases. Its error equation and asymptotic convergence constant are deduced. The family provides the opportunity to obtain derivative-free iterative methods varying in different rates and ranges of convergence. In the numerical examples, the family is not only compared with the related methods for solving nonlinear equations, but also applied in the solution of BVPs of nonlinear ODEs by the finite difference method and the multiple shooting method.     

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.     

一个四阶收敛的牛顿类方法

A fourth-order convergence method of solving roots for nonlinear equation,which is a variant of Newton's method given.Its convergence properties is proved.It is at least fourth-order convergence near simple roots and one order convergence near multiple roots. In the end,numerical tests are given and compared with other known Newton and Newtontype methods.The results show that the proposed method has some more advantages than others.It enriches the methods to find the roots of non-linear equations and it ...     

一类四阶牛顿变形方法

给出非线性方程求根的一类四阶方法,也是牛顿法的变形方法.证明了方法收敛性,它们至少四次收敛到单根,线性收敛到重根.文末给出数值试验,且与牛顿法及其它牛顿变形法做了比较.结果表明方法具有很好的优越性,它丰富了非线性方程求根的方法,在理论上和应用上都有一定的价值.     

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.     

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.     

Third order iterative methods free from second derivative for nonlinear systems

In this work we present a family of predictor-corrector methods free from second derivative for solving nonlinear systems. We prove that the methods of this family are of third order convergence. We also perform numerical tests that allow us to compare these methods with Newton’s method. In addition, the numerical examples improve theoretical results, showing super cubic convergence for some methods of this family.     

An optimal Steffensen-type family for solving nonlinear equations

In this paper, a general family of Steffensen-type methods with optimal order of convergence for solving nonlinear equations is constructed by using Newton’s iteration for the direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub [H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651] that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2^m−1. Its error equations and asymptotic convergence constants are obtained. Finally, it is compared with the related methods for solving nonlinear equations in the numerical examples.     

Two new families of sixth-order methods for solving non-linear equations

In this paper, we developed two new families of sixth-order methods for solving simple roots of non-linear equations. Per iteration these methods require two evaluations of the function and two evaluations of the first-order derivatives, which implies that the efficiency indexes of our methods are 1.565. These methods have more advantages than Newton’s method and other methods with the same convergence order, as shown in the illustration examples. Finally, using the developing methodology described in this paper, two new families of improvements of Jarratt method with sixth-order convergence are derived in a straightforward manner. Notice that Kou’s method in [Jisheng Kou, Yitian Li, An improvement of the Jarratt method, Appl. Math. Comput. 189 (2007) 1816-1821] and Wang’s method in [Xiuhua Wang, Jisheng Kou, Yitian Li, A variant of Jarratt method with sixth-order convergence, Appl. Math. Comput. 204 (2008) 14-19] are the special cases of the new improvements.     

Third and fourth order iterative methods free from second derivative for nonlinear systems

In this work, we develop a family of predictor-corrector methods free from second derivative for solving systems of nonlinear equations. In general, the obtained methods have order of convergence three but, in some particular cases the order is four. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Newton’s classical method and with other recently published methods.     

Newton–harmonic balancing approach for accurate solutions to nonlinear cubic–quintic Duffing oscillators

This paper presents a new approach for solving accurate approximate analytical higher-order solutions for strong nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. The system is conservative and with odd nonlinearity. The new approach couples Newton’s method with harmonic balancing. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton’s method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution. Using the approach, accurate higher-order approximate analytical expressions for period and periodic solution are established. These approximate solutions are valid for small as well as large amplitudes of oscillation. In addition, it is not restricted to the presence of a small parameter such as in the classical perturbation method. Illustrative examples are presented to verify accuracy and explicitness of the approximate solutions. The effect of strong quintic nonlinearity on accuracy as compared to cubic nonlinearity is also discussed.     

Ostrowski’s fourth-order iterative method speedily solves cubic equations of state

Pressure-volume-temperature (P-V-T) data are required in simulating chemical plants because the latter usually involve production, separation, transportation, and storage of fluids. In the absence of actual experimental data, the pertinent mathematical model must rely on phase behaviour prediction by the so-called equations of state (EOS). When the plant model is a combination of differential and algebraic equations, simulation generally relies on numerical integration which proceeds in a piecewise fashion unless an approximate solution is needed at a single point. Needless to say, the constituent algebraic equations must be efficiently re-solved before each update of derivatives. Now, Ostrowski’s fourth-order iterative technique is a partial substitution variant of Newton’s popular second-order method. Although simple and powerful, this two-point variant has been utilised very little since its publication over forty years ago. After a brief introduction to cubic equations of state and their solution, this paper solves five of them. The results clearly demonstrate the superiority of Ostrowski’s method over Newton’s, Halley’s, and Chebyshev’s solvers.     

Some modifications of the quasilinearization method with higher-order convergence for solving nonlinear BVPs

In this paper, modifications of the quasilinearization method with higher-order convergence for solving nonlinear differential equations are constructed. A general technique for systematically obtaining iteration schemes of order m (?>?2) for finding solutions of highly nonlinear differential equations is developed. The proposed iterative schemes have convergence rates of cubic, quartic and quintic orders. These schemes were further applied to bifurcation problems and to obtain critical parameter values for the existence and uniqueness of solutions. The accuracy and validity of the new schemes is tested by finding accurate solutions of the one-dimensional Bratu and Frank-Kamenetzkii equations.     

Third-order iterative methods free from second derivatives for nonlinear equation

In this paper, we suggest and analyze a new two-step iterative method for solving nonlinear equations, which is called the modified Householder method without second derivatives for nonlinear equation. We also prove that the modified method has cubic convergence. Several examples are given to illustrate the efficiency and the performance of the new method. New method can be considered as an alternative to the present cubic convergent methods for solving nonlinear equations.     

Kink solutions for three new fifth order nonlinear equations

In this work we examine three new fifth order nonlinear evolution equations. The simplified form of the Hirota’s direct method is used to derive multiple kink solutions for the first two (1+1)-dimensional equations, and only two soliton solutions for the third (2+1)-dimensional equation. The dispersion relation is the same for the first two equations whereas the third one possesses a different dispersion relation.     

A class of Steffensen type methods with optimal order of convergence

In this paper, a family of Steffensen type methods of fourth-order convergence for solving nonlinear smooth equations is suggested. In the proposed methods, a linear combination of divided differences is used to get a better approximation to the derivative of the given function. Each derivative-free member of the family requires only three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index equal to 1.587. Kung and Traub conjectured that the order of convergence of any multipoint method without memory cannot exceed the bound 2^d-1, where d is the number of functional evaluations per step. The new class of methods agrees with this conjecture for the case d=3. Numerical examples are made to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other ones.     

Weaker conditions for the convergence of Newton’s method

Newton’s method is often used for solving nonlinear equations. In this paper, we show that Newton’s method converges under weaker convergence criteria than those given in earlier studies, such as Argyros (2004) [2, p. 387], Argyros and Hilout (2010)[11, p. 12], Argyros et al. (2011) [12, p. 26], Ortega and Rheinboldt (1970) [26, p. 421], Potra and Pták (1984) [36, p. 22]. These new results are illustrated by several numerical examples, for which the older convergence criteria do not hold but for which our weaker convergence criteria are satisfied.     

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.