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.     

A variant of Chebyshev’s method with sixth-order convergence

In this paper, we present a new variant of Chebyshev’s method for solving non-linear equations. Analysis of convergence shows that the new method has sixth-order convergence. Per iteration the new method requires two evaluations of the function, one of its first derivative and one of its second derivative. Thus the efficiency, in term of function evaluations, of the new method is better than that of Chebyshev’s method. Numerical examples verifying the theory are given.      

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.     

On new third-order convergent iterative formulas

In this paper, we present new iteration methods with cubic convergence for solving nonlinear equations. The main advantage of the new methods are free from second derivatives and it permit that the first derivative is zero in some points. Analysis of efficiency shows that the new methods can compete with Newton’s method and the classical third-order methods. Numerical results indicate that the new methods are effective and have definite practical utility.      

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.     

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.     

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.     

A class of three-point root-solvers of optimal order of convergence

The construction of a class of three-point methods for solving nonlinear equations of the eighth order is presented. These methods are developed by combining fourth order methods from the class of optimal two-point methods and a modified Newton’s method in the third step, obtained by a suitable approximation of the first derivative based on interpolation by a nonlinear fraction. It is proved that the new three-step methods reach the eighth order of convergence using only four function evaluations, which supports the Kung-Traub conjecture on the optimal order of convergence. Numerical examples for the selected special cases of two-step methods are given to demonstrate very fast convergence and a high computational efficiency of the proposed multipoint methods. Some computational aspects and the comparison with existing methods are also included.     

Some variants of Hansen-Patrick method with third and fourth order convergence

Based on Hansen-Patrick method [E. Hansen, M. Patrick, A family of root finding methods, Numer. Math. 27 (1977) 257-269], we derive a two-parameter family of methods for solving nonlinear equations. All the methods of the family have third-order convergence, except one which has the fourth-order convergence. In terms of computational cost, all these methods require evaluations of one function, one first derivative and one second derivative per iteration. Numerical examples are given to support that the methods thus obtained are competitive with other robust methods of similar kind. Moreover, it is shown by way of illustration that the present methods, particularly fourth-order method, are very effective in high precision computations.     

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.     

一个阶为2+\sqrt{6}的Newton改进算法

针对非线性方程求单根问题,提出了一种新的Newton预测-校正格式.通过每步迭代增加计算一个函数值和一阶导数值,使得每步迭代需要估计两个函数值和两个一阶导数值.与标准的Newton算法的二阶收敛速度相比,新算法具有更高阶的收敛速度2+\sqrt{6}.通过测试函数对新算法进行测试, 与相关算法比较,表明算法在迭代次数、运算时间及最优值方面都具有较明显的优势. 最后,将这种新格式推广到多维向量值函数, 采用泰勒公式证明了其收敛性,并给出了两个二维算例来验证其收敛的有效性.     

A note on King's fifth-order family of methods for solving equations

King's two-step fifth-order methods for solving equations require two function and two derivative evaluations per iteration. Using exactly the same information and composing two-point methods of order 2.732 with themselves, other two-step methods can be obtained with asymptotic convergence rates 7.464 which is better than the 5 of King.     

The cubic semilocal convergence on two variants of Newton's method

In this paper, we discuss two variants of Newton's method without using any second derivative for solving nonlinear equations. By using the majorant function and confirming the majorant sequences, we obtain the cubic semilocal convergence and the error estimation in the Kantorovich-type theorems. The numerical examples are presented to support the usefulness and significance.     

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.     

A globally convergent BFGS method for nonlinear monotone equations without any merit functions

Since 1965, there has been significant progress in the theoretical study on quasi-Newton methods for solving nonlinear equations, especially in the local convergence analysis. However, the study on global convergence of quasi-Newton methods is relatively fewer, especially for the BFGS method. To ensure global convergence, some merit function such as the squared norm merit function is typically used. In this paper, we propose an algorithm for solving nonlinear monotone equations, which combines the BFGS method and the hyperplane projection method. We also prove that the proposed BFGS method converges globally if the equation is monotone and Lipschitz continuous without differentiability requirement on the equation, which makes it possible to solve some nonsmooth equations. An attractive property of the proposed method is that its global convergence is independent of any merit function.We also report some numerical results to show efficiency of the proposed method.

     

一类奇性积分方程的多尺度配置快速算法

This paper develops fast multiscale collocation methods for a class of Fredholm integral equations of the second kind with singular kernels. A truncation strategy for the coefficient matrix of the corresponding discrete system is proposed, which forms a basis for fast algorithms. The convergence, stability and computational complexity of these algorithms are analyzed.     

Modified Jarratt method with sixth-order convergence

In this paper, we present a variant of Jarratt method with order of convergence six for solving non-linear equations. Per iteration the method requires two evaluations of the function and two of its first derivatives. The new multistep iteration scheme, based on the new method, is developed and numerical tests verifying the theory are also given.     

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.     

Algorithm for forming derivative-free optimal methods

We develop a simple yet effective and applicable scheme for constructing derivative free optimal iterative methods, consisting of one parameter, for solving nonlinear equations. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on k+1 evaluations could achieve a maximum convergence order of $2^{k}$ . Through the scheme, we construct derivative free optimal iterative methods of orders two, four and eight which request evaluations of two, three and four functions, respectively. The scheme can be further applied to develop iterative methods of even higher orders. An optimal value of the free-parameter is obtained through optimization and this optimal value is applied adaptively to enhance the convergence order without increasing the functional evaluations. Computational results demonstrate that the developed methods are efficient and robust as compared with many well known methods.     

Various Newton-type iterative methods for solving nonlinear equations

The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterative methods for solving nonlinear equations. The ninth order convergent Newton-type iterative method is made derivative free to obtain seventh-order convergent Newton-type iterative method. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterative methods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.