一种求多项式方程根的参数并行加速迭代法

推广了一种在无重根情况下,利用Newton类迭代法对同时求多项式零点的加速的迭代法.讨论了该方法的收敛性和收敛阶;最后给出数值算例表明:计算收敛阶和定理结论是一致的,且本算法具有较大的收敛范围.     

拟等级网格下非线性延迟微分方程间断有限元法

本文利用间断有限元法求解非线性延迟微分方程,在拟等级网格下.给出非线性延迟微分方程间断有限元解的整体收敛阶和局部超收敛阶,数值实验验证了理论结果的正确性.     

一个三阶牛顿变形方法

基于反函数建立的积分方程,结合Simpson公式,给出了一个非线性方程求根的新方法,即为牛顿变形方法.证明了它至少三次收敛到单根,与牛顿法相比,提高了收敛阶和效率指数.文末给出数值试验,且与牛顿法和同类型牛顿变形法做了比较.结果表明方法具有较好的优越性,它丰富了非线性方程求根的方法.     

求解单变量无约束优化问题的一类新割线法

本文基于分式逼近提出了一类求解单变量无约束优化问题的新割线法,给出并证明了该方法的收敛阶是(√2+1).并进一步对新方法的性能进行了分析,给出了新方法、经典的牛顿法和其他修正的割线类方法解单变量无约束优化问题的数值实验.理论和数值结果均表明新的割线法是有效的.     

一种基于随机Taylor展开式的随机微分方程数值解法

给出了一种基于随机Taylor展开式的随机微分方程数值格式,证明了它的均方稳定性。此外,还证明了这种数值格式的均值意义下的局部收敛阶为2,均方意义下的局部收敛阶为1,均方强收敛阶为1.数值实验表明本文的方法比Euler法和Milstein方法具有更好的逼近效果.     

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.     

一种求解非线性方程的修正牛顿迭代法

基于牛顿迭代法,提出了一种求解非线性方程的修正牛顿迭代法,并证明了该方法是3阶收敛的.最后,通过数值实验对比了常见的其他三种类型的迭代法,说明这类修正牛顿迭代法与传统的牛顿迭代法相比,具有更快的收敛速度,从而进一步证实了该方法的有效性.     

Simpson校正公式

给出了Simpson校正公式的截断误差,分析了复化Simpson校正公式的收敛阶.数值算例验证了理论分析的正确性.     

求解非线性方程根四阶收敛的迭代格式

提出了求解非线性方程根新的四阶收敛迭代方法,新方法每次迭代只需要两次函数计算,一次一阶导数值计算,效能指数达到1.587.通过几个数值算例来解释该方法的有效性.     

非光滑函数的分数阶插值公式

本文基于局部分数阶Taylor展开式构造非光滑函数的分数阶插值公式,证明了插值公式的存在和唯一性,给出了分数阶插值的Lagrange表示形式及其误差余项,讨论了一种混合型的分段分数阶插值和整数阶插值的收敛阶.数值算例验证了对于非光滑函数分数阶插值明显优于通常的多项式插值,并说明在实际计算中采用分段混合分数阶和整数阶插值可以使得插值误差在区间上分布均匀,能够极大地提高插值精度.     

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.     

A family of Steffensen type methods with seventh-order convergence

In this paper, based on some known fourth-order Steffensen type methods, we present a family of three-step seventh-order Steffensen type iterative methods for solving nonlinear equations and nonlinear systems. For nonlinear systems, a development of the inverse first-order divided difference operator for multivariable function is applied to prove the order of convergence of the new methods. Numerical experiments with comparison to some existing methods are provided to support the underlying theory.     

改进的两步高阶Euler-Chebyshev方法

给出非线性方程求根的Euler-Chebyshev方法的改进方法,证明了方法的收敛性,它们七次和九次收敛到单根.给出数值试验,且与牛顿法及其它较高阶的方程求根方法做了比较.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.     

New efficient derivative free family of seventh-order methods for solving systems of nonlinear equations

We present a three-step two-parameter family of derivative free methods with seventh-order of convergence for solving systems of nonlinear equations numerically. The proposed methods require evaluation of two central divided differences and inversion of only one matrix per iteration. As a result, the proposed family is more efficient as compared with the existing methods of same order. Numerical examples show that the proposed methods produce approximations of greater accuracy and remarkably reduce the computational time for solving systems of nonlinear equations.     

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.     

A globally convergent Newton method for solving strongly monotone variational inequalities

Variational inequality problems have been used to formulate and study equilibrium problems, which arise in many fields including economics, operations research and regional sciences. For solving variational inequality problems, various iterative methods such as projection methods and the nonlinear Jacobi method have been developed. These methods are convergent to a solution under certain conditions, but their rates of convergence are typically linear. In this paper we propose to modify the Newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length. The purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate. It is then shown that, under the strong monotonicity assumption, the method is globally convergent and, under some additional assumptions, the rate of convergence is quadratic. Limited computational experience indicates the high efficiency of the proposed method.     

A fifth-order family of modified Newton methods

Addition of a correction term every other Newton iteration provides a fifth-order method for finding simple zeros of nonlinear functions. A two-parameter family of such methods is developed. Each family member requires the given function and its derivative to be evaluated at two points per step.Work supported by the British Science Research Council at the University of Dundee, and by the U.S. Atomic Energy Commission.     

A note on some quadrature based three-step iterative methods for non-linear equations

In a recent paper [N.A. Mir, T. Zaman, Some quadrature based three-step iterative methods for non-linear equations, Appl. Math. Comput. 193 (2007) 366-373], some new three-step iterative methods for non-linear equations have been proposed. In this note, we show that the Algorithm 2.2 and Algorithm 2.3 given by the authors have twelfth-order and ninth-order convergence respectively, not seventh-order one as claimed in their work.     

Hansen Patrick迭代的局部收敛性

讨论了Hansen-Patrick迭代的局部特征关系式,引入函数T（t),利用逐步归纳技巧,证明了在α为一定条件下Hansen-Patrick迭代过程对方程f(z)=0零点的局部收敛性。     

求解非线性方程七阶收敛的牛顿迭代修正格式

本文研究了非线性方程求解的问题.利用泰勒公式和耦合方法,获得了一种求解非线性方程的加速收敛的七阶迭代改进格式,该格式不需要计算高阶导数,且具有更大的收敛半径,大大提高了计算效率.