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.     

牛顿方法的两个新格式

给出牛顿迭代方法的两个新格式,S im pson牛顿方法和几何平均牛顿方法,证明了它们至少三次收敛到单根,线性收敛到重根.文末给出数值试验,且与其它已知牛顿法做了比较.结果表明收敛性方法具有较好的优越性,它们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.     

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.     

非线性方程求根的一类预测-校正迭代方法

考虑了非线性方程求根问题,即从一类特殊的积分出发获得了非线性方程求根的方法,所得方法推广了已有结果.将所得方法与变形的牛顿迭代法相结合,获得了非线性方程求根的实用的预测-校正格式,并证明了当β=1/2时格式至少具有局部平方收敛.数值算例表明,所得格式迭代步数少,收敛速度快,是非线性方程求根的有效方法之一.     

Runge-Kutta方法用于非线性方程求根

将Runge-Kutta方法用于非线性方程求根问题,给出二阶,三阶和四阶对应的三个新的方程求根公式,证明了它们至少三次收敛到单根,线性收敛到重根.文末给出数值试验,且与其它已知求根公式做了比较.结果表明此方法具有较好的优越性,它们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.     

几类改进的新的两步六阶Chebyshev-Halley方法

利用权函数法,给出非线性方程求根的Chebyshev-Halley方法的几类改进方法,证明方法六阶收敛到单根.Chebyshev-Halley方法的效率指数为1.442,改进后的两步方法的效率指数为1.565.最后给出数值试验,且与牛顿法,Chebyshev-Halley 方法及其它已知的方程求根方法做了比较.结果表明方法具有一定的优越性.     

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.     

非线性椭圆边值问题的牛顿型迭代法

§1.引言 牛顿迭代法是解非线性方程最著名的方法之一.用牛顿法求解非线性方程,事实上就是通过一系列线性方程的解来逼近原来非线性方程的解.简而言之,就是一种线性化方法.而经典的牛顿法虽有(在一定条件下)平方收敛的性质,但却是局部收敛的.就是说,初始近似要选得足够接近原问题的解,否则可能导致不收敛.后来,人们利用牛顿法     

求解非线性方程的最优8阶史蒂芬森方法

本文研究了非线性方程求根问题.利用权函数方法,获得了一种三步8阶收敛的史蒂芬森型方法.实验结果表明本文提出的方法计算时间少于其它同阶的最优方法.     

求解非线性方程的最优8阶史蒂芬森方法

本文研究了非线性方程求根问题. 利用权函数方法, 获得了一种三步8阶收敛的史蒂芬森型方法. 实验结果表明本文提出的方法计算时间少于其它同阶的最优方法.     

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 ...     

一类三阶牛顿变形方法

A class of third-order convergence methods of solving roots for non-linear equation,which are variant Newton's method,are given.Their convergence properties are proved.They are at least third order convergence near simple root and one order convergence near multiple roots.In the end,numerical tests are given and compared with other known Newton's methods.The results show that the proposed methods have some more advantages than others.They enrich the methods to find the roots of non-linear equations and they are important in both theory and application.     

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.     

An adaptive Newton-method based on a dynamical systems approach

The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.     

Newton and quasi-Newton methods for equations of smooth compositions of semismooth functions

The Newton method and the quasi-Newton method for solving equations of smooth compositions of semismooth functions are proposed. The Q-superlinear convergence of the Newton method and the Q-linear convergence of the quasi-Newton method are proved. The present methods can be more easily implemented than previous ones for this class of nonsmooth equations.     

Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations

Hermitian and skew-Hermitian splitting(HSS) method has been proved quite successfully in solving large sparse non-Hermitian positive definite systems of linear equations. Recently, by making use of HSS method as inner iteration, Newton-HSS method for solving the systems of nonlinear equations with non-Hermitian positive definite Jacobian matrices has been proposed by Bai and Guo. It has shown that the Newton-HSS method outperforms the Newton-USOR and the Newton-GMRES iteration methods. In this paper, a class of modified Newton-HSS methods for solving large systems of nonlinear equations is discussed. In our method, the modified Newton method with R-order of convergence three at least is used to solve the nonlinear equations, and the HSS method is applied to approximately solve the Newton equations. For this class of inexact Newton methods, local and semilocal convergence theorems are proved under suitable conditions. Moreover, a globally convergent modified Newton-HSS method is introduced and a basic global convergence theorem is proved. Numerical results are given to confirm the effectiveness of our method.