求解非线性算子方程的加速迭代格式

研究非线性算子方程的近似求解方法.首先对通常的求解非线性方程加速迭代格式进行推广,得到高阶收敛速度的加速迭代格式,最后把这种加速迭代格式推广到非线性算子方程的求解中去,利用非线性算子的渐进展开,证明了这种加速格式具有三阶的收敛速度.     

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.     

求解非线性方程七阶收敛的牛顿迭代修正格式（英文）

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.     

用修正的平行割线法求解高阶奇异问题

许多实际问题中,方程在解点处的导算子为一高阶奇异算子,如反应扩散系统、优化问题中的歧点等.因此,对于求解高阶奇异问题的研究具有重要的实际意义.利用平行割线法求解高阶奇异问题,得到了渐近收敛速率,最后结合Hilbert空间几何特征,在几乎不增加计算量的前提下,修正了平行割线法,提高了渐近收敛速率.     

基于函数值Pad逼近的[1/n]阶迭代算法

<正>1引言一般的,我们在求解非线性方程的根时,利用最多的是迭代法,其迭代效果也各不一样[1-4].通常,我们在构造非线性方程求根的迭代方法有Newton迭代算法、Halley迭代算法和割线法等,而Newton迭代格式构造简单且收敛速度较快,又被认为是求解一般非线性方程根的最常用方法.在:Newton迭代公式的推导过程中,利用最多的是泰勒展开式法、切线法、积分法[5].本文基于函数值Pad6逼近的行列式表示[6-7],构造出[1/0]、[1/1]、[1/2]阶Pade逼近     

一种基于牛顿迭代改进的新的三阶预估校正格式

基于对牛顿迭代公式的改进及预估校正迭代的思想,提出了一种求解非线性方程的新的三阶预估-校正迭代格式.迭代公式无须计算函数的导数值,且理论上证明了它至少是三阶收敛的.数值实验验证了该迭代公式的有效性.     

关于二步割线法的收敛阶

[1]给出了解非线性议程组的二步割线法的收敛阶是超线性的,[2]又给出了解非线性算子方程的二步割线法具有二阶敛速,本文证明了解非线性方程的二步割线法的收     

关于非线性方程的牛顿迭代格式初始值选取的注记

基于构造非线性方程的牛顿迭代格式简便和牛顿迭代格式具有收敛快的特点,在解决实际问题时,牛顿迭代格式显得尤为重要,但是,牛顿迭代格式的初始值选取具有很大的局限性.利用泰勒级数展开,对牛顿迭代格式的收敛性进行分析,从而提出改进牛顿迭代格式的初始值选取方案,并利用不同的数值算例验证牛顿迭代格式收敛区域的改进方案的可行性,同时数值算例表明该方法具有操作简单的特点.     

一种新的迭代收敛阶数的证明与推广

迭代方法是求解非线性方程近似根的重要方法.本文基于隐函数存在定理,提出了一种新的迭代方法收敛性和收敛阶数的证明方法,并分别对牛顿(Newton)和柯西(Cauchy)迭代方法迭代收敛性和收敛阶数进行了证明.最后,利用本文提出的证明方法,证明了基于三次泰勒(Taylor)展式构成的迭代格式是收敛的,收敛阶数至少为4,并提出猜想,基于n次泰勒展式构成的迭代格式是收敛的,收敛阶数至少为(n+1).     

一类单调变分不等式的非精确交替方向法

交替方向法适合于求解大规模问题.该文对于一类变分不等式提出了一种新的交替方向法.在每步迭代计算中,新方法提出了易于计算的子问题,该子问题由强单调的线性变分不等式和良态的非线性方程系统构成.基于子问题的精确求解,该文证明了算法的收敛性.进一步,又提出了一类非精确交替方向法,每步迭代计算只需非精确求解子问题.在一定的非精确条件下,算法的收敛性得以证明.     

Relationships between different types of initial conditions for simultaneous root finding methods

The construction of initial conditions of an iterative method is one of the most important problems in solving nonlinear equations. In this paper, we obtain relationships between different types of initial conditions that guarantee the convergence of iterative methods for simultaneously finding all zeros of a polynomial. In particular, we show that any local convergence theorem for a simultaneous method can be converted into a convergence theorem with computationally verifiable initial conditions which is of practical importance. Thus, we propose a new approach for obtaining semilocal convergence results for simultaneous methods via local convergence results.     

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.     

Iterative methods for nonlinear systems associated with finite difference approach in stochastic differential equations

We present some iterative methods of different convergence orders for solving systems of nonlinear equations. Their computational complexities are studies. Then, we introduce the method of finite difference for solving stochastic differential equations of Itô-type. Subsequently, our multi-step iterative schemes are employed in this procedure. Several experiments are finally taken into account to show that the presented approach and methods work well.     

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.     

Practical Quasi-Newton algorithms for singular nonlinear systems

Quasi-Newton methods for solving singular systems of nonlinear equations are considered in this paper. Singular roots cause a number of problems in implementation of iterative methods and in general deteriorate the rate of convergence. We propose two modifications of QN methods based on Newton’s and Shamanski’s method for singular problems. The proposed algorithms belong to the class of two-step iterations. Influence of iterative rule for matrix updates and the choice of parameters that keep iterative sequence within convergence region are empirically analyzed and some conclusions are obtained.     

凸约束伪单调方程组的无导数投影算法

基于HS共轭梯度法的结构,本文在弱假设条件下建立了一种求解凸约束伪单调方程组问题的迭代投影算法.该算法不需要利用方程组的任何梯度或Jacobian矩阵信息,因此它适合求解大规模问题.算法在每一次迭代中都能产生充分下降方向,且不依赖于任何线搜索条件.特别是,我们在不需要假设方程组满足Lipschitz条件下建立了算法的全局收敛性和R-线收敛速度.数值结果表明,该算法对于给定的大规模方程组问题是稳定和有效的.     

A technique to choose the most efficient method between secant method and some variants

A few variants of the secant method for solving nonlinear equations are analyzed and studied. In order to compute the local order of convergence of these iterative methods a development of the inverse operator of the first order divided differences of a function of several variables in two points is presented using a direct symbolic computation. The computational efficiency and the approximated computational order of convergence are introduced and computed choosing the most efficient method among the presented ones. Furthermore, we give a technique in order to estimate the computational cost of any iterative method, and this measure allows us to choose the most efficient among them.     

On a simultaneous method of Newton-Weierstrass’ type for finding all zeros of a polynomial

Combining a suitable two-point iterative method for solving nonlinear equations and Weierstrass’ correction, a new iterative method for simultaneous finding all zeros of a polynomial is derived. It is proved that the proposed method possesses a cubic convergence locally. Numerical examples demonstrate a good convergence behavior of this method in a global sense. It is shown that its computational efficiency is higher than the existing derivative-free methods.     

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.     

基于连分式重新推导的Newton迭代公式及其变体

基于Thiele连分式,重新建立了求解非线性方程的经典的Newton迭代公式.为了避免求导数运算,采用差商可以近似代替导数的办法,得到Newton迭代方法的几个变体并给出了其收敛的阶数.最后,数值实例证实了这些迭代格式是有效的.