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.     

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.     

非线性方程的三种牛顿迭代方法

研究了求解非线性方程的三种迭代方法:经典牛顿迭代、五阶牛顿迭代及九阶牛顿迭代.通过设置不同的初始点,详细对比了三种迭代法的收敛性.     

开普勒定律的数学解释及现代证明

牛顿和开普勒关于行星运动的数学解释是科学史上极其重要的两大成就.牛顿对开普勒定律的解释,虽然包含了许多微积分的基本思想,其推理还是用的相似三角形和几何学[1].这里,我们可以给出一个现代的证明方法来解释牛顿的推算.     

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.     

基于模量重构的张量互补问题的光滑牛顿算法（英文）

近年来,张量作为矩阵的推广,得到了广泛的研究.在众多张量相关的问题中,张量互补问题(TCP)是许多学者研究的一个重要领域,人们提出了许多解决TCP的方法.本文在强P-张量张量和光滑逼近函数的基础上,提出一种基于基于模的重构的TCP光滑牛顿算法,证明光滑牛顿方法是全局收敛的.数值算例验证了光滑牛顿算法的有效性.     

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.     

无约束最优化问题中一类新黄族及其性质

本文提出了一类新的用于解决无约束最优化问题的拟牛顿方法,并证明了这样的性质,在 精确线性搜索条件下,每一步该族所有方法所产生的迭代方向和迭代点列仅依赖于参数ρ.该方 法可视为拟牛顿方法中黄族的推广.     

牛顿(Newton)后向插指

利用牛顿后向插指算子将牛顿(Newton)插值推广到牛顿后向插指,并得到了牛顿后向插指算子与牛顿后向插值算子之间的关系.     

结合滤子技术的牛顿折线法及其实现

成功将多维滤子技术应用到牛顿折线法,提出了多维滤子牛顿折线法.新算法增加了牛顿点以及信赖域的试探点被接收作为下一步迭代点的几率.在一定的假设条件下证明了算法的全局收敛性.数值试验表明,滤子牛顿折线法适合于求解等势线呈峡谷状的函数.     

具有参数平方收敛的线性插值迭代类

提出了一类具有参数平方收敛的求解非线性方程的线性插值迭代法,方法以Newton法和Steffensen法为其特例,并且给出了该类方法的最佳迭代参数.数值试验表明,选用最佳迭代参数或其近似值的新方法比Newton法和Steffensen方法更有效.     

A novel method for robust minimisation of univariate functions with quadratic convergence

We describe a novel method for minimisation of univariate functions which exhibits an essentially quadratic convergence and whose convergence interval is only limited by the existence of near maxima. Minimisation is achieved through a fixed-point iterative algorithm, involving only the first and second-order derivatives, that eliminates the effects of near inflexion points on convergence, as usually observed in other minimisation methods based on the quadratic approximation. Comparative numerical studies against the standard quadratic and Brent's methods demonstrate clearly the high robustness, high precision and convergence rate of the new method, even when a finite difference approximation is used in the evaluation of the second-order derivative.     

具有参数的不带有导数的平方收敛的迭代法

1.引 言 考虑数值求解非线性方程 f(x)=0, (1)其中实值函数f(x)在实零点x*的某邻域U(x*)内连续可微且f'(x)≠0. 牛顿法是科学与工程计算中数值求解(1)的常用数值方法.虽然它一般至少是二阶收敛的,但它需要调用导数值,这使其应用受到限制.我们修改牛顿法,用割线代替切线可得不带     

A modified accelerated monotone iterative method for finite difference reaction-diffusion-convection equations

This paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction-diffusion-convection equations with nonlinear boundary conditions. A modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time-dependent problem and its corresponding steady-state problem. This method leads to a simple and yet efficient linear iterative algorithm. It yields two sequences of iterations that converge monotonically from above and below, respectively, to a unique solution of the system. The monotone property of the iterations gives concurrently improving upper and lower bounds for the solution. It is shown that the rate of convergence for the sum of the two sequences is quadratic. Under an additional requirement, quadratic convergence is attained for one of these two sequences. In contrast with the existing accelerated monotone iterative methods, our new method avoids computing local maxima in the construction of these sequences. An application using a model problem gives numerical results that illustrate the effectiveness of the proposed method.     

Improving order and efficiency: Composition with a modified Newton’s method

In this paper a zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to compose a given iterative method with a modified Newton’s method that introduces just one evaluation of the function. To carry out this procedure some classical methods with different orders of convergence are used to obtain new methods that can be generalized in Banach spaces.     

PRECONDITIONING BLOCK LANCZOS ALGORITHM FOR SOLVING SYMMETRIC EIGENVALUE PROBLEMS

1. Introduction;The Lanczos process is an effective method [1, 2, 14, 21] for computing a feweigenValues and corresponding eigenvectors of a large sparse symmetric matrix A ERnxn. If it is practical to factor the matrix A -- PI for one or more values of p near thedesired eigenvalues, the Lanczos method can be used with the inverted operator andconvergence will be very rapid[5,10,22]. In practical applications, however, the matrixA is usually large and sparse, so factoring A is either impos…     

Accelerated monotone iterative methods for finite difference equations of reaction-diffusion

This paper is concerned with numerical methods for a finite difference system of reaction-diffusion-convection equation under nonlinear boundary condition. Various monotone iterative methods are presented, and each of these methods leads to an existence-comparison theorem as well as a computational algorithm for numerical solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function. Application is given to a model problem from chemical engineering, and some numerical results, including a test problem with known analytical solution, are presented to illustrate the various rates of convergence of the iterations. Received November 2, 1995 / Revised version received February 10, 1997     

A very simple SQCQP method for a class of smooth convex constrained minimization problems with nice convergence results

We introduce a new and very simple algorithm for a class of smooth convex constrained minimization problems which is an iterative scheme related to sequential quadratically constrained quadratic programming methods, called sequential simple quadratic method (SSQM). The computational simplicity of SSQM, which uses first-order information, makes it suitable for large scale problems. Theoretical results under standard assumptions are given proving that the whole sequence built by the algorithm converges to a solution and becomes feasible after a finite number of iterations. When in addition the objective function is strongly convex then asymptotic linear rate of convergence is established.     

A method for obtaining iterative formulas of order three

An improved method for the order of convergence of iterative formulas of order two is given. Using this method, new third-order modifications of Newton’s method are derived. A comparison with other methods is given.     

A Parameterized Class of Complex Nonsymmetric Algebraic Riccati Equations

In this paper, by introducing a definition of parameterized comparison matrix of a given complex square matrix, the solvability of a parameterized class of complex nonsymmetric algebraic Riccati equations (NAREs) is discussed. The existence and uniqueness of the extremal solutions of the NAREs is proved. Some classical numerical methods can be applied to compute the extremal solutions of the NAREs, mainly including the Schur method, the basic fixed-point iterative methods, Newton's method and the doubling algorithms. Furthermore, the linear convergence of the basic fixed-point iterative methods and the quadratic convergence of Newton's method and the doubling algorithms are also shown. Moreover, some concrete parameter selection strategies in complex number field for the doubling algorithms are also given. Numerical experiments demonstrate that our numerical methods are effective.