牛顿法

But it is a hard task to solve this equation, one reason is that the TV-norm is nondifferentiable when |▽_u|= 0, so we cant apply a linearizationtechnique such as the Newton method. The other reason is that the Euler-Lagrange equation has a highly nonlinear term. The Newton method for such equation is known to have a very small domain of convergence, and it is very difficult to make sure that the initial value such as the observed image belongs to this domain. So the Newton method is not an ideal method for solving this equation.

但是求解它比较困难，主要是因为TV-范数在|▽u|=0处不可微，不能用诸如牛顿法之类的方法将其线性化；且Euler-Lagrange方程含有一个高度非线性的项，牛顿法只有局部收敛性，对于高度非线性问题它的收敛域很小，因此难以保证所取的初始点在它的收敛域内，故一般不用牛顿法直接求解。

The derivatives of the nonsmooth equations are computed to make the generalized derivative matrix nonsingular. Then the nonsmmoth damped Newton method is used to solve them.Analytic solutions of the KdV equation with variable coefficients: Shallow water wave problems belong to problems of free surface wave, and can be molded by the nonsmooth equations model given above in principle.

在前面建立的非光滑方程组数学模型和固定网格法基础上，利用广义导数的概念给出了求解渗流自由面的一种新方法-----非光滑阻尼牛顿法，该法是对非光滑方程组求导，适当的处理广义导数矩阵使其非奇异，利用非光滑牛顿法求解。

These strategies include: an improved adaptive movement penalty method which prevent ill condition matrix based numerical instability, a network topology based identification method which prevent too much tuning and a limited times skipping method of trial iteration.

在上篇中，本文给出了以下内容：第二章详细描述了最优潮流牛顿算法的实现，着重分析了牛顿法实际应用中存在着的两个主要问题：数值不稳定性和如何快速正确地识别起作用的不等式约束集，并提出了相应的处理对策。

It has been prepared for the investigation of inexact quasi-newton method for operator equations in infinite dimensional space.

文献[1]对有限维空间中非线性方程组的不精确拟牛顿法的收敛性进行了研究，为无穷维空间上的算子方程的不精确拟牛顿法的研究作好了准备。

To obtain quadratical convergence, however, strict complementarity condition at the Danskin point was used. The condition is too strict to be satisfied in many practical problems such as discrete semi-infinite minimax problem. Another kind of Newton method for finite minimax problems was presented by E. Polak and, without strict complementarity at the Danskin point, superlinear convergence (of order 3/2) was proven.

Polak等人提出了一种直接求解极大极小问题的二阶收敛的牛顿法，但是为获得二阶收敛速度要求在Danskin点处满足严格互补条件，这个条件太强，很多实际问题尤其是半无限极大极小问题的离散化不满足该条件；他们又给出另外一种牛顿法，在不假设严格互补条件成立的情况下，证明了它的超线性（3/2阶）收敛性。

Fist, comprehensive analysis on the current study on data processing is made, and characters of Newton methods about nonlinear surveying and mapping data processing are discussed, and then new solutions to parameters estimate with multi-sources, multi-types, multi-dimension, multi-precision bynonlinear least square are presented such as PSB algorithm, digital continuation and generalized digital continuation algorithm, cone model method, tensor analysis method, GCMA(mixed algorithm of gradient method and conjugate gradient method), combining algorithm based on Newton method and gradient method and confidence region and so on, and a new fast difference iterative algorithm is proposed towards parameters estimation containing random parameters in nonlinear models, and a new solutions to nonlinear least squares surveying and mapping adjustment by parameters estimation both considering the random and nonrandom parameters is presented after studying on nonlinear data processing in deformation monitoring, and at last primary analysis on error propagation of spatial data is made and approximate error propagation formula and error analysis formula to length and area are proposed.

首先比较全面分析了目前测量数据处理理论的研究现状，讨论了牛顿类方法在处理非线性模型参数估计方面的特点，研究并给出了求解多源、多类型、多维、多精度、非线性最小二乘测量平差参数估计的若干种新方法，这包括PSB算法、数值延拓及其广义数值延拓算法、锥模型法、张量分析法、基于最速下降法和牛顿法的组合算法、基于最速下降

It is proved that Newton iteration method is still quadratically convergent under a weak condition, and a new point is added to Newton s method , i.

本文讨论求解非线性方程的牛顿法，证明牛顿法在一个弱条件下仍保持局部二阶收敛性，给出牛顿法的一点改进，即一个不带导数的单参数的二阶收敛的迭代法，而且分别得到这两种迭代法的收敛因子，最后进行数值实验

A multi-distance measurements method is employed in this paper. A single parameter fitting method and a double parameters (reduced scattering coefficient and amplitude factor) fitting method on the basis of the least square Newton fitting algorithm is used to derive the optical parameters respectively. For the finite-thickness homogenous tissue, it can be concluded that the rest terms in expressions of diffusion reflectance and transmittance for the corresponding tissue geometries that subtract the exponential term contained source-detector distance depend on the detected position weakly.

在多程测量的基础上，使用单参数最小二乘牛顿法拟合方法和双参数（约化散射系数、幅度因子）最小二乘牛顿法拟合方法，得到有限厚均匀组织的反射率分布和透射率分布公式中除去包含测量位置的指数项的其它部分对测量位置皆具有弱依赖性的结论，并对拟合结果进行了分析讨论；而对于有限厚平板状双层组织，可根据对反射测量结果及透射测量结果进行拟合所得的约化散射系数间的差异及常数因子的正负判断双层组织的可能结构分布。

The first part of this paper goes to a thoroughreview of studies on methods for nonlinear least squares problems, which are then,from the perspective of arithmetic design, classified into five major types: methodsbased on quasi-Newton equation, hybrid methods, factorized quasi-Newton method,trust-region methods, and self-scaling method. The second part of this paper turns thefocus toward a type of new algorithms for nonlinear least squares problems based onnew quasi-Newton equation.

本文的前半部分通过对非线性最小二乘各种求解方法的回顾，从算法设计的角度将求解方法划分为五大类：基于拟牛顿修正的方法，混合法，分解拟牛顿法，信赖域算法以及具有乘积结构类算法，本文后半部分给出了基于新拟牛顿方程的一类求解非线性最小二乘的新算法。

First introducing the basic principle and several classical distort of Newtons method.

首先介绍牛顿法的基本原理及几个经典的牛顿法变形，然后系统地对其局部收敛性质进行分析，最后拓展牛顿法所体现的基本思想，改进牛顿法，介绍了割线法，抛物线法，指数迭代法和导数迭代法。

Singer Leona Lewis and former Led Zeppelin guitarist Jimmy Page emerged as the bus transformed into a grass-covered carnival float, and the pair combined for a rendition of "Whole Lotta Love".

歌手leona刘易斯和前率领的飞艇的吉他手吉米页出现巴士转化为基层所涵盖的嘉年华花车，和一双合并为一移交&整个lotta爱&。

This is Kate, and that's Erin.

这是凯特，那个是爱朗。