Triangle Splitting迭代方法是求解大型稀疏非Hermitian正定线性代数方程组的一种有效迭代算法。为了有效求解大型稀疏且Jacobi矩阵为非Hermitian正定的非线性代数方程组，本文将Triangle Splitting迭代方法作为不精确Newton方法的内迭代求解器，构造了不精确Newton-Triangle Splitting迭代方法。在适当的约束条件下，给出了该方法的两类局部收敛性定理。通过数值实验结果验证了该方法的可行性和有效性，并说明了该方法在计算时间和迭代次数方面比Newton-BTSS迭代方法更有优势。

The Triangle Splitting iteration method is an effective iteration method for solving large-scale sparse non-Hermitian positive definite system of linear algebraic equations. By mak-ing use of the Triangle Splitting iteration method on non-Hermitian positive definite matrices as the inner solver of the inexact Newton method, we establish a class of inexact Newton-Triangle Splitting iteration methods for solving the large-scale sparse system of nonlinear algebraic equa-tions with positive definite Jacobian matrices in the paper. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions. The numerical results are given to examine their feasibility and effectiveness. The numerical im-plementations also show that the Newton-Triangle Splitting methods have advantages over Newton-BTSS methods with less computation time and iteration steps.

针对n阶变截面压杆稳定性问题,基于一般的压杆稳定性挠曲线方程建立了n阶变截面压杆的微分方程组,结合相邻阶的边界条件,求解方程组获取递推关系式,即关于临界载荷的方程,应用Newton迭代法对方程进行迭代求解。以1200t全地面起重机伸缩臂为例,分别采用Newton迭代法、弦截法及有限元法计算了伸缩臂的临界载荷,对比分析结果证明了所推导出的临界载荷关系式的合理性及Newton迭代法的高效性。

For the stability of n-order variable cross-section compression bars,the differential equations concerning the flexible curve equation were established.Based on the boundary conditions of neighbored-order,a critical load coefficient equation was built by solving the differential equations. And Newton iteration method was applied to calculate the critical load.As a case of telescopic boom for 1200 ton all-terrain crane,comparison results of calculation were analyzed among Newton iteration method,secant method and stability of ANSYS,which indicate the rationality of the critical load coef-ficient equation and efficiency of Newton iteration method.

根据Hertz接触理论和刚性套圈理论，建立了轴向受载时高速滚动轴承的力学模型，得到了组成该模型的动态特性方程组。针对传统Newton-Raphson迭代方法对所建立的动态特性方程组求解过程中对初值敏感、不易收敛和振荡的问题，提出了基于遗传算法的求解方法，并将所得结果与传统Newton-Raphson方法结果进行了对比。结果表明，遗传算法可以有效求解高速滚动轴承的动态特性方程组，避免了传统方法的缺点，提高了编程效率。

Based on Hertz contact theory and rigid races method,a mechanical model of high-speed rolling bearing under axial load was established,and the dynamic characteristics equations for the model were obtained.In order to avoid sensitivity to initial val-ue,non-convergence and oscillation problem during solving the established dynamic characteristics equations by using Newton-Raphson iteration method,a solution method based on genetic algorithm was proposed.The comparision results with the tradi-tional Newton-Raphson method show that genetic algorithm is more effective to solve dynamic equations of high-speed rolling bearing with higher program efficiency.

作为Newton多项式插值在重节点情形时的推广,Newton-Hermite多项式插值是很常用的切触线性插值,它建立在广义差商基础之上,广义差商能被递归地计算并产生有用的中间结果。Newton-Hermite插值实际上是基于点的插值,可以通过增加新的节点来获得一个新的插值多项式。这里将基于点的插值推广到基于块的插值。受现代建筑设计的启发,将插值点集划分为一些子集(块),然后将在每个子集上选择切触插值,线性或有理插值,最后用类似于Newton-Hermite插值的格式进行装配。显然,在切触有理插值上提供了灵活的选择,这里也包括它的特殊情形Newton-Hermite多项式插值。本文介绍了所谓的基于块的广义差商并给出递归算法,给出的数值例子说明了方法的有效性。

As the generalization of New ton''s polynomial interpolation so as to accommodate repeated abscissae ,New ton-Hermite polynomial interpolation may be the favourite osculatory linear interpolation in the sense that is built up by means of the generalized divided differences which can be calculated recursively and produce useful intermediate results .However Newton-Hermite interpolation is in fact point based interpolation since a new interpolating polynomial is obtained by adding a new support point into the current set of support points once at a time .In this paper we extend the point based interpolation to the block based interpolation .Inspired by the idea of the modern architectural design ,we first divide the original set of support points into some subsets (blocks) ,then construct each block by using whatever osculatory interpolation means ,linear or rational and finally assemble these blocks by New ton-Hermite''s method to shape the w hole interpolation scheme .Clearly

考虑Black-Scholes模型下美式看跌期权的定价问题。采用有限差分法和 Newton法耦合求解Black-Scholes方程，得到了期权价格和最佳实施边界的数值逼近结果。数值实验验证了算法的有效性。

This paper deals with the American put option pricing problem governed by the Black-Scholes equation.Applying finite difference method coupled with Newton’s method to solve the Black-Scholes equation,we can get the numerical approximations of the option price and the optimal exercise boundary simultaneously.Numerical experiments verify the efficiency of the method.