对角和

Chapter 9: We report a large quantity of numerical experiments of 13 different algebraic multigrid algorithms for solving the Poisson equation, anisotropic equation, equation with cross-derivative terms, general matrix problems with large off-diagonal positive entries, biharmonic equation, Toeplity matrix, elasticity systems, finite element discretization of the Laplacian and even 3D problems. Particular attention is focused on asymptotic convergence factors and CPU-time consumed. Numerical results for many different types of practical problems demonstrate the efficiency and robustness of the proposed algebraic multigrid methods.

第九章：在各种代数多重网格算法的基础上，进行了大量的数值试验，具体给出了十三种不同的代数多重网格方法求解泊松方程，各向异性方程，带混合导数项的方程，带有大的非对角正元素的一般矩阵问题，重调和方程，托普利兹矩阵，弹性力学方程组，拉普拉斯算子的有限元离散，甚至三维问题的较为丰富的数值结果，重点关注它们的渐近收敛因子和所需的CPU时间，来源于不同类型问题的计算结果既为代数多重网格理论分析和算法的改进提供了很实用的资料，同时也证实了本文给出的代数多重网格算法的效绩和稳健性。

Secondly, we discuss the boundary value problem of two dimensional biharmonic equation in a rectangular field and its variational problem,discretize it by using dual tensor product of and the direct product and lining up of matrices, we get some special matrices which are the presupposition in exploring the fast computation, then solve the system of linear equations.

其次本文讨论了矩形域上二维双调和方程边值问题及其相应的变分问题，利用二元张量积小波分析和矩阵的直积、拉直技巧将变分问题离散化，从而使求解偏微分方程问题变为求解线性方程组的问题。一维情况两组基的张量积下得到的系数矩阵分别为块状七对角阵和稀疏矩阵。这些特殊结构为以后快速算法的研究打下一个基础。

Tridiagonal block matrix elemination and sweep algorithm is successfully applied to solve the large matrix involved; central difference method is supplemented to Houbolt method to form a time integration method with estimation and correction, to solve the time integration of second order implicit non-linear equation; simplification of system equation by decoupling between the physical coordinates x and y further improves the calculation accuracy and speed.

其中：成功地引入块三对角矩阵追赶法求解上述问题构成的大型矩阵；提出中心差分法+Houbolt法的预估计—校正时间积分法，以解决二阶隐式非线性方程的时间积分问题；系统方程在物理坐标x和y上的"解耦"简化，使计算速度和计算精度得以进一步的提高。

In the proposed method the data from state estimation or power flows during larger time interval is used to calculate nodal voltages and the initial value of current transitive-relationship matrix; the variation of diagonal elements of this matrix reflects the variations of the detailed models of system components, network topology and parameters after the perturbation, then by use of the PMU measured data in millisecond level the primary diagonal element can be modified.

该方法采用时间间隔较大的状态估计或潮流数据计算节点电压和电流传递关系矩阵的初值，并利用该矩阵主对角元的变化反映受扰后系统各元件的详细模型、网络拓扑和参数的改变，然后使用相量测量单元毫秒级的量测数据对其进行修正，从而为在实际应用中将相量测量单元与SCADA系统相结合做出了有益探索。

In this paper, the problems of correlation characterizations of complex symmetric matrices are deduced and analyzed, which include the eigenvalue, the condition number and the problem of diagonalization.

本论文分析和阐述了复对称矩阵在相关方面的性质和问题，包括特征值、条件数和对角化等问题。

The thesis considers the methods for judginggeneralized diagonally dominant matrix 、M-matrix which canbe viewed as an extension and improvement of thecorresponding result recently.

本文主要研究了广义对角占优矩阵和 M-矩阵的判定条件和数值判定方法，改进和推广了近期的一些研究结果。

Based on the research of the multi-component refinery process, mechanism model and MESH equations are given. The bubble-point method and tridiagonal matrix method which is solved by Gaussian elimination algorithm are proposed.

本文针对多元精馏过程的常压塔进行了分析和研究，建立了平衡级机理模型和MESH方程组，提出采用三对角矩阵法和泡点法相结合作为模型求解的解决方案。

The completely diagonalized Hamiltonian matrices of 3d2/3d8 ion configuration in the trigonal symmetry have been established by irreducible representation method. Taken into account the influence of the spin-spin coupling interactions omitted in previous publications, the spectra and the constants of crystal structure and zero-field splitting parameters of CsNiCl3 crystal and CsNiCl3: Mg2+ crystal are calculated and the influence of the doping to the spectra, the constants of crystal structure, zero-field splitting parameters and Jahn-Teller effect are studied. The results show that doping can change the structure of crystal which brings about the changes in the spectra and zero-field splitting parameters as well as Jahn-Teller effect.

应用不可约张量理论构造了三角对称晶场中3d2/3d8态离子的45阶可完全对角化的微扰哈密顿矩阵，在考虑了以前工作中被忽略的自旋-自旋耦合作用的基础上计算了CsNiCl3晶体和CsNiCl3：Mg2+晶体的基态能级、晶体结构、零场分裂参量和Jahn-Teller效应，研究了掺入Mg2+对CsNiCl3晶体的光谱、零场分裂参量及Jahn-Teller效应的影响和自旋单重态对基态能级的贡献，发现掺杂使得晶体结构产生畸变，从而改变晶体光谱的精细结构和零场分裂参量，不改变Jahn-Teller效应的分裂规律但改变分裂的大小。

The completely diagonalized Hamiltonian matrices of 3d2／3d8 ion configuration in the trigonal symmetry sites have been established by irreducible representation method. Taken into account microscopic magnetic interactions, the spectral fine structure, local structure and zero-field splitting parameters of V3+:α-Al2O3 and Ni2+:α-Al2O3 crystals are investigated as well as the influence of the doping.

应用不可约张量方法构造了三角晶场中3d2/3d8态离子的45阶可完全对角化的能量哈密顿矩阵，在考虑了微小磁相互作用的基础上计算了V3+：α-Al2O3和Ni2+：α-Al2O3晶体的光谱精细结构、晶体局域结构和零场分裂参量，研究了掺入Ni2+和V3+对α-Al2O3晶体光谱精细结构、晶体局域结构和零场分裂参量的影响。

First of all, the concept of S-N curve and the procedures of p-S-N curve fatigue test, data processing were introduced. And then, these procedures were taken into practice on 4 kinds of aluminum alloy welded joints: corner joint, butt joint, cruciform joints and double-sided lap joint.

文中首先介绍了S-N曲线的概念和p-S-N曲线疲劳试验方法和数据处理方法，并按照这些方法对角接、对接、十字接和双面搭接四种铝合金焊接接头试样进行了疲劳试验，得出了这四种焊接接头的中值S-N曲线和置信度为95%，存活率分别为90%、99%下的p-S-N曲线及其方程。

Methods: Five patients with parkinsonism or dystonia were assigned to general anesthesia using an modified endotracheal tube.

本实验依照人体实验之相关规定进行，五位患有帕金森氏症或肌张力异常的病人接受神经立体定位手术。

If you can benefit from this book, it is our honour.

如果您能从本书获益，这将是我们的荣幸。