对角元素

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。

With the LU decomposition of the block tridiagonal matrix, an explicit expression of the block inverse elements is obtained.

由块三对角矩阵的LU分解，得到了其逆矩阵块元素的显式表达式。

When scaled factor circulant matrices are nonsingular, we can find the single solution of the scaled factor circulant matrix equation; When scaled factor circulant matrices are singular, we can find the special as well as the general solution of the scaled factor circulant matrix equation; There is only an error of approximation when the fast algorithm is implemented on computers, and only the elements in the first row of the scaled factor circulant matrix and the constants in the diagonal matrix are needed by the fast algorithm.

当鳞状因子循环矩阵非奇异时，该快速算法求出线性方程组的唯一解；当鳞状因子循环矩阵奇异时，该快速算法求出线性方程组的特解与通解。该快速算法仅用到鳞状因子循环矩阵的第一行元素及对角矩阵中的对角上的常数进行计算。在计算机上实现时只有舍入误差。

A determinant whose only nonzero elements occur in square blocks centered about the principal diagonal is in block- diagonal form.

仅有的非零元素集中于主对角线的方块中的行列式是分块对角形式的。

For example: Causes the diagonal matrix aspect using the matrix similarity the operation simplification, as well as using the matrix similarity determined transports in the mathematics problem in the matrix unknown element and so on.

利用矩阵的相似性使得对角矩阵方面的运算更简单化，以及利用矩阵的相似性来确定运算题中的矩阵中的未知元素等。

In chapter two, by using the elements of the matrix we first construct some multiplier factors, then, use the properties of αdiagonally dominant matrix and the techniques of inequalities, we give some new determination conditions for generalized strictly diagonally dominant matrix, these theory have improved some existing results.

在第二章中，利用矩阵某些元素，构造出了几个乘积因子，然后利用α-对角占优矩阵的一些性质，结合放缩不等式的技巧，给出了广义严格对角占优矩阵的几个新的判定条件，改善了已有的某些结果。

By using the properties of part matrix elements in N1 , and by looking for positive diagonal matrix factors, we present some new sufficient conditions of generalized strictly diagonally dominant matrix and improve the recent results.

第二章在行非严格对角占优集N_1划分为N_1~(1)与N_1~(2)的直和N_1~(1)⊕N_1~(2)的条件下，利用下标在N_1上部分矩阵元素的性质，寻求正对角矩阵因子，给出了广义严格对角占优矩阵的几个新的充分条件，同时改进了近期的一些结果。

In chapter three, we divide the rows of matrix into types finer than those in chapter one. Again by looking for positive diagonal matrix factors, we obtain several new criteria for identifying generalized strictly diagonally dominant matrix, improving some existing criteria.

第三章在下标集N 递进式划分的条件下，根据递进后，某些下标集上部分矩阵元素的性质，寻求正对角矩阵因子，获得了广义严格对角占优矩阵的几个新判别法，同时改进了一些已有结果。

Examples illustrate that some special properties of the diagonally dominant matrix with chain of non-zero elements probably doesn't come into existence for quasi diagonally dominant matrix with chain of non-zero elements.

譬如目前关于拟对角占优矩阵的判定定理就有数十条之多[6-7]，至今仍有许多相关结论发表，其中杨志明[1]给出了拟具非零元素链对角占优矩阵的定义，并就这类矩阵的特征值的分布情况进行了讨论。

The characteristic vector of the real symmetric matrix should be found out, which is orthogonalized and normalized to a standard orthogonal base and is used as row vector to construct the transformation matrix P, so the P~(-1)AP can be made into diagonal matrix.

实对称矩阵A经相似变换P-1AP可化为对角矩阵，在x =Py 下，不一定能化A的二次型为标准型；应寻求对称矩阵A的特征向量，将其正交化并单位化作为标准正交基，作为列向量构造变换矩阵P，可使P-1AP=Λ为对角阵，在x =Py 下，要将二次型化为标准型，且二次项系数即为对角阵Λ主对角线上元素。

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如果你不幸失恋了，请告诉我，我会帮助你摆脱困境，真的，请联系我啦！

China's plan to cut energy intensity by 20 percent and pollutant discharges by 10 percent between 2006 and 2010 is a case in point.

中国计划在2006年到2010间降低20%的能源强度和减少10%的主要污染物排放，就是一个这样的例子。