MATHEMATICAL ANALYSIS AND NUMERICAL ANALYSIS OF SEVERAL KINDS OF DIFFERENTIAL EQUATIONS.

为了解决由奇异扰动问题离散化导出的非对称线性系统，我们提出了一种具有多个松弛参数的广义SOR方法，即具有排序的改进SOR方法，并研究了其理论和实际应用。在三对角矩阵的情况下，检查了参数的最佳选择：结果表明，对于根据应用于系统的高斯消元法的主元计算的一对参数值，迭代矩阵的谱半径减小到零。还提出了迭代的排序和起始向量的正确选择。我们将上述方法应用于二维情况，并针对分块三叉矩阵提出了“带排序的自适应改进分块SOR方法”。该方法的要点是不仅为每个块而且为每次迭代更改多个松弛参数。如果对于块矩阵均为n*n矩阵的n*n块三对角矩阵选择特殊的多重松弛参数并与该方法一起使用，则该迭代方法最多收敛n^2次迭代。我们还提出了带序的改进SSOR方法，对于三对角系统最多仅收敛一次迭代，对于块三对角系统最多收敛n次。还考虑了带序改进SOR方法的广义收敛定理，并研究了矩阵成为广义对角占优矩阵的充分必要条件。利用矩阵的“基本LUL分解”符号，我们给出了一些技术来获得特殊的多重松弛参数，使得对于Hessenberg矩阵和一类矩阵，迭代矩阵的谱半径为零。

To solve non-symmetric linear systems derived from the discretization of singular pertur-bation problems, we propose a generalized SOR method with multiple relaxation parameters, that is the improved SOR method with orderings and study its theory and practical use.In the case of tridiagonal matrices, optimal choices of the parameters are examined : It is shown that the spectral radius of the iterative matrix is reduced to zero for a pair of parameter values which are computed from the pivots of the Gaussian elimination applied to the system. A proper choice of orderings and starting vectors for the iteration is also proposed.We apply the above method to two-dimensional cases, and propose the "adaptive improved block SOR method with orderings" for block tridiafonal matrices. The point of this method is to change the multiple relaxation parameters not only for each block but also for each iteration. If special multiple relaxation parameters are selected and used with this method for an n * n block tridiagonal matrix whose block matrices are all n * n matrices, then this iterative method converges at most n^2 iterations.We also proposed the improved SSOR method with orderings, which converges at most only one iteration for a tridiagonal system, and n iterations for a block tridiagonal system.The generalized convergence theorems to the improved SOR method with orderings are also considered, and we study necessary and sufficient conditions for a matrix to be a generalized diagonally dominant.Using the notation 'basic LUL factorization' of matrices, we give some techniques to obtain special multiple relaxation parameters such that the spectral radius of the iterative matrix is zero for the Hessenberg matrices and a class of matrices.