The synthetic research for the basic iterative method for the non-diagonal dominant matrix.

(1)开发新的预处理矩阵我们开发了新的预处理矩阵。采用新的预处理矩阵可以加快经典高斯-赛德尔方法的收敛速度。此外，还可以求解非对角占优矩阵，而不可能采用迭代法。另一方面，Kohno开发了其他类型的预处理矩阵，并获得了许多有效的结果。与ICCG方法相比，该方法在迭代次数和CPU时间上都具有优异的性能。(2) h矩阵判据的建立。h矩阵对于用迭代法求解线性代数方程组是非常重要的。但是对于h矩阵没有判据。我们开发了迭代准则作为新的准则。这种方法完全是我们独创的。然而，在对所有h矩阵进行判断时，需要大量的迭代次数。为了克服这一缺陷，我们发现了新的迭代准则(3)。对于涉及的矩阵为p矩阵的线性互补问题，我们给出了新的误差界。严格误差界的计算可以转化为p矩阵线性区间系统。此外，对于涉及的矩阵是具有正对角线的h矩阵，可以通过求解一个比Mathias-Pang误差界更尖锐的线性方程组来找到误差界。

(1)Development of the new preconditioning matrixWe developed the new preconditioning matrix. Using the new preconditioning matrix can be accelerate of convergence of the classical Gauss-Seidel method. Moreover, it is possible to solve a non-diagonal dominant matrix which is impossible to apply the iterative method. On the other hand, Kohno developed other type's preconditioning matrix and obtained many effective results. The proposed method is excellent in iterative numbers and CPU time as compared with the ICCG method.(2)Development of criterion of the H-matrix.The H-matrix is very important for the solution of linear systems of algebraic equations by the iterative methods. But there is not criterion for the H-matrix. We developed the iterative criterion as new criterion. This method is our completely original. However, many iteration numbers are needed in judge to all H-matrix. In order to conquer this fault, we discovered new iterative criterion(3)We give new error bounds for the linear complementarity problem where the involved matrix is a P-matrix. Computation of rigorous error bounds can be turned into P-matrix linear interval system. Moreover, for the involved matrix being an H-matrix with positive diagonals, an error bound can be found by solving a linear system of equations which is sharper than the Mathias-Pang error bound.