Using the general method of Owe Axelsson given in 1986 for incomplete factorization of M‐matrices in block‐matrix form, we give a recursive approach to construct incomplete block‐matrix factorization of M‐matrices by proposing a two‐step iterative method for the approximation of the inverse of diagonal pivoting block matrices at each stage of the recursion. For various predescribed tolerances in the accuracy of the approximation of the inverses, the obtained incomplete block‐matrix factorizations are used to precondition the iterative methods as one‐step stationary iterative (OSSI) method and biconjugate gradient stabilized method (BI‐CGSTAB). Certain applications are conducted on M‐matrices occurring from the discretization of two Dirichlet boundary value problems of Laplace's equation on a rectangle using finite difference method. Numerical results justify that the given incomplete block‐matrix factorization of M‐matrices using the two‐step iterative method to approximate the inverse of diagonal pivoting block matrices at each stage of the recursion give robust preconditioners. The obtained results are presented through tables and figures.

中文翻译：

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使用矩阵迭代和预处理的两步迭代方法对M矩阵进行不完整的块矩阵分解

使用1986年给出的Owe Axelsson的一般方法对块矩阵形式的M矩阵进行不完全因子分解，我们提出了一种递归方法，通过提出一种两步迭代的近似方法来构造M矩阵的不完全块矩阵因式分解。在递归的每个阶段对角枢转块矩阵的逆函数。对于上述逆近似精度中的各种公差，使用获得的不完整块矩阵分解对迭代方法进行预处理，使其成为单步平稳迭代（OSSI）方法和双共轭梯度稳定方法（BI-CGSTAB）。某些应用程序在M上进行矩阵是通过使用有限差分法将矩形上的拉普拉斯方程的两个Dirichlet边值问题离散化而产生的。数值结果证明，在递归的每个阶段，使用两步迭代方法逼近对角枢轴块矩阵的逆，可以得到给定的M矩阵不完全块矩阵分解。所得结果通过表格和数字表示。