Incomplete block‐matrix factorization of M‐matrices using two‐step iterative method for matrix inversion and preconditioning

Abstract

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.