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.

Two Stage Implicit Method on Hexagonal Grids for Approximating the First Derivatives of the Solution to the Heat Equation

“…To solve the obtained linear algebraic system of equations, we applied incomplete block-matrix factorization of the block tridiagonal stiffness matrices which are symmetric M−matrices for the all considered pairs of (h, τ) using two-step iterative method for matrix inversion. Then these incomplete block-matrix factorizations are used as preconditioners for the conjugate gradient method as given in [34] (see also [35,36] ). We use the following notations in tables and figures: obtained by using the two stage implicit method is given by…”

“…In Section 5, a numerical example is constructed to support the theoretical results. We applied incomplete block preconditioning given in [34] (see also [35,36]) for the conjugate gradient method to solve the obtained algebraic systems of linear equations for various values of r. In Section 6, conclusions and some remarks are given. γ j is the boundary of D and denote by D = D ∪ S the closure of D. Let Q T = D × (0, T), with lateral surface S T more precisely the set of points (x, t), x = (x 1 , x 2 ) ∈ S and t ∈ [0, T] also Q T is the closure of Q T .…”

Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods

“…Further, Mathematica is used for the realization of the algorithms in machine precision. Also we used preconditioned conjugate gradient method with the preconditioning approach given in Buranay and Iyikal [29] (see also Concus et al [30] and Axelsson [31]). We define the following:…”

“…Numerical examples are given and for the solution of the obtained algebraic linear systems preconditioned conjugate gradient method is used. The incomplete block matrix factorization of the M-matrices given in Buranay and Iyikal [29] (see also Concus et al [30], Axelsson [31]) is applied for the preconditioning.…”

Numerical Solution of Sylvester Equation Based on Iterative Predictor-Corrector Method

“…e given algorithm was applied to solve Fredholm integral equations of first kind in [22]. Another recursive approach for constructing incomplete block-matrix factorization of the M-matrices by iterative two-step method for the approximation of the inverse of pivoting the diagonal block-matrices at each stage of the recursion was given in [23].…”