Iterative Methods for Solving Singular and Rectangular Systems

Abstract

The thesis considers the numerical treatment of singular and rectangular linear algebraic systems by iterative techniques. Thefull rank rectangular systemA X=b, A∈ R^(m×n), rank(A) = n, is reformulated to introduce square non-singular system of order m+n through partitioning the coefficient matrix into two parts A1, A2 with non-singular part A1. The 2-block KSOR and the 3-block KSOR methods are introduced corresponding to those of the SOR versions. The parameters of the problem are discussed to insure convergence and the selection of their optimal values are considered. Also, the four block SOR method is discussed and the four block KSOR method is introduced for singular systems. Semi convergence properties for iterative methods are discussed. Application of the basic iterative methods (JOR, SOR and KSOR) are applied on the singular system appears in the discretization of Poisson’s equation with Neumann and periodic boundary conditions. Moreover, the concepts of preconditioned matrices are considered. Application of the theoretical results and comparison of the performance is considered with numerical examples. Keywords: SOR, KSOR, 2-block SOR, 3-block SOR, 4-block SOR, rectangular system, singular system, least squares, preconditioned