A Weak Regular Splitting For Singular Linear Systems ∗

In this paper, the semiconvergence of a proper weak regular splitting method for the singular linear algebraic system Ax = b with a singular M-matrix A is discussed. The extrapolated iterative method based on the weak regular splitting iterative method is also constructed. Let us consider the solution of a linear system of equations Ax = b (1) by iterative methods. Here A is an n × n singular matrix, x, b ∈ R n with b known and x unknown. We assume that the system (1) is solvable, i.e., it has at least a solution, which is equivalent to b ∈ R(A), where R(A) denotes the range of A. In order to solve the system (1) with iterative methods usually we construct the iteration of the form x 0 ∈ R n; x k+1 = Tx k + c, k =0, 1, 2,.... (2) Here the question is the construction of the iterative matrix T and the vector c. The usual approach is through the splitting of the coefficient matrix A: by the formulas A = M − N, T = M −1 N, c = M −1 b, where M is nonsingular. It is well known that for singular system the iterative method (2) is semiconvergent if, and only if, the associated convergence factor υ(T) ≡ max{|λ|,λ ∈ σ(T)\{1}} < 1, (3) and the elementary divisors associated with µ =1 ∈ σ(T) are linear, i.e.