Comparison of Convergence of General Stationary Iterative Methods for Singular Matrices

New comparison theorems are presented comparing the asymptotic convergence factor of iterative methods for the solution of consistent (as well as inconsistent) singular systems of linear equations. The asymptotic convergence factor of the iteration matrix T is the quantity γ(T ) = max{|λ|, λ ∈ σ(T ), λ = 1}, where σ(T ) is the spectrum of T . In the new theorems, no restrictions are imposed on the projections associated with the two iteration matrices being compared. The splittings of the well-known example of Kaufman [SIAM J. Sci. Statist. Comput., 4 (1983), pp. 525–552] satisfy the hypotheses of the new theorems.