Comparison theorems for the convergence factor of iterative methods for singular matrices

AbstractIterative methods for the solution of consistent singular systems of linear equations are governed by the convergence factor of the iteration matrix T, i.e., by the quantity γ(T)=max{|λ|,λ∈σ(T),λ≠1}, where σ(T) is the spectrum of T. Theorems are presented comparing the convergence factor of two iterative methods. The comparison is based on the relationship between the matrices of the splittings. A cone other than the usual nonnegative hyperoctant is used to define the order used in this comparison. Although this cone is based on the (unknown) projection onto the null-space of a matrix, the characterization provided in the paper allows, in specific instances, the cone to be readily computable