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

Iterative 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 gamma(T) = max{\lambda \, lambda is an element of s igma(T), lambda not equal 1}, where sigma(T) is the spectrum of T. Theorems are presented comparing the convergence factor of two iterative methods, T he comparison is based on the relationship between the matrices of the spli ttings. A cone other than the usual nonnegative hyperoctant is used to defi ne the order used in this comparison. Although this cone is based on the (u nknown) projection onto the null-space of a matrix, the characterization pr ovided in the paper allows, in specific instances, the cone to be readily c omputable. (C) 2000 Elsevier Science Inc. All rights reserved.