Convergence and quotient convergence of iterative methods for solving singular linear equations with index one

Singular systems with index one arise in many applications, such as Markov chain modelling. In this paper, we use the group inverse to characterize the convergence and quotient convergence properties of stationary iterative schemes for solving consistent singular linear systems when the index of the coefficient matrix equals one. We give necessary and sufficient conditions for the convergence of stationary iterative methods for such problems. Next we show that for the stationary iterative method, the convergence and the quotient convergence are equivalent.