Quotient convergence and multi-splitting methods for solving singular linear equations

In this paper, we use the group inverse to characterize the quotient convergence of an iterative method for solving consistent singular linear systems, when the matrix index equals one. Next, we show that for stationary splitting iterative methods, the convergence and the quotient convergence are equivalent, which was first proved in [7]. Lastly, we propose a (multi-)splitting iterative method A=F–G, where the splitting matrix F may be singular, endowed with group inverse, by using F# as a solution tool for any iteration. In this direction, sufficient conditions for the quotient convergence of these methods are given. Then, by using the equivalence between convergence and quotient convergence, the classical convergence of these methods is proved. These latter results generalize Cao’s result, which was given for nonsingular splitting matrices F.