breakdowns of rrgmres and dgmres

The gmres method is one of the most common iterative methods to solve linear systems of equations with an n × n large nonsingular matrix . when the matrix is singular, the gmres method may break down before determining an acceptable approximate solution. the rrgmres and dgmres are modified gmres restricting the krylov subspaces within the range of a and the range of a^m to make solutions more stable, where m is the index of a. the aim of this paper is to characterize breakdowns and least square solutions of the rrgmres and dgmres algorithms for solving a singular linear system equations.