Iterative refinement of the solution of ill-conditioned linear system of equations

We first present a generalized two-step iterative refinement procedure to solve ill-conditioned linear system of equations. Afterward, the idea of an orthogonal projection technique together with a basic stationery iterative method are utilized to construct a new efficient and neat hybrid algorithm for solving the mentioned problem. Numerical examples are examined to demonstrate the feasibility of proposed algorithms and their superiority to some of existing approaches for solving ill-conditioned linear system of equations.