On the iterative refinement of the solution of ill-conditioned linear system of equations

ABSTRACT

Recently, Salkuyeh and Fahim [A new iterative refinement of the solution of ill-conditioned linear system of equations, Int. Comput. Math. 88(5) (2011), pp. 950–956] have proposed a two-step iterative refinement of the solution of an ill-conditioned linear system of equations. In this paper, we first present a generalized two-step iterative refinement procedure to solve ill-conditioned linear system of equations and study its convergence properties. Afterward, it is shown that the idea of an orthogonal projection technique together with a basic stationary iterative method can be utilized to construct a new efficient and neat hybrid algorithm for solving the mentioned problem. The convergence of the offered hybrid approach is also established. 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.

Keywords:

• Iterative method
• iterative refinement
• convergence
• projection technique

2010 AMS Subject Classification:

• 65F10

Acknowledgments

The authors would like to express their heartfelt thanks to the anonymous referees for their valuable suggestions and constructive comments which have improved the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.