Preconditioned generalized mixed-type splitting iterative method for solving weighted least-squares problems

^†Dedicated to Professor Mahmoud Mohseni Moghadam on the occasion of his 65th birthday.

Abstract

Recently, Shen et al. [Preconditioned iterative methods for solving weighted linear least squares problems, Appl. Math. Mech.-Engl. Ed. 33(3) (2012), pp. 375–384] have considered four kinds of preconditioned generalized accelerated overrelaxation (GAOR) methods for solving the linear systems based on a class of weighted least-squares problems and examined their convergence rates. More recently, Yun [Comparison results on the preconditioned GAOR method for generalized least squares problems, Int. J. Comput. Math. 89 (2012), pp. 2094–2105] has focused on the same problem and suggested three different types of preconditioned GAOR methods and studied their convergence properties. In this paper, we first introduce the generalized mixed-type splitting (GMTS) iterative method for solving the linear systems corresponding to the weighted least-squares problems. The GMTS iterative method exploits auxiliary matrices L₁ and D₁ which gratify certain conditions. In order to improve the convergence rate of the GMTS method, different types of preconditioners are applied. In addition, the convergence of the (preconditioned) GMTS iterative methods is discussed. It is both theoretically and experimentally demonstrated that by appropriate choices of auxiliary matrices L₁ and D₁, the (preconditioned) GMTS method outperforms the (preconditioned) GAOR method.

Keywords:

• linear system
• iterative method
• preconditioner
• convergence rate

2010 AMS Subject Classification:

• 65F10

Acknowledgements

The authors would like to express their sincere gratitude to Dr. Davod Khojasteh Salkuyeh and the anonymous referees for their valuable suggestions and constructive comments which have improved the quality of the paper.

Notes

^† Dedicated to Professor Mahmoud Mohseni Moghadam on the occasion of his 65th birthday.