A novel class of block methods based on the block AA^T-Lanczos bi-orthogonalization process for matrix equations

Abstract

In this paper, we introduce a block AA ^T-Lanczos bi-orthogonalization process. Based on this new process, the block bi-conjugate residual (Bl-BCR) method is derived, which is also a generalization of bi-conjugate residual method. In order to accelerate the rate of convergence, we generate a stabilized and more smoothly converging variant of Bl-BCR using formal matrix-valued orthogonal polynomials. Finally, numerical experiments illustrate the effectiveness of these block methods.

Keywords:

• Bl-BCG
• block Krylov subspace
• nonsymmetric linear systems
• AA ^T-Lanczos
• multiple right-hand sides

2010 AMS Subject Classifications:

• 65F10
• 65F50
• 65N22

Acknowledgements

The authors thank the referees for their valuable suggestions and detailed comments which greatly improved the presentation of our paper. Jianhua Zhang also thanks his Ph.D. supervisor Prof. Hua Dai for fruitful discussions on this work. Jianhua Zhang and Jing Zhao acknowledge the support of Natural Science Foundation from Anhui Province Education Department under Grant No. KJ2012B508.