A symmetric positive definite preconditioner for saddle-point problems

Abstract

For the classical saddle-point problem, we present precisely two intervals containing the positive and the negative eigenvalues of the preconditioned matrix, respectively, when the inexact version of the symmetric positive definite preconditioner introduced in Section 2.1 of Gill et al. [Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl. 13 (1992), pp. 292–311] is employed. The model of Stokes problem is used to test the effectiveness of the presented bounds as well as the quality of the symmetric positive definite preconditioner.

Keywords:

• Saddle-point problem
• preconditioner
• eigenvalue
• MINRES

2000 AMS Subject Classifications :

• 65F10
• 65F50
• 65N22

Acknowledgements

We are grateful to the referees and the editors for their helpful suggestions to improve the quality of this paper. We thank Dr. Jian-Song Zhang for very useful discussions and suggestions. The work of S.-Q. Shen was supported by the NSFC Tianyuan Mathematics Youth Fund (10926086) and the Fundamental Research Funds for the Central Universities. The work of T.-Z. Huang was supported by 973 Program (2007CB311002), NSFC (60973015), Sichuan Province Sci. and Tech. Research Project (2009SPT-1).