Effect on Spectral Properties by the Splitting Correction Preconditioning for Linear Systems that Arise from Periodic Boundary Problems
In this paper, the spectral properties of the preconditioned systems by the “Splitting Correction (SC)”, proposed by the present authors, are studied and it is conjectured that the degeneracy not the clustering of the eiganvalues plays an important role in the convergence. The SC preconditioner is one of new pre-conditioners based on block factorization for solving linear systems that arise from periodic boundary problems. From the viewpoint of the convergence of residual norm, the behaviors of the residual norm of the conjugate gradient (CG) method preconditioned by the SC and the block incomplete Cholesky (block IC) are very peculiar. Generally, the convergence of the CG method depends on spectral properties, such as the clustering and the degeneracy of the eigenvalues, of the coefficient matrix. Some numerical results suggest that the fast convergence of the SC is due not to the clustering but to the degeneracy of the eigenvalues of the preconditioned coefficient matrix.
Unable to display preview. Download preview PDF.
- 1.Axelsson O., Incomplete Block Matrix Factorization Preconditioning Methods. The Ultimate Answer?, J. Comp. Appl. Math., 12and13, pp. 3–18, 1985.Google Scholar
- 2.Barrett R., et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, USA, 1994.Google Scholar
- 3.Bruaset A. M., A Survey of Preconditioned Iterative Methods, Longman Scientific and Technical, 1995.Google Scholar
- 7.Golub G. H. and van Loan C. F., Matrix Computations, Johns Hopkins University Press, 1996.Google Scholar
- 9.Itoh S., Zhang S. -L., Oyanagi Y. and Natori M., Splitting Correction Pre-conditioner for Linear Systems that Arise from Periodic Boundary Problems, ISE-Technical Report, ISE-TR-00–170, University of Tsukuba, June, 2000.Google Scholar
- 10.Kelley C. T., Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, 1995.Google Scholar
- 12.Natori M., Numerical Analysis and its Applications, Corona Publishing, Tokyo, 1990 (in Japanese).Google Scholar