Abstract
We propose block ILU (incomplete LU) factorization preconditioners for a nonsymmetric block-tridiagonal M-matrix whose computation can be done in parallel based on matrix blocks. Some theoretical properties for these block ILU factorization preconditioners are studied and then we describe how to construct them effectively for a special type of matrix. We also discuss a parallelization of the preconditioner solver step used in nonstationary iterative methods with the block ILU preconditioners. Numerical results of the right preconditioned BiCGSTAB method using the block ILU preconditioners are compared with those of the right preconditioned BiCGSTAB using a standard ILU factorization preconditioner to see how effective the block ILU preconditioners are.
Similar content being viewed by others
REFERENCES
O. Axelsson, Incomplete block matrix factorization preconditioning methods. The ultimate answer?, J. Comput. Appl. Math., 12–13 (1985), pp. 3–18.
O. Axelsson, A general incomplete block-matrix factorization method, Linear Algebra Appl., 74(1986), pp. 179–190.
O. Axelsson, Iterative Solution Methods, Cambridge University Press, New York, 1994.
R. Barrett et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, PA, 1994.
R. Beauwens and M. Bouzid, On sparse block factorization, iterative methods, SIAM J. Numer. Anal., 24(1987), pp. 1066–1076.
P. Concus, G. H. Golub, and G. Meurant, Block preconditioning for the CG method, SIAM J. Sci. Stat. Comput., 6(1985), pp. 220–252.
R. Fletcher, Conjugate Gradient Methods for indefinite systems, in Proceedings of the Dundee Biennal Conference on Numerical Analysis 1974, G. A. Watson ed., Springer-Verlag, Berlin, 1975, pp. 73–89.
G. H. Golub and C. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 3rd ed., 1996.
J. A. Meijerink and H. A. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comput. 31(1977), pp. 148–162.
Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, MA, 1996.
Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7(1986), pp. 856–869.
P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 10(1989), pp. 36–52.
H. A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAMJ. Sci. Statist. Comput., 13 (1992), pp. 631–644.
R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yun, J.H. Block ILU Preconditioners for a Nonsymmetric Block-Tridiagonal M-Matrix. BIT Numerical Mathematics 40, 583–605 (2000). https://doi.org/10.1023/A:1022328131952
Issue Date:
DOI: https://doi.org/10.1023/A:1022328131952