GPBi-CG: Generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems

SIAM Journal on Scientific Computing

Volumn 18, Issue 2, 1997, Pages 537-551

  Zhang, Shao Liang ^a  

^a UNIVERSITY OF TSUKUBA   (Japan)

Author keywords

Bi CG; Bi CGSTAB; Bi CGSTAB2; CGS; Nonsymmetric linear systems; Producttype methods; Residual polynomial; Restructuring Bi CG; Three term recurrence relations

Indexed keywords

GENERALIZED PRODUCT TYPE METHODS; NONSYMMETRIC LINEAR SYSTEMS; THREE TERM RECURRENCE RELATIONS;

CONVERGENCE OF NUMERICAL METHODS; LINEAR SYSTEMS; MATRIX ALGEBRA; POLYNOMIALS; PROBLEM SOLVING; VECTORS;

MATHEMATICAL TECHNIQUES;

EID: 0031082725     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827592236313     Document Type: Article

References (18)

1. A. BAYLISS, C. I. GLODSTEIN, AND E. TURKEL, An iterative method for the Helmholtz equation, J. Comput. Phys., 49 (1983), pp. 443-457.
2. M. DRIESSEN AND H. A. VAN DER VORST, Bi-CGSTAB in semiconductor modeling, in Simulation of Semiconductor Devices and Processes 4, W. Fichtner, ed., Hartung-Gorre, Zurich, 1991, pp. 45-54.
3. S. C. EISENSTAT, H. C. ELMAN, AND M. H. SCHULTZ, Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 20 (1983), pp. 345-357.
4. R. FLETCHER, Conjugate gradient methods for indefinite systems, Lecture Notes in Math. 506, Springer-Verlag, Berlin, Heidelberg, New York, 1976, pp. 73-89.
5. M. H. GUTKNECHT, Variants of Bi-CGSTAB for matrices with complex spectrum, SIAM J. Sci. Comput., 14 (1993), pp. 1020-1033.
6. S. FUJINO AND S. L. ZHANG, Analysis on convergence behavior of the CGS and Bi-CGSTAB method, in Computer Arithmetic and Enclosure Methods (IMACS), L. Atanassova, ed., North-Holland, New York, 1992, pp. 381-390.
7. M. R. HESTENES AND E. STIEFEL, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, 49 (1952), pp. 409-435.
8. C. LANCZOS, Solution of systems of linear eguations by minimized iterations, J. Res. Nat. Bur. Standards, 49 (1952), pp. 33-53.
9. J. A. MEIJERINK AND H. A. VAN DER VORST, An iterative solution method for linear sys-tems of which the coefficient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148-162.
10. J. A. MEIJBRINK AND H. A. VAN DER VORST, Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems, J. Comput. Phys., 44 (1981), pp. 134-155.
11. N. M. NACHTIGAL, S. C. REDDY, AND L. N. TREFETHEN, How fast are nonsymmetric matrix iterations?, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 778-795.
12. 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.
13. G. L. G. SLEIJPEN AND D. R. FOKKEMA, BICGSTAB(L) for linear equations involving unsymmetric matrices with complex spectrum, Elec. Trans. Numer. Anal., 1 (1993), pp. 11-32.
14. P. SONNEVELD, CGS, A fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 36-52.
15. E. L. STIEFEL, Kernel polynomials in linear algebra and their numerical applications, in Further Contributions to the Determination of Eigenvalues, NBS Applied Math. Ser. 49, U. S. Government Printing Office, Washington, D.C., 1958, pp. 1-22.
16. H. A. VAN DER VORST, Preconditioning by Incomplete Decompositions, Ph.D. thesis, University of Utrecht, The Netherlands, December 16, 1982.
17. H. A. VAN DER VORST, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 631-644.
18. S. L. ZHANG AND S. FUJINO, Observations on convergence behaviours of the Bi-CG, CGS, and Bi-CGSTAB from separating rounding errors, Trans. of Japan SIAM, 3 (1993), pp. 135-146.