A New Adaptive GMRES Algorithm for Achieving High Accuracy

Numerical Linear Algebra with Applications

Volumn 5, Issue 4, 1998, Pages 275-297

  Sosonkina, Maria ^a   Watson, Layne T ^a   Kapania, Rakesh K ^a   Walker, Homer F ^b  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

^b Worcester Polytechnic Institute   (United States)

Author keywords

GMRES method; Homotopy curve tracking; Krylov subspace methods; Nonsymmetric linear systems; Sparse matrix

Indexed keywords

EID: 0032341591     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1099-1506(199807/08)5:4<275::AID-NLA131>3.0.CO;2-B     Document Type: Article

References (26)

1. C. H. Bischof and P. T. P. Tang. Robust incremental condition estimation. Tech. Rep. CS-91-133, LAPACK Working Note 33, Computer Science Department, University of Tennessee, May, 1991.
2. O. Axelsson and M. Nikolova. A generalized conjugate gradient minimum residual method (GCG-MR) with variable preconditioners and a relation between residuals of the GCG-MR and GCG-OR methods. Comm. Appl. Anal., 1, 371-388, 1997.
3. Å. Björck. Solving linear least squares problems by Gram-Schmidt orthogonalization. BIT, 7, 1-21, 1967.
4. P. N. Brown and H. F. Walker. GMRES on (nearly) singular systems. SIAM J. Matrix Anal. Appl., 18, 37-51, 1997.
5. C. DeSa, K. M. Irani, C. J. Ribbens, L. T. Watson and H. F. Walker. Preconditioned iterative methods for homotopy curve tracking. SIAM J. Sci. Stat. Comput., 13, 30-46, 1992.
6. R. W. Freund and N. M. Nachtigal. An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, part II. Tech. Rep. 90.46, RIACS, NASA Ames Research Center, November, 1990.
7. R. W. Freund and N. M. Nachtigal. QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math., 60, 315-339, 1991.
8. P. E. Gill and W. Murray. Newton-type methods for unconstrained and linearly constrained optimization. Math. Programming, 7, 311-350, 1974.
9. G. H. Golub and C. F. Van Loan. Matrix Computations, 2nd edition. Johns Hopkins University Press, Baltimore, MD, 1989.
10. A. Greenbaum, M. Rozloẑník and Z. Strakoš. Numerical behaviour of the modified Gram-Schmidt GMRES implementation. BIT, 37, 706-719, 1997.
11. S. M. Holzer, R. H. Plaut, A. E. Somers, Jr. and W. S. White. Stability of lattice structures under combined loads. J. Engrg. Mech., 106, 289-305, 1980.
12. K. M. Irani, M. P. Kamat, C. J. Ribbens, H. F. Walker and L. T. Watson. Experiments with conjugate gradient algorithms for homotopy curve tracking. SIAM J. Optim., 1, 222-251, 1991.
13. W. Joubert. On the convergence behavior of the restarted GMRES algorithm for solving non-symmetric linear systems. Numer. Linear Algebra Appl., 1, 427-447, 1994.
14. R. K. Kapania and T. Y. Yang. Formulation of an imperfect quadrilateral doubly curved shell element for postbuckling analysis. AIAA J., 24, 310-311, 1986.
15. W. D. McQuain, R. C. Melville, C. J. Ribbens and L. T. Watson. Preconditioned iterative methods for sparse linear algebra problems arising in circuit simulation. Comput. Math. Appl., 27, 25-45, 1994.
16. R. C. Melville, S. Moinian, P. Feldmann and L. T. Watson. Sframe: an efficient system for detailed DC simulation of bipolar analog integrated circuits using continuation methods. Analog Integrated Circuits Signal Processing, 3, 163-180, 1993.
17. R. C. Melville, Lj. Trajković, S.-C. Fang and L. T. Watson. Artificial parameter homotopy methods for the DC operating point problem. IEEE Trans. Computer-Aided Design, 12, 861-877, 1993.
18. Y. Saad. A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput., 14, 461-469, 1993.
19. Y. Saad and M. H. Schultz. GMRES: a generalized minimal residual method for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7, 856-869, 1986.
20. H. A. van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Numer. Linear Algebra Appl., 1, 369-386, 1994.
21. H. A. van der Vorst and C. Vuik. The superlinear convergence behaviour of GMRES. J. Comp. Appl. Math., 48, 327-341, 1993.
22. H. F. Walker. Implementation of the GMRES method using Householder transformations. SIAM J. Sci. Stat. Comput., 9, 152-163, 1988.
23. L. T. Watson. Globally convergent homotopy methods: a tutorial. Appl. Math. Comput., 31BK, 369-396, 1989.
24. L. T. Watson, S. M. Holzer and M. C. Hansen. Tracking nonlinear equilibrium paths by a homotopy method. Nonlinear Anal., 7, 1271-1282, 1983.
25. L. T. Watson, S. C. Billups and A. P. Morgan. Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms. ACM Trans. Math. Software., 13, 281-310, 1987.
26. T. Y. Yang, R. K. Kapania and S. Saigal. Accurate rigid-body modes representation for a nonlinear curved thin-shell element. AIAA J.. 27, 211-218, 1989.