Convergence properties of preconditioned hermitian and skew-hermitian splitting methods for non-hermitian positive semidefinite matrices

Mathematics of Computation

Volumn 76, Issue 257, 2007, Pages 287-298

  Bai, Zhong Zhi ^a,b   Golub, Gene H ^c   Li, Chi Kwong ^d  

^a FUDAN UNIVERSITY   (China)

^b CHINESE ACADEMY OF SCIENCES   (China)

^c Stanford University   (United States)

^d The College of William and Mary   (United States)

Author keywords

Convergence; Hermitian and skew Hermitian splitting; Non Hermitian matrix; Positive semidefinite matrix; Splitting iteration method

Indexed keywords

EID: 52049090886     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-06-01892-8     Document Type: Article

References (29)

1. Z.-Z. Bai, A class of parallel decomposition-type relaxation methods for large sparse systems of linear equations, Linear Algebra Appl., 282(1998), 1-24. MR1648288 (99j:65077)
2. Z.-Z. Bai and G.H. Golub, Generalized preconditioned Hermitian and skew-Hermitian splitting methods for saddle-point problems, Technical Report SCCM-04-07, Scientific Computing and Computational Mathematics Program, Department of Computer Science, Stanford University, 2004.
3. Z.-Z. Bai, G.H. Golub and M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24(2003), 603-626. MR1972670 (2004c:65023)
4. Z.-Z. Bai, G.H. Golub and J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98(2004), 132. MR2076052 (2005h:65041)
5. Z.-Z. Bai and Y.-F. Su, On the convergence of a class of parallel decomposition-type relaxation methods, Appl. Math. Comput., 81:1(1997), 1-21. MR1417759 (97g:65097)
6. Z.-Z. Bai and G.-Q. Li, Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal., 23:4(2003), 561-580. MR2011340 (2004i:65025)
7. Z.-Z. Bai and R. Nabben, Some properties of the block matrices in the parallel decompositiontype relaxation methods, Appl. Numer. Math., 29:2(1999), 167-170. MR1666521 (99k:65038)
8. Z.-Z. Bai and M.K. Ng, On inexact preconditioners for nonsymmetric matrices, SIAM J. Sci. Comput., 26:5(2005), 1710-1724. MR2142592 (2006d:65026)
9. Z.-Z. Bai, B.N. Parlett and Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102:1(2005), 1-38.
10. Z.-Z. Bai and Z.-Q. Wang, Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems, J. Comput. Appl. Math., 187(2006), 202-226. MR2185669
11. M. Benzi and G.H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix. Anal. Appl., 26(2004), 20-41. MR2112850 (2005j:65026)
12. M. Bertaccini, G.H. Golub, S. Serra-Capizzano and C.T. Possio, Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation, Numer. Math., 99:3(2005), 441-484. MR2117735
13. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York and London, 1991. MR1115205 (92d:65187)
14. 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), 345-357. MR0694523 (84h:65030)
15. H.C. Elman and M.H. Schultz, Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations, SIAM J. Numer. Anal., 23(1986), 44-57. MR0821905 (87g:65124)
16. H.C. Elman, D.J. Silvester and A.J. Wathen, Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math., 90(2002), 665-688. MR1888834 (2002m:76071)
17. M. Fortin and R. Glowinski, Augmented Lagrangian Methods, Applications to the Numerical Solution of Boundary Value Problems, North-Holland, Amsterdam, 1983. MR0724072 (85a:49004)
18. G.H. Golub and C. Greif, On solving block-structured indefinite linear systems, SIAM J. Sci. Comput., 24(2003), 2076-2092. MR2005622 (2004h:65030)
19. G.H. Golub, C. Greif and J.M. Varah, Block orderings for tensor-product grids in two and three dimensions, Numer. Algorithms, 30(2002), 93-111. MR1917293 (2003f:65182)
20. G.H. Golub and D. Vanderstraeten, On the preconditioning of matrices with skew-symmetric splittings, Numer. Algorithms, 25(2000), 223-239. MR1827156 (2002e:65067)
21. G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd Edition, The Johns Hopkins University Press, Baltimore, 1996. MR1417720 (97g:65006)
22. G.H. Golub and A.J. Wathen, An iteration for indefinite systems and its application to the Navier-Stokes equations, SIAM J. Sci. Comput., 19(1998), 530-539. MR1618828 (99d:65107)
23. H. Modaressi and P. Aubert, A diffuse element-finite element technique for transient coupled analysis, Intern. J. Numer. Methods Engrg., 39(1996), 3809-3838. MR1417791 (97i:76077)
24. J.J. Modi, Parallel Algorithms and Matrix Computation, In Oxford Applied Mathematics and Computing Science Series, J.N. Buxton, R.F. Churchhouse and A.B. Tayler eds., Clarendon Press, Oxford, 1988.
25. M.F. Murphy, G.H. Golub and A.J. Wathen, A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21(2000), 1969-1972. MR1762024 (2001a:65055)
26. C.-L. Wang and Z.-Z. Bai, Sufficient conditions for the convergent splittings of non-Hermitian positive definite matrices, Linear Algebra Appl., 330(2001), 215-218. MR1826657 (2002a:65059)
27. A.J.Wathen and D.J. Silvester, Fast iterative solution of stabilized Stokes systems. Part I: Using simple diagonal preconditioners, SIAM J. Numer. Anal., 30(1993), 630-649. MR1220644 (94a:65060)
28. S.L. Weissman, High-accuracy low-order three-dimensional brick elements, Intern. J. Numer. Methods Engrg., 39(1996), 2337-2361. MR1399024 (97b:73094)
29. J.-X. Zhao, W.-G. Wang and W.-Q. Ren, Stability of the matrix factorization for solving block tridiagonal symmetric indefinite linear systems, BIT, 44(2004), 181-188. MR2057369 (2005a:65030)