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Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices
Author(s):
Zhong-Zhi
Bai;
Gene
H.
Golub;
Chi-Kwong
Li.
Journal:
Math. Comp.
76
(2007),
287-298.
MSC (2000):
Primary 65F10, 65F50
Posted:
August 31, 2006
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Abstract:
For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skew-Hermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skew-Hermitian splitting iteration methods.
References:
-
- 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. MR 1648288 (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. MR 1972670 (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), 1-32. MR 2076052 (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. MR 1417759 (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. MR 2011340 (2004i:65025)
- 7.
- Z.-Z. Bai and R. Nabben, Some properties of the block matrices in the parallel decomposition-type relaxation methods, Appl. Numer. Math., 29:2(1999), 167-170. MR 1666521 (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. MR 2142592 (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. MR 2185669
- 11.
- M. Benzi and G.H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix. Anal. Appl., 26(2004), 20-41. MR 2112850 (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. MR 2117735
- 13.
- F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York and London, 1991. MR 1115205 (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. MR 0694523 (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. MR 0821905 (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. MR 1888834 (2002m:76071)
- 17.
- M. Fortin and R. Glowinski, Augmented Lagrangian Methods, Applications to the Numerical Solution of Boundary Value Problems, North-Holland, Amsterdam, 1983. MR 0724072 (85a:49004)
- 18.
- G.H. Golub and C. Greif, On solving block-structured indefinite linear systems, SIAM J. Sci. Comput., 24(2003), 2076-2092. MR 2005622 (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.MR 1917293 (2003f:65182)
- 20.
- G.H. Golub and D. Vanderstraeten, On the preconditioning of matrices with skew-symmetric splittings, Numer. Algorithms, 25(2000), 223-239.MR 1827156 (2002e:65067)
- 21.
- G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd Edition, The Johns Hopkins University Press, Baltimore, 1996. MR 1417720 (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. MR 1618828 (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. MR 1417791 (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. MR 1762024 (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. MR 1826657 (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. MR 1220644 (94a:65060)
- 28.
- S.L. Weissman, High-accuracy low-order three-dimensional brick elements, Intern. J. Numer. Methods Engrg., 39(1996), 2337-2361. MR 1399024 (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. MR 2057369 (2005a:65030)
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Additional Information:
Zhong-Zhi
Bai
Affiliation:
Department of Mathematics, Fudan University, Shanghai 200433, People's Republic of China, and State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, People's Republic of China
Email:
bzz@lsec.cc.ac.cn
Gene
H.
Golub
Affiliation:
Scientific Computing and Computational Mathematics Program, Department of Computer Science, Stanford University, Stanford, California 94305-9025
Email:
golub@sccm.stanford.edu
Chi-Kwong
Li
Affiliation:
Department of Mathematics, The College of William & Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795
Email:
ckli@math.wm.edu
DOI:
10.1090/S0025-5718-06-01892-8
PII:
S 0025-5718(06)01892-8
Keywords:
Non-Hermitian matrix,
positive semidefinite matrix,
Hermitian and skew-Hermitian splitting,
splitting iteration method,
convergence.
Received by editor(s):
February 11, 2005
Posted:
August 31, 2006
Additional Notes:
The work of the first author was supported by The Special Funds For Major State Basic Research Projects (No. G1999032803), The National Basic Research Program (No.~2005CB321702), The China NNSF Outstanding Young Scientist Foundation (No.~10525102) and The National Natural Science Foundation (No.~10471146), P.R. China, and The 2004 Ky and Yu-Fen Fan Fund Travel Grant of American Mathematical Society
The work of the second author was in part supported by the Department of Energy: DE-FC02-01ER41177
The research of the third author was partially supported by an NSF grant.
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Copyright
2006,
American Mathematical Society
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