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Krylov subspace solvers and preconditioners. (English) Zbl 1406.65021
The author offers a special point of view to the subject of numerical solution of systems of linear equations (SLE), with particular emphasis on the Krylov subspace method for the SLE with a symmetric positive definite matrix. The conjugate gradient method is studied in detail and good upper bounds of the distance between \(k\)-th iterate and the exact solution of the SLE are obtained. Good examples to illustrate a super-linear convergence behavior are provided. Analogous questions for the SLE with the “preconditioner” matrix and with a “general” matrix are considered. The investigations are of interest to the following topics: solving large, sparse systems of linear equations.
65F10 Iterative numerical methods for linear systems
65F08 Preconditioners for iterative methods
65F50 Computational methods for sparse matrices
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[1] O. Axelsson. Iterative Solution Methods. Cambridge University Press, Cambridge, UK, 1994. · Zbl 0795.65014
[2] O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numer. Math., 48:479–498, 1986. · Zbl 0564.65016
[3] R. Barrett, M. Berry, T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia, 1994.
[4] A. Bj¨orck and T. Elfving. Accelerated projection methods for computing pseudo-inverse solution of systems of linear equations. BIT, 19:145–163, 1979.
[5] E.K. Blum. Numerical Analysis and Computation, Theory and Practice. Addison-Wesley, Reading, 1972. · Zbl 0273.65001
[6] A.M. Bruaset. A Survey of Preconditioned Iterative Methods. Pitman research notes in mathematics series 328. Longman Scientific and Technical, Harlow, 1995.
[7] B.A. Carr´e. The determination of the optimum accelerating factor for successive over-relaxation. Computer Journal, 4:73–78, 1961.
[8] M. Eiermann, W. Niethammer, and R.S. Varga. A study of semiiterative methods for nonsymmetric systems of linear equations. Numer. Math., 47:505–533, 1985. · Zbl 0585.65025
[9] S.C. Eisenstat. Efficient implementation of a class of preconditioned conjugate gradient methods. SIAM J. Sci. Stat. Comput., 2:1–4, 1981. · Zbl 0474.65020
[10] S.C. Eisenstat, H.C. Elman, and M.H. Schultz. Variable iterative methods for nonsymmetric systems of linear equations. SIAM J. Num. Anal., 20:345–357, 1983. · Zbl 0524.65019
[11] M. Embree. The Tortoise and the Hare restart GMRES. SAIM Review, 45:259–266, 2003. · Zbl 1027.65039
[12] Y.A. Erlangga, C.W. Oosterlee, and C. Vuik. A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput., 27:1471–1492, 2006. · Zbl 1095.65109
[13] V. Faber and T. Manteuffel. Necessary and sufficient conditions for the existence of a conjugate gradient method. SIAM J. Num. Anal., 21:356–362, 1984. · Zbl 0546.65010
[14] R. Fletcher. Factorizing symmetric indefinite matrices. Lin. Alg. and its Appl., 14:257–277, 1976. · Zbl 0336.65022
[15] R.W. Freund, G.H. Golub, and N.M. Nachtigal. Iterative solution of linear systems. In A. Iserles, editor, Acta Numerica, pages 57–100. Cambridge University Press, Cambridge, UK, 1992. · Zbl 0762.65019
[16] R.W. Freund, M.H. Gutknecht, and N.M. Nachtigal. An implimentation of the look-ahead Lanczos algorithm for non-Hermitian matrices. SIAM J. Sci. Comp., 14:137–156, 1993. ESAIM: PROCEEDINGS AND SURVEYS43 · Zbl 0770.65022
[17] R.W. Freund and N.M. Nachtigal. QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math., 60:315–339, 1991. · Zbl 0754.65034
[18] T. Ginsburg. The conjugate gradient method. In J.H. Wilkinson and C. Reinsch, editors, Handbook for Automatic Computation, 2, Linear Algebra, pages 57–69, Berlin, 1971. Springer.
[19] G.H. Golub and C.F. van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore, 1996. Third edition.
[20] G.H. Golub and R.S. Varga. Chebychev semi-iterative methods, successive over-relaxation iterative methods and second order Richardson iterative methods. Part I and II. Numer. Math., 3:147–156, 157–168, 1961. · Zbl 0099.10903
[21] A. Greenbaum. Iterative Methods for Solving Linear Systems. Frontiers in applied mathmatics 17. SIAM, Philadelphia, 1997. · Zbl 0883.65022
[22] A. Greenbaum, V. Ptak, and Z. Strakos. Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl., 17:465–469, 1996. · Zbl 0857.65029
[23] I.A. Gustafsson. A class of first order factorization methods. BIT, 18:142–156, 1978. · Zbl 0386.65006
[24] L.A. Hageman and D.M. Young. Applied Iterative Methods. Academic Press, New York, 1981. · Zbl 0459.65014
[25] M.R. Hestenes and E. Stiefel. Methods of Conjugate Gradients for Solving Linear Systems. J. Res. Nat. Bur. Stand., 49:409– 436, 1952. · Zbl 0048.09901
[26] R.A. Horn and C.R. Johnson. Matrix Analysis (second edition). Cambridge University Press, Cambrige, 2013.
[27] E.F. Kaasschieter. Preconditioned conjugate gradients for solving singular systems. J. of Comp. Appl. Math., 24:265–275, 1988. · Zbl 0659.65031
[28] T.A. Manteuffel. The Tchebychev iteration for nonsymmetric linear systems. Numer. Math., 28:307–327, 1977. · Zbl 0361.65024
[29] 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. Comp., 31:148–162, 1977. · Zbl 0349.65020
[30] N.M. Nachtigal, S.C. Reddy, and L.N. Trefethen. How fast are non symmetric matrix iterations. SIAM J. Matrix Anal. Appl., 13:778–795, 1992. · Zbl 0754.65036
[31] C.C. Paige and M.A. Saunders. Solution of sparse indefinite system of linear equations. SIAM J. Num. Anal., 12:617–629, 1975. · Zbl 0319.65025
[32] C.C. Paige and M.A. Saunders. LSQR: an algorithm for sparse linear equations and sparse least square problem. ACM Trans. Math. Softw., 8:44–71, 1982. · Zbl 0478.65016
[33] B.N. Parlett, D.R. Taylor, and Z.A. Liu. A look-ahead Lanczos algorithm for unsymmetric matrices. Math. Comp., 44:105–124, 1985. · Zbl 0564.65022
[34] T. F. Chan J. Demmel J. Donato J. Dongarra V. Eijkhout R. Pozo C. Romine R. Barrett, M. Berry and H. Van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition. SIAM, Philadelphia, 1994. · Zbl 0814.65030
[35] J.K. Reid. The use of conjugate for systems of linear equations posessing property A. SIAM J. Num. Anal., 9:325–332, 1972. · Zbl 0259.65037
[36] Y. Saad. Preconditioning techniques for non symmetric and indefinite linear system. J. Comp. Appl. Math., 24:89–105, 1988. · Zbl 0662.65028
[37] Y. Saad. A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Stat. Comput., 14:461–469, 1993. · Zbl 0780.65022
[38] Y. Saad. Iterative Methods for Sparse Linear Systems. PWS Publishing, Boston, 1996. · Zbl 1031.65047
[39] Y. Saad. Iterative methods for sparse linear systems, Second Edition. SIAM, Philadelphia, 2003. · Zbl 1031.65046
[40] Y. Saad and M.H. Schultz. GMRES: a generalized minimal residual algorithm for solving non-symmetric linear systems. SIAM J. Sci. Stat. Comp., 7:856–869, 1986. · Zbl 0599.65018
[41] P. Sonneveld. CGS: a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 10:36–52, 1989. · Zbl 0666.65029
[42] C. Oosterlee U. Trottenberg and A. Sch¨uller. Multigrid. Academic Press, San Diego, 2001.
[43] A. van der Sluis. Conditioning, equilibration, and pivoting in linear algebraic systems. Numer. Math., 15:74–86, 1970. · Zbl 0182.49002
[44] A. van der Sluis and H.A. van der Vorst. The rate of convergence of conjugate gradients. Numer. Math., 48:543–560, 1986. · Zbl 0596.65015
[45] H.A. van der Vorst. High performance preconditioning. SIAM J. Sci. Stat. Comp., 10:1174–1185, 1989. · Zbl 0693.65027
[46] H.A. van der Vorst. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comp., 13:631–644, 1992. · Zbl 0761.65023
[47] H.A. van der Vorst. Iterative Krylov Methods for Large Linear Systems. Cambridge Monographs on Applied and Computational Mathematics, 13. Cambridge University Press, Cambridge, 2003. · Zbl 1023.65027
[48] H.A. van der Vorst and C. Vuik. The superlinear convergence behaviour of GMRES. J. Comput. Appl. Math., 48:327–341, 1993. · Zbl 0797.65026
[49] H.A. van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Num. Lin. Alg. Appl., 1:369–386, 1994. · Zbl 0839.65040
[50] R. S. Varga. Matrix Iterative Analysis. Springer, Berlin, 2000. · Zbl 0998.65505
[51] R.S. Varga. Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, N.J., 1962. · Zbl 0133.08602
[52] C. Vuik. Solution of the discretized incompressible Navier-Stokes equations with the GMRES method. Int. J. Num. Meth. in Fluids, 16:507–523, 1993. · Zbl 0825.76552
[53] E.L. Wachspress. Iterative Solution of Elliptic Systems. Prentice-Hall, Englewood Cliffs, 1966. · Zbl 0161.12203
[54] D.M. Young. Iterative Solution of Large Linear Systems. Academic Press, New York, 1971. · Zbl 0231.65034
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