| Title:
|
Combining the preconditioned conjugate gradient method and a matrix iterative method (English) |
| Author:
|
Zítko, Jan |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
41 |
| Issue:
|
1 |
| Year:
|
1996 |
| Pages:
|
19-39 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The preconditioned conjugate gradient method for solving the system of linear algebraic equations with a positive definite matrix is investigated. The initial approximation for conjugate gradient is constructed as a result of a matrix iteration method after $m$ steps. The behaviour of the error vector for such a combined method is studied and special numerical tests and conclusions are made. (English) |
| Keyword:
|
conjugate gradients |
| Keyword:
|
preconditioning |
| Keyword:
|
iterative method |
| Keyword:
|
numerical experiments |
| MSC:
|
65F10 |
| MSC:
|
65F35 |
| idZBL:
|
Zbl 0847.65016 |
| idMR:
|
MR1365137 |
| DOI:
|
10.21136/AM.1996.134311 |
| . |
| Date available:
|
2009-09-22T17:50:01Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134311 |
| . |
| Reference:
|
[G-L] G.H. Golub, C.F. Van Loan: Matrix Computation.The John Hopkins University Press, Baltimore, 1984. |
| Reference:
|
[E-G] H.C. Elman, G.H. Golub: Block Iterative Methods for Cyclically Reduced Non-Self-Adjoint Elliptic Problems.Chapter 6 in the book “Iterative Methods for Large Linear Systems” edited by David R. Kincaid and Linda J. Hayes, Center for Numerical Analysis The University of Texas at Austin, Academic Press, 1989. |
| Reference:
|
[He] P. Henrici: The Quotient-Difference Algorithm.Further Contribution to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues, Vol. 49, National Bureau of Standards Applied Mathematics Series, 1958. Zbl 0136.12803, MR 0094901 |
| Reference:
|
[D.O’L] D.P. O’Leary: The Block Conjugate Gradient Algorithm and Related Methods., Linear Algebra Appl. 29 (1980), 293–322. MR 0562766, 10.1016/0024-3795(80)90247-5 |
| Reference:
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| Reference:
|
[Si-F-Sm] A. Sidi, W.F. Ford, D.A. Smith: Acceleration of Convergence of Vector Sequences.SIAM J. Numer. Anal. 23 (1986), no. 1, 178–196. MR 0821914, 10.1137/0723013 |
| Reference:
|
[Si 86] A. Sidi: Convergence and Stability Properties of Minimal Polynomial and Reduced Rank Extrapolation Algorithms., SIAM J. Numer. Anal. 23 (1986), no. 1, 197–209. Zbl 0612.65001, MR 0821915, 10.1137/0723014 |
| Reference:
|
[S-S 86] Y. Saad, M.H. Schultz: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems., SIAM J. Sci. Stat. Comput. 7 (1986), no. 3, 856–869. MR 0848568, 10.1137/0907058 |
| Reference:
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| Reference:
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[V-V 93] H.A. Van der Vorst, C. Vuik: The superlinear convergence behaviour of GMRES.J. Comput. Appl. Math. 48 (1993), 327–341. MR 1252545, 10.1016/0377-0427(93)90028-A |
| Reference:
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[Y] D.M. Young: Iterative solution of large linear systems.Academic Press, New York-London, 1971. Zbl 0231.65034, MR 0305568 |
| Reference:
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| Reference:
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| Reference:
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[Zi 92] J. Zítko: Numerical experiments with extrapolated procedures.Programy a algoritmy numerické matematiky 6, Sborník kursu, Bratříkov 1992, pp. 178–187. (Czech) |
| Reference:
|
[Zi 93] J. Zítko: Combining the preconditioned conjugate gradient method and the norm-reducing matrix iterative method.Technical report No 106/93, Prague 1993, pp. 1–17. |
| . |
