Previous |  Up |  Next


conjugate gradients; preconditioning; iterative method; numerical experiments
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.
[G-L] G.H. Golub, C.F. Van Loan: Matrix Computation. The John Hopkins University Press, Baltimore, 1984.
[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.
[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. MR 0094901 | Zbl 0136.12803
[D.O’L] D.P. O’Leary: The Block Conjugate Gradient Algorithm and Related Methods.  Linear Algebra Appl. 29 (1980), 293–322. DOI 10.1016/0024-3795(80)90247-5 | MR 0562766
[Si 88] A. Sidi: Extrapolation vs. Projection Methods for Linear Systems of Equations. J. Comput. Appl. Math. 22 (1988), 71–88. DOI 10.1016/0377-0427(88)90289-0 | MR 0948887 | Zbl 0646.65031
[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. DOI 10.1137/0723013 | MR 0821914
[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. DOI 10.1137/0723014 | MR 0821915 | Zbl 0612.65001
[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. DOI 10.1137/0907058 | MR 0848568
[V] R.L. Varga: Matrix iterative analysis. Prentice-Hall Englewood Clifs, New Jersey, 1962. MR 0158502
[V-V 93] H.A. Van der Vorst, C. Vuik: The superlinear convergence behaviour of GMRES. J. Comput. Appl. Math. 48 (1993), 327–341. DOI 10.1016/0377-0427(93)90028-A | MR 1252545
[Y] D.M. Young: Iterative solution of large linear systems. Academic Press, New York-London, 1971. MR 0305568 | Zbl 0231.65034
[Zi 83] J. Zítko: Improving the Convergence of Iterative Methods. Apl. Mat. 28 (1983), 215–229. MR 0701740
[Zi 84] J. Zítko: Convergence of Extrapolation Coefficients.  Apl. Mat. 29 (1984), 114–133. MR 0738497
[Zi 92] J. Zítko: Numerical experiments with extrapolated procedures. Programy a algoritmy numerické matematiky 6, Sborník kursu, Bratříkov 1992, pp. 178–187. (Czech)
[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.
Partner of
EuDML logo