| Title:
|
Padesát let metody sdružených gradientů aneb Zvládnou počítače soustavy milionů rovnic o milionech neznámých? (Czech) |
| Title:
|
Fifty years of the method of conjugate gradients or Will computers cope with (English) |
| Author:
|
Brandts, Jan |
| Author:
|
Křížek, Michal |
| Language:
|
Czech |
| Journal:
|
Pokroky matematiky, fyziky a astronomie |
| ISSN:
|
0032-2423 |
| Volume:
|
47 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
103-113 |
| . |
| Category:
|
math |
| . |
| MSC:
|
01A60 |
| MSC:
|
65-03 |
| MSC:
|
65F10 |
| idZBL:
|
Zbl 1051.65029 |
| . |
| Date available:
|
2010-12-11T19:15:29Z |
| Last updated:
|
2012-08-26 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141120 |
| . |
| Reference:
|
[1] Arnoldi, W.: The principle of minimized iterations in the solution of the matrix eigenvalue problem.Quart. Appl. Math. 9 (1951), 17–29. Zbl 0042.12801, MR 0042792 |
| Reference:
|
[2] Axelsson, O.: Iterative solution methods.Cambridge Univ. Press, Cambridge 1994. Zbl 0813.15021, MR 1276069 |
| Reference:
|
[3] Axelsson, O.: Optimal preconditioners based on rate of convergence estimates for the conjugate gradient method.LN of IMAMM 99 (eds. Míka, S., Brandner, M.), Univ. of West Bohemia, Pilsen 1999, 5–56. |
| Reference:
|
[4] Axelsson, O., Barker, V. A.: Finite element solution of boundary value problems. Theory and computation.Academic Press, New York 1984. Zbl 0537.65072, MR 0758437 |
| Reference:
|
[5] Beckermann, B., Kuijlaars, A. B. J.: Superlinear convergence of conjugate gradients.SIAM J. Numer. Anal. 39 (2001), 300–329. Zbl 0997.65058, MR 1860727 |
| Reference:
|
[6] Benzi, M., Tůma, M.: A sparse approximate inverse preconditioner for nonsymmetric linear systems.SIAM J. Sci. Comput. 19 (1998), 968–994. MR 1616710 |
| Reference:
|
[7] Brandts, J.: Explanation of a phenomenon witnessed in pre-processed GMRES.ENUMATH 99 — Proc. of the 3rd European Conf. on Numer. Math. and Advanced Applications, Jyväskylä, Finland 1999, ed. by Neittaanmäki, P. et al., World Scientific, Singapore 2000, 440–447. |
| Reference:
|
[8] Brandts, J.: Riccati algorithms for eigenvalues and invariant subspaces of large and sparse matrices.Accepted by Linear Algebra Appl. in 2001. |
| Reference:
|
[9] Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progression.J. Symbolic Comput. 9 (1990), 251–280. MR 1056627 |
| Reference:
|
[10] Daniel, J. W.: The conjugate gradient method for linear and nonlinear operator equations.SIAM J. Numer. Anal. 4 (1967), 10–26. Zbl 0154.40302, MR 0217987 |
| Reference:
|
[11] Golub, G. H., Loan, Ch. F. van: Matrix computation.The John Hopkins Univ. Press, Baltimore 1984. |
| Reference:
|
[12] Hestenes, M. R., Stiefel, E.: Methods of conjugate gradients for solving linear systems.J. Res. Nat. Bur. Standards 49 (1952), 409–436. Zbl 0048.09901, MR 0060307 |
| Reference:
|
[13] Jennings, A.: Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method.J. Inst. Maths Applics 20 (1977), 61–72. Zbl 0364.65028, MR 0451658 |
| Reference:
|
[14] Kaniel, S.: Estimates for some computational techniques in linear algebra.Math. Comp. 20 (1966), 369–378. MR 0234618 |
| Reference:
|
[15] Křížek, M., Neittaanmäki, P.: Finite Element Approximation of Variational Problems and Applications.Longman Scientific & Technical, Harlow; John Wiley & Sons, New York 1990. MR 1066462 |
| Reference:
|
[16] Křížek, M., Neittaanmäki, P.: Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications.Kluwer Academic Publishers, Dordrecht 1996. MR 1431889 |
| Reference:
|
[17] Křížek, M., Segeth, K.: Numerické modelování problémů elektrotechniky.Karolinum, Praha 2001. |
| Reference:
|
[18] Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators.J. Res. Nat. Bur. Standards 45 (1950), 255–282. MR 0042791 |
| Reference:
|
[19] Lanczos, C.: Solution of systems of linear equations by minimized iterations.J. Res. Nat. Bur. Standards 49 (1952), 33–53. MR 0051583 |
| Reference:
|
[20] Saad, Y., Schultz, M. H.: GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems.SIAM J. Sci. Stat. Comput. 7 (1986), 856–869. MR 0848568 |
| Reference:
|
[21] Simoncini, V.: A stabilized QMR version of block BiCG.SIAM J. Matrix Anal. Appl. 18 (1997), 419–434. Zbl 0872.65024, MR 1437340 |
| Reference:
|
[22] Simoncini, V.: On the convergence of restarted Krylov subspace methods.SIAM J. Matrix Anal. Appl. 22 (2000), 430–452. Zbl 0969.65023, MR 1780193 |
| Reference:
|
[23] Sluis, A. van der, Vorst, H. van der: The rate of convergence of conjugate gradients.Numer. Math. 48 (1986), 543–560. MR 0839616 |
| Reference:
|
[24] Sorensen, D. C.: Implicit application of polynomial filters in a $k$-step Arnoldi method.SIAM J. Matrix Anal. Appl. 13 (1992), 357–385. Zbl 0763.65025, MR 1146670 |
| Reference:
|
[25] Strassen, V.: Gaussian elimination is not optimal.Numer. Math. 13 (1969), 354–356. Zbl 0185.40101, MR 0248973 |
| Reference:
|
[26] Strassen, V.: Algebraic complexity theory.Handbook of Theoretical Computer Science, Vol. A (ed. van Leeuwen, J.), Elsevier, Amsterdam 1990, 634–672. Zbl 0900.68247, MR 1127177 |
| Reference:
|
[27] Vorst, H. van der: Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems.SIAM J. Sci. Statist. Comput. 13 (1992), 631–644. Zbl 0761.65023, MR 1149111 |
| Reference:
|
[28] Verzuh, F. M.: The solution of simultaneous linear equations with the aid of the 602 calculating punch.Math. Comp. (Math. Tables and other Aids to Computation) 3 (1949), 453–462. MR 0030816 |
| Reference:
|
[29] Zítko, J.: Combining the preconditioned conjugate gradient method and a matrix iterative method.Appl. Math. 41 (1996), 19–39. MR 1365137 |
| Reference:
|
[30] Zowe, J.: Nondifferentiable optimization — a motivation and short introduction in the subgradient and the bundle concept.ASI Proc. Comp. Math. Progr. 1985, 323–356. MR 0820049 |
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