Fifty years of the method of conjugate gradients or Will computers cope with

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1. Arnoldi, W., The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quart. Appl. Math. 9 (1951), 17–29. (1951) Zbl0042.12801MR0042792
2. Axelsson, O., Iterative solution methods, Cambridge Univ. Press, Cambridge 1994. (1994) Zbl0813.15021MR1276069
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. (1999) 
4. Axelsson, O., Barker, V. A., Finite element solution of boundary value problems. Theory and computation, Academic Press, New York 1984. (1984) Zbl0537.65072MR0758437
5. Beckermann, B., Kuijlaars, A. B. J., Superlinear convergence of conjugate gradients, SIAM J. Numer. Anal. 39 (2001), 300–329. (2001) Zbl0997.65058MR1860727
6. Benzi, M., Tůma, M., A sparse approximate inverse preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput. 19 (1998), 968–994. (1998) Zbl0930.65027MR1616710
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. (1999) Zbl0969.65027
8. Brandts, J., Riccati algorithms for eigenvalues and invariant subspaces of large and sparse matrices, Accepted by Linear Algebra Appl. in 2001. (2001) 
9. Coppersmith, D., Winograd, S., Matrix multiplication via arithmetic progression, J. Symbolic Comput. 9 (1990), 251–280. (1990) Zbl0702.65046MR1056627
10. Daniel, J. W., The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal. 4 (1967), 10–26. (1967) Zbl0154.40302MR0217987
11. Golub, G. H., Loan, Ch. F. van, Matrix computation, The John Hopkins Univ. Press, Baltimore 1984. (1984) 
12. Hestenes, M. R., Stiefel, E., Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards 49 (1952), 409–436. (1952) Zbl0048.09901MR0060307
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. (1977) Zbl0364.65028MR0451658
14. Kaniel, S., Estimates for some computational techniques in linear algebra, Math. Comp. 20 (1966), 369–378. (1966) Zbl0156.16202MR0234618
15. Křížek, M., Neittaanmäki, P., Finite Element Approximation of Variational Problems and Applications, Longman Scientific & Technical, Harlow; John Wiley & Sons, New York 1990. (1990) Zbl0708.65106MR1066462
16. Křížek, M., Neittaanmäki, P., Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications, Kluwer Academic Publishers, Dordrecht 1996. (1996) Zbl0859.65128MR1431889
17. Křížek, M., Segeth, K., Numerické modelování problémů elektrotechniky, Karolinum, Praha 2001. (2001) 
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. (1950) Zbl0045.39702MR0042791
19. Lanczos, C., Solution of systems of linear equations by minimized iterations, J. Res. Nat. Bur. Standards 49 (1952), 33–53. (1952) MR0051583
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. (1986) Zbl0599.65018MR0848568
21. Simoncini, V., A stabilized QMR version of block BiCG, SIAM J. Matrix Anal. Appl. 18 (1997), 419–434. (1997) Zbl0872.65024MR1437340
22. Simoncini, V., On the convergence of restarted Krylov subspace methods, SIAM J. Matrix Anal. Appl. 22 (2000), 430–452. (2000) Zbl0969.65023MR1780193
23. Sluis, A. van der, Vorst, H. van der, The rate of convergence of conjugate gradients, Numer. Math. 48 (1986), 543–560. (1986) Zbl0596.65015MR0839616
24. Sorensen, D. C., Implicit application of polynomial filters in a $k$-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13 (1992), 357–385. (1992) Zbl0763.65025MR1146670
25. Strassen, V., Gaussian elimination is not optimal, Numer. Math. 13 (1969), 354–356. (1969) Zbl0185.40101MR0248973
26. Strassen, V., Algebraic complexity theory, Handbook of Theoretical Computer Science, Vol. A (ed. van Leeuwen, J.), Elsevier, Amsterdam 1990, 634–672. (1990) Zbl0900.68247MR1127177
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. (1992) Zbl0761.65023MR1149111
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. (1949) MR0030816
29. Zítko, J., Combining the preconditioned conjugate gradient method and a matrix iterative method, Appl. Math. 41 (1996), 19–39. (1996) Zbl0847.65016MR1365137
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. (1985) MR0820049

1. Zdeněk Strakoš, Metoda konjugovaných gradientů jako dobrodružství jdoucí přes staletí