Geodesic
On solving indefinite symmetric linear systems by means of the Lanczos method
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 39 (1999) no. 3, pp. 371-377 
A. Greenbaum; V. L. Druskin; L. A. Knizhnerman. On solving indefinite symmetric linear systems by means of the Lanczos method. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 39 (1999) no. 3, pp. 371-377. http://geodesic.mathdoc.fr/item/ZVMMF_1999_39_3_a2/
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     title = {On solving indefinite symmetric linear systems by means of the {Lanczos} method},
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Voir la notice de l'article provenant de la source Math-Net.Ru

A new result on the distribution of Ritz values at two consecutive steps of a simple Lanczos process is proved. It is used in showing why the Lanczos method effectively solves indefinite symmetric linear systems in computer arithmetic. The result of a numerical experiment is presented.

[1] Parlett B. N., The symmetric eigenvalue problem, Prentice Hall, N.J., Englewood Cliffs, 1980 | MR | Zbl

[2] Paige C. C., “Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix”, J. Inst. Math. Appl., 18 (1976), 341–349 | DOI | MR | Zbl

[3] Paige C. C., “Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem”, Linear Algebra and Appl., 34 (1980), 235–258 | DOI | MR | Zbl

[4] Druskin V. L., Knizhnerman L. A., “Otsenki oshibok v prostom protsesse Lantsosha pri vychislenii funktsii ot simmetrichnykh matrits i sobstvennykh znachenii”, Zh. vychisl. matem. i matem. fiz., 31:7 (1991), 970–983 | MR

[5] Druskin V., Knizhnerman L., “Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic”, Numer. Linear Algebra and Appl., 2 (1995), 205–219 | DOI | MR

[6] Golub G. H., Strakoš Z., “Estimates in quadratic formulas”, Numer. Algorithms, 8 (1994), 241–268 | DOI | MR | Zbl

[7] Greenbaum A., “Behavior of slightly perturbed Lanczos and conjugate gradient recurrences”, Linear Algebra and Appl., 11 (1989), 7–63 | DOI | MR

[8] Greenbaum A., Strakoš Z., “Predicting the behavior of finite precision Lanczos and conjugate gradient computations”, SIAM J. Matrix Analys. Appl., 13 (1992), 121–137 | DOI | MR | Zbl

[9] Knizhnerman L. A., “Kachestvo approksimatsii k khorosho otdelennomu sobstvennomu znacheniyu i raspolozhenie “chisel Rittsa” v prostom protsesse Lantsosha”, Zh. vychisl. matem. i matem. fiz., 35:10 (1995), 1459–1475 | MR | Zbl

[10] Paige C. C., Proc. Cornelius Lanczos Internat. Centenary Conf. 12–17 Dec. 1993, PA, USA: SIAM, 1993

[11] Paige C. C., Saunders M. A., “Solution of sparse indefinite systems of linear equations”, SIAM J. Nunier. Analys., 11 (1974), 197–209 | DOI | MR | Zbl

[12] Modersitzki J., Conf. Linear and Nonlinear Conjugate Gradient Methods (Seattle, WA, USA, 1995)

[13] Knizhnerman L., On adaptation of the Lanczos method to the spectrum: Res. Note, Ridgefield: Schlumberger-Doll Res., May 12, 1995, 18 pp.

[14] Van der Sluis A., van der Vorst H. A., “The convergence behavior of Ritz values in the presence of close eigenvalues”, Linear Algebra and Appl., 88–89 (1987), 651–694 | DOI | MR | Zbl

[15] Druskin V., Greenbaum A., Knizhnerman L., “Using nonorthogonal Lanczos vectors in the computation of matrix functions”, SIAM J. Sci. Comput., 19 (1998), 38–54 | DOI | MR | Zbl

[16] Cullum J., Greenbaum A., “Relations between Galerkin and norm-minimizing iterative methods for solving linear systems”, SIAM J. Matrix Analys. and Appl., 17 (1996), 223–247 | DOI | MR | Zbl