On stabilization and convergence of clustered Ritz values in the Lanczos method

SIAM Journal on Matrix Analysis and Applications

Volumn 27, Issue 3, 2005, Pages 891-908

  Wülling, Wolfgang ^a  

^a BIELEFELD UNIVERSITY   (Germany)

Author keywords

Eigenvalues; Lanczos algorithm; Ritz values; Tridiagonal matrix

Indexed keywords

APPROXIMATION THEORY; CALCULATIONS; CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; MATRIX ALGEBRA; MEASUREMENT THEORY; STABILIZATION;

EIGENVALUES; LANCZOS ALGORITHM; RITZ VALUES; TRIDIAGONAL MATRIX;

ALGORITHMS;

EID: 33646777312     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/040608908     Document Type: Article

References (24)

1. L. V. AHLFORS, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1978.
2. Z. BAI, Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalue problem, Math. Comp., 62 (1994), pp. 209-226.
3. H. FASSBENDER, Error analysis of the symplectic Lanczos method for the symplectic eigenvalue problem, BIT, 40 (2000), pp. 471-496.
4. A. GREENBAUM, Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences, Linear Algebra Appl., 113 (1989), pp. 7-63.
5. A. GREENBAUM AND Z. STRAKOŠ, Predicting the behavior of finite precision Lanczos and conjugate gradient computations, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 121-137.
6. L. A. KNIZHNERMAN, The quality of approximations to a well-isolated eigenvalue and the arrangement of Ritz numbers in a simple Lanczos process, Comput. Math. Math. Phys., 36 (1995), pp. 1176-1187.
7. C. LANCZOS, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards, 46 (1960), pp. 266-282.
8. C. C. PAIGE, The Computation of Eigenvalues and Eigenvectors of Very Large Sparse Matrices, Ph.D. thesis, London University, Institute of Computer Science, London, England, 1971.
9. C. C. PAIGE, Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix, J. Inst. Math. Appl., 18 (1976), pp. 341-349.
10. C. C. PAIGE, Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem, Linear Algebra Appl., 34 (1980), pp. 235-258.
11. D. N. PARLETT, The Symmetric Eigenvalue Problem, Prentice Hall, Englewood Cliffs, NJ, 1980.
12. B. N. PARLETT, Misconvergence in the Lanczos Algorithm, Oxford Univ. Press, New York, 1990.
13. B. N. PARLETT AND J. LE, Forward instability of tridiagonal QR, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 279-316.
14. B. N. PARLETT, Do we fully understand the symmetric Lanczos algorithm yet?, Proceedings of the Cornelius Lanczos International Centenary Conference, Raleigh, NC, 1993, pp. 93-107, SIAM, Philadelphia, 1994.
15. B. N. PARLETT, Invariant subspaces for tightly clustered eigenvalues of tridiagonals, BIT, 36 (1996), pp. 642-662.
16. D. C. SORENSEN, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 367-386.
17. Z. STRAKOŠ, On the real convergence rate of the conjugate gradient method, Linear Algebra Appl., 164-166 (1991), pp. 636-649.
18. Z. STRAKOŠ AND A. GREENBAUM, Open Questions in the Convergence Analysis of the Lanczos Process for the Real Symmetric Eigenvalue Problem, IMA Research Report 934, IMA Workshop on Iterative Methods, University of Minnesota, Minneapolis, MN, 1992.
19. R. C. THOMPSON AND P. MCENTEGGERT, Principal submatrices, II: The upper and lower quadratic inequalities, Linear Algebra Appl., 1 (1968), pp. 211-243.
20. J. H. WILKINSON, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1966.
21. K. WU AND H. SIMON, Thick-restart Lanczos method for large symmetric eigenvalue problems, SIAM J. Matrix Anal. Appl., 22 (2000), pp. 602-616.
22. Q. YE, On close eigenvalues of tridiagonal matrices, Numer. Math., 70 (1996), pp. 607-614.
23. J. P. M. ZEMKE, Krylov Subspace Methods in Finite Precision: A Unified Approach, Dissertation, Technische Universität Hamburg-Harburg, Hamburg, Germany, 2003.
24. J. P. M. ZEMKE, (Hessenberg) Eigenvalue-Eigenmatrix Relations, Report 78 des Arbeitsbereiches Mathematik 4-13, Technische Universität Hamburg-Harburg, Hamburg, Germany, 2004.