Block Krylov-Schur method for large symmetric eigenvalue problems

Numerical Algorithms

Volumn 47, Issue 4, 2008, Pages 341-359

  Zhou, Yunkai ^a   Saad, Yousef ^b  

^a SOUTHERN METHODIST UNIVERSITY   (United States)

^b University of Minnesota   (United States)

Author keywords

Block method; Implicit restart; Krylov Schur; Lanczos

Indexed keywords

EID: 42149133054     PISSN: 10171398     EISSN: 15729265     Source Type: Journal    
DOI: 10.1007/s11075-008-9192-9     Document Type: Article

References (38)

1. Baglama, J., Calvetti, D., Reichel, L.: IRBL: an implicitly restarted block-lanczos method for large-scale Hermitian eigenproblems. SIAM J. Sci. Comput. 24, 1650-1677 (2003)
2. Baglama, J., Calvetti, D., Reichel, L.: Irbleigs: a MATLAB program for computing a few eigenpairs of a large sparse Hermitian matrix. ACM Trans. Math. Software 5, 337-348 (2003)
3. Bai, Z., Day, D., Ye, Q.: ABLE: an adaptive block Lanczos method for non-Hermitian eigenvalue problems. SIAM J. Matrix Anal. Appl. 20, 1060-1082 (1999)
4. Bischof, C.H., Quintana-Ortí, G.: Computing rank-revealing QR factorizations of dense matrices. ACM Trans. Math. Software 24, 226-253 (1998)
5. Chelikowsky, J.R., Troullier, N., Saad, Y.: Finite-difference- pseudopotential method: electronic structure calculations without a basis. Phys. Rev. Lett. 72, 1240-1243 (1994)
6. Dai, H., Lancaster, P.: Preconditioning block Lanczos algorithm for solving symmetric eigenvalue problems. J. Comput. Math. 18(4), 365-374 (2000)
7. Daniel, J., Gragg, W.B., Kaufman, L., Stewart, G.W.: Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comp. 30, 772-795 (1976)
8. Freund, R.W.: Band Lanczos method. In: Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.), Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, pp. 80-88. SIAM, Philadelphia (2000)
9. Golub, G.H., Luk, F.T., Overton, M.L.: A block Lanczos method for computing the singular values and corresponding singular vectors of a matrix. ACM Trans. Math. Software 7, 149-169 (1981)
10. Grimes, R.G., Lewis, J.G., Simon, H.D.: A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems. SIAM J. Matrix Anal. Appl. 15, 228-272 (1994)
11. Knyazev, A.V.: Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23(2), 517-541 (2001)
12. Knyazev, A.V., Neymeyr, K.: Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method. ETNA 15, 38-55 (2003)
13. Kohn, W., Sham, L.J.: Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133-A1138 (1965)
14. Kronik, L., Makmal, A., Tiago, M., Alemany, M., Jain, M., Huang, X., Saad, Y., Chelikowsky, J.: PARSEC-the pseudopotential algorithm for real-space electronic structure calculations: recent advances and novel applications to nano-structures. Phys. Status Solidi B 243, 1063-1079 (2006)
15. Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Standards 45, 255-282 (1950)
16. Lehoucq, R., Sorensen, D.C.: Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Anal. Appl. 17, 789-821 (1996)
17. Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK user's guide: solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, Philadelphia. Available at http://www.caam.rice.edu/software/ARPACK/ (1998)
18. Miminis, G., Paige, C.: Implicit shifting in the QR and related algorithms. SIAM J. Matrix Anal. Appl. 12, 385-400 (1991)
19. Parlett, B.N.: The symmetric eigenvalue problem. No. 20 in Classics in Applied Mathematics. SIAM, Philadelphia, PA (1998)
20. Parlett, B.N., Le, J.: Forward instability of tridiagonal QR. SIAM J. Matrix Anal. Appl. 4, 279-316 (1993)
21. Saad, Y.: On the rates of convergence of the Lanczos and the block Lanczos methods. SIAM J. Numer. Anal. 17, 687-706 (1980)
22. Y. Saad 1992 Numerical methods for large eigenvalue problems John Wiley New York Available at http://www.cs.umn.edu/~saad/books.html
23. Saad, Y.: Numerical methods for large eigenvalue problems. John Wiley, New York. Available at http://www.cs.umn.edu/~saad/books.html (1992)
24. Sadkane, M.: Block-Arnoldi and Davidson methods for unsymmetric large eigenvalue problems. Numerische Mathematik 64, 195-212 (1993a)
25. Sadkane, M.: A block Arnoldi-Chebyshev method for computing leading eigenpairs of large sparse unsymmetric matrices. Numerische Mathematik 64, 181-194 (1993b)
26. Sadkane, M., Sidje, R.B.: Implementation of a variable block Davidson method with deflation for solving large sparse eigenproblems. Numer. Algor. 20(2), 217-240 (1999)
27. Scott, D.S.: Block Lanczos software for symmetric eigenvalue problems. Technical report ORNL/CSD-48, Oak Ridge National Laboratory, Oak Ridge, TN (1979)
28. Sorensen, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357-385 (1992)
29. Sorensen, D.C.: Deflation for implicitly restarted Arnoldi methods. Technical report CAAM:TR98-12, Rice University (1998)
30. Stathopoulos, A.: Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part I: seeking one eigenvalue. SIAM J. Sci. Comput. 29(2), 481-514 (2007)
31. Stathopoulos, A., McCombs, J.R.: Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part II: seeking many eigenvalues. SIAM J. Sci. Comput. 29(5), 2162-2188 (2007)
32. Stathopoulos, A., Saad, Y., Wu, K.: Dynamic thick restarting of the Davidson and the implicitly restarted Arnoldi methods. SIAM J. Sci. Comput. 19, 227-245 (1998)
33. Stewart, G.W.: A Krylov-Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23, 601-614 (2001)
34. Stewart, G.W.: Matrix Algorithms II: Eigensystems. SIAM, Philadelphia (2001)
35. Watkins, D.S.: Forward stability and transmission of shifts in the QR algorithm. SIAM J. Matrix Anal. Appl. 16, 469-487 (1995)
36. Wu, K., Simon, H.: Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22, 602-616 (2000)
37. Ye, Q.: An adaptive block Lanczos algorithm. Numer. Algor. 12, 97-110 (1996)
38. Zhou, Y.: A block Chebyshev-Davidson method with inner-outer restart for large eigenvalue problems. Technical report, Southern Methodist University (to be submitted)