Skip to main content
SpringerLink
Log in

Block Krylov–Schur method for large symmetric eigenvalue problems

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Stewart’s Krylov–Schur algorithm offers two advantages over Sorensen’s implicitly restarted Arnoldi (IRA) algorithm. The first is ease of deflation of converged Ritz vectors, the second is the avoidance of the potential forward instability of the QR algorithm. In this paper we develop a block version of the Krylov–Schur algorithm for symmetric eigenproblems. Details of this block algorithm are discussed, including how to handle rank deficient cases and how to use varying block sizes. Numerical results on the efficiency of the block Krylov–Schur method are reported.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bischof, C.H., Quintana-Ortí, G.: Computing rank-revealing QR factorizations of dense matrices. ACM Trans. Math. Software 24, 226–253 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  6. Dai, H., Lancaster, P.: Preconditioning block Lanczos algorithm for solving symmetric eigenvalue problems. J. Comput. Math. 18(4), 365–374 (2000)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  13. Kohn, W., Sham, L.J.: Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133–A1138 (1965)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  16. Lehoucq, R., Sorensen, D.C.: Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Anal. Appl. 17, 789–821 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Parlett, B.N.: The symmetric eigenvalue problem. No. 20 in Classics in Applied Mathematics. SIAM, Philadelphia, PA (1998)

    Google Scholar 

  20. Parlett, B.N., Le, J.: Forward instability of tridiagonal QR. SIAM J. Matrix Anal. Appl. 4, 279–316 (1993)

    Article  MathSciNet  Google Scholar 

  21. Saad, Y.: On the rates of convergence of the Lanczos and the block Lanczos methods. SIAM J. Numer. Anal. 17, 687–706 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  22. Saad, Y.: Numerical methods for large eigenvalue problems. John Wiley, New York. Available at http://www.cs.umn.edu/~saad/books.html (1992)

    MATH  Google Scholar 

  23. Sadkane, M.: Block-Arnoldi and Davidson methods for unsymmetric large eigenvalue problems. Numerische Mathematik 64, 195–212 (1993a)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sadkane, M.: A block Arnoldi–Chebyshev method for computing leading eigenpairs of large sparse unsymmetric matrices. Numerische Mathematik 64, 181–194 (1993b)

    Article  MathSciNet  MATH  Google Scholar 

  25. 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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Scott, D.S.: Block Lanczos software for symmetric eigenvalue problems. Technical report ORNL/CSD-48, Oak Ridge National Laboratory, Oak Ridge, TN (1979)

  27. Sorensen, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sorensen, D.C.: Deflation for implicitly restarted Arnoldi methods. Technical report CAAM:TR98-12, Rice University (1998)

  29. 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)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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)

    Article  MathSciNet  Google Scholar 

  31. 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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Stewart, G.W.: A Krylov–Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23, 601–614 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Stewart, G.W.: Matrix Algorithms II: Eigensystems. SIAM, Philadelphia (2001)

    MATH  Google Scholar 

  34. Watkins, D.S.: Forward stability and transmission of shifts in the QR algorithm. SIAM J. Matrix Anal. Appl. 16, 469–487 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wu, K., Simon, H.: Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22, 602–616 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ye, Q.: An adaptive block Lanczos algorithm. Numer. Algor. 12, 97–110 (1996)

    Article  MATH  Google Scholar 

  37. Zhou, Y.: A block Chebyshev–Davidson method with inner-outer restart for large eigenvalue problems. Technical report, Southern Methodist University (to be submitted)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Southern Methodist University, Dallas, TX, 75275, USA

    Yunkai Zhou

  2. Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN, 55455, USA

    Yousef Saad

Authors
  1. Yunkai Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yousef Saad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunkai Zhou.

Additional information

Work supported by US Department of Energy under contract DE-FG02-03ER25585, by NSF grants ITR-0428774 and CMMI-0727194, and by the Minnesota Supercomputing Institute.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, Y., Saad, Y. Block Krylov–Schur method for large symmetric eigenvalue problems. Numer Algor 47, 341–359 (2008). https://doi.org/10.1007/s11075-008-9192-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-008-9192-9

Keywords

Search

Navigation