A highly parallel explicitly restarted Lanczos algorithm

  • M. Szularz
  • J. Weston
  • M. Clint
  • K. Murphy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1184)


The Lanczos algorithm is one of the principal methods for the computation of a few of the extreme eigenvalues and their corresponding eigenvectors of large, usually sparse, real symmetric matrices. In this paper a single-vector Lanczos method based on a simple restarting strategy is proposed. The algorithm finds one eigenpair at a time using a deflation technique in which each Lanczos vector generated is orthogonalized against all previously converged eigenvectors. The approach taken yields a fixed k-step restarting scheme which permits the reorthogonalization between the Lanczos vectors to be almost completely eliminated. The orthogonalization strategy developed falls naturally into the class of selective orthogonalization strategies as described by Simon.

A ’reverse communication’ implementation of the algorithm on an MPP Connection Machine CM-200 with 8K processors is discussed. Test results using examples from the Harwell-Boeing collection of sparse matrices show the method to be very effective when compared with Sorensen's state of the art routine taken from the ARPACK library. Advantages of the algorithm include its guaranteed convergence, the ease with which it copes with genuinely multiple eigenvalues and fixed storage requirements.

Key words

Lanczos algorithm restarting deflation reorthogonalization MPP 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Daniel, J., Gragg, W.B., Kaufman, L., and Stewart, G.W., (1976),'Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization', Math. Comp., 30, 772–795.Google Scholar
  2. 2.
    Duff, I.S., Grimes, R.G., and Lewis, J.G., (1992),’ User's Guide for the Harwell-Boeing Sparse Matrix Collection’ (release I), available online ftp Scholar
  3. 3.
    Golub, G., and Van Loan C.F., (1989),’ Matrix Computations', John Hopkins University Press, London.Google Scholar
  4. 4.
    Lehoucq, R., Sorensen, D.C., and Vu, P.A., (1994), SSAUPD: Fortran subroutines for solving large scale eigenvalue problems, Release 2.1, available from in the scalapack directory.Google Scholar
  5. 5.
    Paige, C.C., (1976),’ Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix', J. Inst. Math. Applic., 18, 341–349.Google Scholar
  6. 6.
    Parlett, B.N., (1980),’ The Symmetric Eigenvalue Problem', Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
  7. 7.
    Parlett, B.N., and Scott, D.S., (1979),’ The Lanczos algorithm with selective orthogonalization', Math. Comput., 33, 217–238.Google Scholar
  8. 8.
    Rutter, J.D., (1994),’ A Serial Implementation of Cuppen's Divide and Conquer Algorithm for the Symmetric Eigenvalue Problem', LAPACK Working Note 69.Google Scholar
  9. 9.
    Simon, H.D., (1984),’ Analysis of the Symmetric Lanczos Algorithm with Reorthogonalization Methods',Linear Algebra Appl., 61, 101–131.Google Scholar
  10. 10.
    Sorensen, D.C., (1992),’ Implicit Application of Polynomial Filters in a k-step Arnoldi Method',SIAM J. Matrix Anal. Appl., 13, 357–385.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • M. Szularz
    • 1
  • J. Weston
    • 1
  • M. Clint
    • 2
  • K. Murphy
    • 2
  1. 1.School of Information & Software EngineeringUniversity of UlsterColeraineNorthern Ireland
  2. 2.Department of Computer ScienceThe Queen's University of BelfastBelfastNorthern Ireland

Personalised recommendations