We compare several iterative eigenvalue algorithms applicable to large sparse symmetric matrices, of so high an order that the matrix A cannot be stored in the memory of the computer, while it is easy to compute y = Ax for a given vector x.

Our main interest is focused to the Lanczos algorithm, and algorithms that apply an optimization strategy to the Rayleigh quotient, mainly steepest descent, multiple step steepest descent and conjugate gradients. We study the rates of convergence, nd find that the theoretical bounds for all the algorithms is determined by the separation of the extreme eigenvalue from the rest of the spectrum of the matrix.

We discuss computer implementation and report several numerical tests. These show a marked superiority for the Lanczos algorithm compared to the others. Of the other algorithms, the c-g algorithm also works well in wellconditioned cases, and it can be realized by a very simple program.


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  1. 1.
    Agoshkov, V.I., and Ju. A. Kuznetsov: Metod Lanczosa v problems sobstvennyc znachenii (Lanczos method for the eigenvalue problem) (Russian) in G. I. Marchuk (ed), Comp. Meth Linear Algebra, Novosibirsk (1972).Google Scholar
  2. 2.
    Alsen Britt Marie: Multiple step gradient iterative methods for computing eigen-values of large symmetric matrices. Techn. Rep. UMINF-15. 71,Univ. of Ume..Google Scholar
  3. 3.
    Andersson, I.: Experiments with the conjugate gradient algorithm for the determination of eigenvalues of symmetric matrices. Tech. Rep. UMINF-4. 71, Univ. of Ume..Google Scholar
  4. 4.
    Bradbury, W. W., and R. Fletcher: New iterative methods for solution of the eigen-problem. Num. Math. 9 (1966), 259–267.CrossRefGoogle Scholar
  5. 5.
    Buffoni, G.: Evaluation of eigensolutions of discrete space diffusion equation. Calcolo 4 (1967), 169–177.CrossRefGoogle Scholar
  6. 6.
    Byström, P.: SOR iterations for the computation of eigenvalues of a symmetric matrix. Tech. Rep. UMINF-29.72.Google Scholar
  7. 7.
    Cohen, A. I.: Rate of convergence of several conjugate gradient algorithms. SIAM J. Num. An. 9 (1972), 248–259.CrossRefGoogle Scholar
  8. 8.
    Curtis, A. R., and J. K. Reid: The solution of large sparse unsymmetric systems of linear equations. Proc. IFIP congr. (1971), Ljubljana TA-1, 1–5.Google Scholar
  9. 9.
    Daniel, J. W.: The Approximate minimization of functionals. Prentice Hall, Englewood Cliffs (1971).Google Scholar
  10. 10.
    Davis, C., and W. M. Kahan: The rotation of eigenvectors by a perturbation III. SIAM J. Num. An. 7 (1970), 1–46.CrossRefGoogle Scholar
  11. 11.
    Engeli, M., Ginsburg, Th., Rutishauser, H., and E. Stiefel: Refined Iterative methods for computation of the solution and the eigenvalues of self adjoint boundary value problems. Birkhäuser Verlag Basel (1959).Google Scholar
  12. 12.
    Faddev, D. K., and V. N. Faddeeva: Computational Methods of Linear Algebra. (Transl.) Freeman & Co., San Francisco (1963).Google Scholar
  13. 13.
    Fletcher, R., and C. M. Reeves: Function minimization by conjugate gradients. Comp. J. 7 (1964), 149–153.CrossRefGoogle Scholar
  14. 14.
    Forsythe, G. E.: On the asymptotic directions of the s-dimensional optimum gradient method. Num. Math. 11 (1968), 57–76.CrossRefGoogle Scholar
  15. 15.
    Fox, R. L., and M. P. Kapoor: A minimization method for the solution of the eigen-problem arising in structural dynamics. 2nd Conf. on Matrix meth. in struct. Engnr. Wright Patterson AFB, Ohio, 15–17. oct. 1968, Clearinghouse AD 703685, 271–306.Google Scholar
  16. 16.
    Fried, I.: Gradient methods for finite element eigenproblems. AIAA Jour. 7 (1969), 739–741.CrossRefGoogle Scholar
  17. 17.
    Fried, I.: Optimal gradient minimization scheme for finite element eigenproblems. J. Sound. Vib. 20 (1972), 333–342.CrossRefGoogle Scholar
  18. 18.
    Fried, I.: Bounds on the extremal eigenvalues of the finite element stiffness and mass matrices and their spectral condition number. J. Sound. Vib. 22 (1972), 407–418.CrossRefGoogle Scholar
  19. 19.
    Geradin, M.: The computational efficiency of a new minimization algorithm for eigen-value analysis. J. Sound. Vib. 19 (1971), 319–331.CrossRefGoogle Scholar
  20. 20.
    Geradin, M.: Analyse dynamique duals des structures par la méthode des éléments finis. Diss. Univ. de Liège, Belgium, (1972).Google Scholar
  21. 21.
    Golub, G. H., Underwood, R., and J. H. Wilkinson: The Lanczos algorithm for the symmetric Ax = λBx problem. Techn. Rep. STAN-CS-72–270, Stanford University.Google Scholar
  22. 22.
    Golub, G. H.: Some uses of the Lanczos Algorithm in numerical linear algebra. Conf. Num. An. Dublin Aug (1972).Google Scholar
  23. 23.
    Hestenes, M. R.: Multiplier and gradient methods. Jour. Opt. Theory Appl. 4 (1969), 303–320.CrossRefGoogle Scholar
  24. 24.
    Hestenes, M. R., and W. Karush: A method of gradients for the calculation of the characteristic roots and vectors of a real symmetric matrix. NBS Jour. Res. 47 (1952), 45–61.Google Scholar
  25. 25.
    Hestenes, M. R., and E. Stiefel: Methods of conjugate gradients for solving linear systems. NBS Jour. Res. 49 (1952), 409–436.Google Scholar
  26. 26.
    Householder, A. S.: The Theory of matrices in numerical analysis. Blaisdell, New York (1964).Google Scholar
  27. 27.
    Kahan, W.: Relaxation methods for an eigenproblem. Tech. Rep. CS-44, Comp. Sci. Dept. Stanford University (1966).Google Scholar
  28. 28.
    Kammerer, W. J., and M. Z. Nashed: On the convergence of the conjugate gradient method for singular linear operator equation. SIAM J. Num. An. 9 (1972), 165–181.CrossRefGoogle Scholar
  29. 29.
    Kaniel, S.: Estimates for some computational techniques in linear algebra. Math. Comp. 20 (1966), 369–378.CrossRefGoogle Scholar
  30. 30.
    Karush, W.: An iterative method for finding characteristic vectors of a symmetric matrix. Pacific J. Math. 1 (1951), 233–248.Google Scholar
  31. 31.
    Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. NBS J. Res. 45 (1950), 255–282.Google Scholar
  32. 32.
    Miele, A., and J. W. Cantrell: Memory gradient method for the minimization of functions. Symp. Optimization, Lecture Notes Math. 132, Springer-Verlag, BerlinHeidelberg-New York (1970), 252–263.Google Scholar
  33. 33.
    Nesbet, R. K.: Algorithm for diagonalization of large matrices. J. Chem. Physics 43 (1965), 311–312.CrossRefGoogle Scholar
  34. 34.
    Paige, C. C.: Practical use of the symmetric Lanczos process with re-orthogonalization. BIT 10 (1970), 183–195.CrossRefGoogle Scholar
  35. 35.
    Paige, C. C.: The computation of eigenvalues and eigenvectors of very large sparse matrices. Diss. London Univ. Institute of Computer Science (1971).Google Scholar
  36. 36.
    Paige, C. C.: Eigenvalues of perturbed hermitian matrices. (to appear).Google Scholar
  37. 37.
    Polak, E. et G. Ribiére: Note sur la convergence de méthodes de diréctions con-jugées. Rev. Franc. Inf. Rech. Oper. 16-R1 (1969), 35–43.Google Scholar
  38. 38.
    Polyak, B. T.: The conjugate gradient method in extremal problems. Zh.VychisL Mat. mat. Fiz. 9 (1969), 807–821. Trnasl. USSR Comp. Mat. Mat. Phys. 7, 4, 94–112.Google Scholar
  39. 39.
    Reid, J. K.: On the method of conjugate gradients for the solution of large sparse systems of linear equations. In J. K. Reid (Ed.) Large Sparse Sets of Linear Equations. Acad. Press, London (1971), 231–254.Google Scholar
  40. 40.
    Ruhe, A, and T. Wiberg: The method of conjugate gradients used in inverse iteration. BIT 12 (1972).Google Scholar
  41. 41.
    Ruhe, A., and T. Wiberg: Computing the dominant eigenvalue and the corresponding eigenvector by a Lanczos-c-g iteration. (to Appear in BIT).Google Scholar
  42. 42.
    Rutishauser, H.: Computational aspects of F. L. Bauer’s simultaneous iteration method. Num. Math. 13 (1969), 4–13.CrossRefGoogle Scholar
  43. 43.
    Rutishauser, H.: Simultaneous iteration method for symmetric matrices. Handbook of Automatic Computation 2, Linear Algebra, Springer-Verlag (1971), 284302.Google Scholar
  44. 44.
    Shavitt, I.: Modification of Nesbet’s algorithm for the iterative evaluation of eigenvalues and eigenvectors of large matrices. Jour. Comp. Phys. 6 (1970), 124–130.CrossRefGoogle Scholar
  45. 45.
    Weaver, W., and D. M. Yoshida: The eigenvalue problem for banded matrices. Computers & Structures 1 (1971), 651–664.CrossRefGoogle Scholar
  46. 46.
    Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965).Google Scholar
  47. 47.
    Wilkinson, J. H., and C. Reinsch (Eds): Handbook for Autom. Computation Vol II, Linear Algebra. Springer-Verlag, Berlin-Heidelberg-New York (1971).Google Scholar
  48. 48.
    Zingmark, S.: Beräkning av flera egenvärden med iterativ Lanczosalgoritm. (in Swedish) Tech. Rep. UMINF-25.72.Google Scholar

Copyright information

© Springer Basel AG 1974

Authors and Affiliations

  • A. Ruhe
    • 1
  1. 1.UmeaSchweden

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