Large Symmetric Eigenvalue Problems

  • Efstratios Gallopoulos
  • Bernard Philippe
  • Ahmed H. Sameh
Part of the Scientific Computation book series (SCIENTCOMP)


In this chapter we consider the following problems.


  1. 1.
    Parlett, B.N.: The Symmetric Eigenvalue Problem. Prentice Hall, Englewood Cliffs (1980)MATHGoogle Scholar
  2. 2.
    Bauer, F.: Das verfahren der treppeniteration und verwandte verfahren zur losung algebraischer eigenwertprobleme. ZAMP 8, 214–235 (1957)CrossRefMATHGoogle Scholar
  3. 3.
    Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford University Press, New York (1965)Google Scholar
  4. 4.
    Rutishauser, H.: Simultaneous iteration method for symmetric matrices. Numer. Math. 16, 205–223 (1970)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Stewart, G.W.: Simultaneous iterations for computing invariant subspaces of non-Hermitian matrices. Numer. Math. 25, 123–136 (1976)CrossRefMATHGoogle Scholar
  6. 6.
    Stewart, W.J., Jennings, A.: Algorithm 570: LOPSI: a simultaneous iteration method for real matrices [F2]. ACM Trans. Math. Softw. 7(2), 230–232 (1981). doi: 10.1145/355945.355952 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Saad, Y.: Numerical Methods for Large Eigenvalue Problems. Halstead Press, New York (1992)MATHGoogle Scholar
  8. 8.
    Sameh, H., Lermit, J., Noh, K.: On the intermediate eigenvalues of symmetric sparse matrices. BIT 185–191 (1975)Google Scholar
  9. 9.
    Bunch, J., Kaufman, K.: Some stable methods for calculating inertia and solving symmetric linear systems. Math. Comput. 31, 162–179 (1977)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Golub, G., Van Loan, C.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)MATHGoogle Scholar
  11. 11.
    Duff, I., Gould, N.I.M., Reid, J.K., Scott, J.A., Turner, K.: The factorization of sparse symmetric indefinite matrices. IMA J. Numer. Anal. 11, 181–204 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Duff, I.: MA57-a code for the solution of sparse symmetric definite and indefinite systems. ACM TOMS 118–144 (2004)Google Scholar
  13. 13.
    Kalamboukis, T.: Tridiagonalization of band symmetric matrices for vector computers. Comput. Math. Appl. 19, 29–34 (1990)CrossRefGoogle Scholar
  14. 14.
    Lang, B.: A parallel algorithm for reducing symmetric banded matrices to tridiagonal form. SIAM J. Sci. Comput. 14(6), 1320–1338 (1993). doi: 10.1137/0914078 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Philippe, B., Vital, B.: Parallel implementations for solving generalized eigenvalue problems with symmetric sparse matrices. Appl. Numer. Math. 12, 391–402 (1993)CrossRefMATHGoogle Scholar
  16. 16.
    Carey, C., Chen, H.C., Golub, G., Sameh, A.: A new approach for solving symmetric eigenvalue problems. Comput. Sys. Eng. 3(6), 671–679 (1992)CrossRefGoogle Scholar
  17. 17.
    Golub, G., Underwood, R.: The block Lanczos method for computing eigenvalues. In: Rice, J. (ed.) Mathematical Software III, pp. 364–377. Academic Press, New York (1977)Google Scholar
  18. 18.
    Underwood, R.: An iterative block Lanczos method for the solution of large sparse symmetric eigenproblems. Technical Report STAN-CS-75-496, Computer Science, Stanford University, Stanford (1975)Google Scholar
  19. 19.
    Kaniel, S.: Estimates for some computational techniques in linear algebra. Math. Comput. 20, 369–378 (1966)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Paige, C.: The computation of eigenvalues and eigenvectors of very large sparse matrices. Ph.D. thesis, London University, London (1971)Google Scholar
  21. 21.
    Meurant, G.: The Lanczos and Conjugate Gradient Algorithms: from Theory to Finite Precision Computations (Software, Environments, and Tools). SIAM, Philadelphia (2006)CrossRefMATHGoogle Scholar
  22. 22.
    Paige, C.C.: Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem. Linear Algebra Appl. 34, 235–258 (1980)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Cullum, J.K., Willoughby, R.A.: Lanczos Algorithms for Large Symmetric Eigenvalue Computations. SIAM, Philadelphia (2002)CrossRefMATHGoogle Scholar
  24. 24.
    Lehoucq, R., Sorensen, D.: Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Anal. Appl. 17, 789–821 (1996)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lehoucq, R., Sorensen, D., Yang, C.: ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia (1998)CrossRefMATHGoogle Scholar
  26. 26.
    Calvetti, D., Reichel, L., Sorensen, D.C.: An implicitly restarted Lanczos method for large symmetric eigenvalue problems. Electron. Trans. Numer. Anal. 2, 1–21 (1994)MathSciNetMATHGoogle Scholar
  27. 27.
    Sorensen, D.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Lewis, J.G.: Algorithms for sparse matrix eigenvalue problems. Technical Report STAN-CS-77-595, Department of Computer Science, Stanford University, Palo Alto (1977)Google Scholar
  29. 29.
    Ruhe, A.: Implementation aspects of band Lanczos algorithms for computation of eigenvalues of large sparse symmetric matrices. Math. Comput. 33, 680–687 (1979)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Scott, D.: Block lanczos software for symmetric eigenvalue problems. Technical Report ORNL/CSD-48, Oak Ridge National Laboratory, Oak Ridge (1979)Google Scholar
  31. 31.
    Baglama, J., Calvetti, D., Reichel, L.: IRBL: an implicitly restarted block Lanczos method for large-scale Hermitian eigenproblems. SIAM J. Sci. Comput. 24(5), 1650–1677 (2003)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Chen, H.C., Sameh, A.: Numerical linear algebra algorithms on the cedar system. In: Noor, A. (ed.) Parallel Computations and Their Impact on Mechanics, Applied Mechanics Division, vol. 86, pp. 101–125. American Society of Mechanical Engineers (1987)Google Scholar
  33. 33.
    Chen, H.C.: The sas domain decomposition method. Ph.D. thesis, University of Illinois at Urbana-Champaign (1988)Google Scholar
  34. 34.
    Davidson, E.: The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 17, 817–825 (1975)CrossRefGoogle Scholar
  35. 35.
    Morgan, R., Scott, D.: Generalizations of Davidson’s method for computing eigenvalues of sparse symmetric matrices. SIAM J. Sci. Stat. Comput. 7, 817–825 (1986)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Crouzeix, M., Philippe, B., Sadkane, M.: The Davidson method. SIAM J. Sci. Comput. 15, 62–76 (1994)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Sameh, A.H., Wisniewski, J.A.: A trace minimization algorithm for the generalized eigenvalue problem. SIAM J. Numer. Anal. 19(6), 1243–1259 (1982)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Sameh, A., Tong, Z.: The trace minimization method for the symmetric generalized eigenvalue problem. J. Comput. Appl. Math. 123, 155–170 (2000)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Sleijpen, G., van der Vorst, H.: A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 401–425 (1996)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Simoncini, V., Eldén, L.: Inexact Rayleigh quotient-type methods for eigenvalue computations. BIT Numer. Math. 42(1), 159–182 (2002). doi: 10.1023/A:1021930421106 CrossRefMATHGoogle Scholar
  41. 41.
    Bathe, K., Wilson, E.: Large eigenvalue problems in dynamic analysis. ASCE J. Eng. Mech. Div. 98, 1471–1485 (1972)Google Scholar
  42. 42.
    Bathe, K., Wilson, E.: Solution methods for eigenvalue problems in structural mechanics. Int. J. Numer. Methods Eng. 6, 213–226 (1973)CrossRefMATHGoogle Scholar
  43. 43.
    Grimm, R., Greene, J., Johnson, J.: Computation of the magnetohydrodynamic spectrum in axissymmetric toroidal confinement systems. Method Comput. Phys. 16 (1976)Google Scholar
  44. 44.
    Gruber, R.: Finite hybrid elements to compute the ideal magnetohydrodynamic spectrum of an axisymmetric plasma. J. Comput. Phys. 26, 379–389 (1978)CrossRefMATHGoogle Scholar
  45. 45.
    Stewart, G.: A bibliographical tour of the large, sparse generalized eigenvalue problems. In: Banch, J., Rose, D. (eds.) Sparse Matrix Computations, pp. 113–130. Academic Press, New York (1976)Google Scholar
  46. 46.
    van der Vorst, H., Golub, G.: One hundred and fifty years old and still alive: eigenproblems. In: Duff, I., Watson, G. (eds.) The State of the Art in Numerical Analysis, pp. 93–119. Clarendon Press, Oxford (1997)Google Scholar
  47. 47.
    Rutishauser, H.: Computational aspects of f. l. bauer’s simultaneous iteration method. Numerische Mathematik 13(1), 4–13 (1969). doi: 10.1007/BF02165269 MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Clint, M., Jennings, A.: The evaluation of eigenvalues and eigenvectors of real symmetric matrices by simultaneous iteration. Computers 13, 76–80 (1970)MathSciNetMATHGoogle Scholar
  49. 49.
    Levin, A.: On a method for the solution of a partial eigenvalue problem. J. Comput. Math. Math. Phys. 5, 206–212 (1965)CrossRefGoogle Scholar
  50. 50.
    Stewart, G.: Accelerating the orthogonal iteration for the eigenvalues of a Hermitian matrix. Numer. Math. 13, 362–376 (1969)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159, 119–128 (2003)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Tang, P., Polizzi, E.: FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection. SIAM J. Matrix Anal. Appl. 35(2), 354–390 (2014)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl. Bur. Stand. 45, 225–280 (1950)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Fokkema, D.R., Sleijpen, G.A.G., van der Vorst, H.A.: Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20(1), 94–125 (1998)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Sleijpen, G., Booten, A., Fokkema, D., van der Vorst, H.: Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36, 595–633 (1996)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Cullum, J., Willoughby, R.: Lanczos and the computation in specified intervals of the spectrum of large, sparse, real symmetric matrices. In: Duff, I., Stewart, G. (eds.) Proceedings of the Sparse Matrix 1978. SIAM (1979)Google Scholar
  57. 57.
    Parlett, B., Scott, D.: The Lanczos algorithm with selective orthogonalization. Math. Comput. 33, 217–238 (1979)MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Simon, H.: The Lanczos algorithm with partial reorthogonalization. Math. Comput. 42, 115–142 (1984)CrossRefMATHGoogle Scholar
  59. 59.
    Cullum, J., Willoughby, R.: Computing eigenvalues of very large symmetric matrices—an implementation of a Lanczos algorithm with no reorthogonalization. J. Comput. Phys. 44, 329–358 (1984)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Ericsson, T., Ruhe, A.: The spectral transformation Lanczos method for the solution of large sparse generalized symmetric eigenvalue problems. Math. Comput. 35, 1251–1268 (1980)MathSciNetMATHGoogle Scholar
  61. 61.
    Grimes, R., Lewis, J., Simon, H.: A shifted block Lanczos algorithm for solving sparse symmetric generalized Eigenproblems. SIAM J. Matrix Anal. Appl. 15, 228–272 (1994)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Kalamboukis, T.: A Lanczos-type algorithm for the generalized eigenvalue problem \(ax = \lambda b x\). J. Comput. Phys. 53, 82–89 (1984)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Liu, B.: The simultaneous expansion for the solution of several of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. In: Moler, C., Shavitt, I. (eds.) Numerical Algorithms in Chemistry: Algebraic Method, pp. 49–53. University of California, Lawrence Berkeley Laboratory (1978)Google Scholar
  64. 64.
    Stathopoulos, A., Saad, Y., Fischer, C.: Robust preconditioning of large, sparse, symmetric eigenvalue problems. J. Comput. Appl. Math. 197–215 (1995)Google Scholar
  65. 65.
    Wu, K.: Preconditioning techniques for large eigenvalue problems. Ph.D. thesis, University of Minnesota (1997)Google Scholar
  66. 66.
    Jacobi, C.: Über ein leichtes verfahren die in der theorie der säculärstörungen vorkom menden gleichungen numerisch aufzulösen. Crelle’s J. für reine und angewandte Mathematik 30, 51–94 (1846)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Beckenbach, E., Bellman, R.: Inequalities. Springer, New York (1965)CrossRefGoogle Scholar
  68. 68.
    Kantorovic̆, L.: Functional analysis and applied mathematics (Russian). Uspekhi Mat. Nauk. 3, 9–185 (1948)Google Scholar
  69. 69.
    Newman, M.: Kantorovich’s inequality. J. Res. Natl. Bur. Stand. B. Math. Math. Phys. 64B(1), 33–34 (1959). CrossRefGoogle Scholar
  70. 70.
    Benzi, M., Golub, G., Liesen, J.: Numerical solution of Saddle-point problems. Acta Numerica pp. 1–137 (2005)Google Scholar
  71. 71.
    Elman, H., Silvester, D., Wathen, A.: Performance and analysis of Saddle-Point preconditioners for the discrete steady-state Navier-Stokes equations. Numer. Math. 90, 641–664 (2002)MathSciNetCrossRefMATHGoogle Scholar
  72. 72.
    Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12(4), 617–629 (1975)MathSciNetCrossRefMATHGoogle Scholar
  73. 73.
    Daniel, J., Gragg, W., Kaufman, L., Stewart, G.: Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comput. 136, 772–795 (1976)MathSciNetGoogle Scholar
  74. 74.
    Sun, J.G.: Condition number and backward error for the generalized singular value decomposition. SIAM J. Matrix Anal. Appl. 22(2), 323–341 (2000)MathSciNetCrossRefMATHGoogle Scholar
  75. 75.
    Stewart, G.G., Sun, J.: Matrix Perturbation Theory. Academic Press, Boston (1990)MATHGoogle Scholar
  76. 76.
    Demmel, J., Gu, M., Eisenstat, S.I., Slapničar, K., Veselić, Z.D.: Computing the singular value decomposition with high accuracy. Linear Algebra Appl. 299(1–3), 21–80 (1999)MathSciNetCrossRefMATHGoogle Scholar
  77. 77.
    Sun, J.: A note on simple non-zero singular values. J. Comput. Math. 6(3), 258–266 (1988)MathSciNetMATHGoogle Scholar
  78. 78.
    Berry, M., Sameh, A.: An overview of parallel algorithms for the singular value and symmetric eigenvalue problems. J. Comput. Appl. Math. 27, 191–213 (1989)MathSciNetCrossRefMATHGoogle Scholar
  79. 79.
    Dongarra, J., Sorensen, D.C.: A fully parallel algorithm for the symmetric eigenvalue problem. SIAM J. Sci. Stat. Comput. 8(2), s139–s154 (1987)MathSciNetCrossRefMATHGoogle Scholar
  80. 80.
    Golub, G., Reinsch, C.: Singular Value Decomposition and Least Squares Solutions. Springer (1971)Google Scholar
  81. 81.
    Golub, G., Luk, F., Overton, M.: A block Lanczos method for computing the singular values and corresponding singular vectors of a matrix. ACM Trans. Math. Softw. 7, 149–169 (1981)MathSciNetCrossRefMATHGoogle Scholar
  82. 82.
    Berry, M.: Large scale singular value decomposition. Int. J. Supercomput. Appl. 6, 13–49 (1992)Google Scholar
  83. 83.
    Wilkinson, J.: Inverse Iteration in Theory and in Practice. Academic Press (1972)Google Scholar
  84. 84.
    Philippe, B., Sadkane, M.: Computation of the fundamental singular subspace of a large matrix. Linear Algebra Appl. 257, 77–104 (1997)MathSciNetCrossRefMATHGoogle Scholar
  85. 85.
    Hochstenbach, M.: A Jacobi-Davidson type SVD method. SIAM J. Sci. Comput. 23(2), 606–628 (2001)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Efstratios Gallopoulos
    • 1
  • Bernard Philippe
    • 2
  • Ahmed H. Sameh
    • 3
  1. 1.Computer Engineering and Informatics DepartmentUniversity of PatrasPatrasGreece
  2. 2.Campus de BeaulieuINRIA/IRISARennes CedexFrance
  3. 3.Department of Computer SciencePurdue UniversityWest LafayetteUSA

Personalised recommendations