Numerical Algorithms

, Volume 70, Issue 2, pp 407–426 | Cite as

A key to choose subspace size in implicitly restarted Arnoldi method

  • S. A. Shahzadeh Fazeli
  • Nahid EmadEmail author
  • Zifan Liu
Original Paper


The implicitly restarted Arnoldi method (IRAM) computes some eigenpairs of large sparse non Hermitian matrices. However, the size of the subspace in this method is chosen empirically. A poor choice of this size could lead to the non-convergence of the method. In this paper we propose a technique to improve the choice of the size of subspace. This approach, called multiple implicitly restarted Arnoldi method with nested subspaces (MIRAMns) is based on the projection of the problem on several nested subspaces instead of a single one. Thus, it takes advantage of several different sized subspaces. MIRAMns updates the restarting vector of an IRAM by taking the eigen-information of interest obtained in all subspaces into account. With almost the same complexity as IRAM, according to our experiments, MIRAMns improves the convergence of IRAM.


Large eigenproblems Krylov subspace size Arnoldi method Implicit restarting 


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  1. 1.
    Arnoldi, W.E.: The Principle of Minimized Iteration in the Solution of Matrix Eigenvalue Problems. Quart. J. Appl. Math. 9, 17–29 (1951)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bai, Z., Day, D., Demmel, J., Dongarra, J.J.: A Test Matrix Collection for Non-Hermitian Eigenvalue Problems,
  3. 3.
    Baker, A.H., Jessup, E.R., KoleV, Tz. V.: A Simple Strategy for Varying the Restart Parameter in GMRES(M). J. Comput. Appl. Math. 230(2), 751–761 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chatelain, F.: Eigenvalues of matrices. Wiley (1993)Google Scholar
  5. 5.
    Daniel, J.W., Gragg, W.B., Kaufman, L., Stewart, G.W.: Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization, Math. Comp (1976)Google Scholar
  6. 6.
    Dookhitram, K., Boojhawon, R., Bhuruth, M.: A new method for accelerating Arnoldi algorithms for large scale Eigenproblems. Math. Comput. Simul. 2, 387–401 (2010)MathSciNetGoogle Scholar
  7. 7.
    Duff, I.S., Scott, J.A.: Computing selected eigenvalues of large sparse unsymmetric matrices using subspace iteration 19, 137–159 (1993)Google Scholar
  8. 8.
    Emad, N., Petiton, S., Edjlali, G.: Multiple Explicitly Restarted Arnoldi Method for Solving Large Eigenproblems. SIAM J. Sci. Comput. (SJSC) 27(1), 253–277 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Embree, M.: TheTortoise and the Hare Restart GMRES. SIAM Rev. 45(2), 259–266 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lehoucq, R.B.: Analysis and Implementation of an Implicitly Restarted Iteration, PhD thesis, Rice University, Houston, Texas (1995)Google Scholar
  11. 11.
    Lehoucq, R.B., Sorensen, D.C.: Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Analysis and Applications 17(4), 789–821 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK User’s Guide: Solution of Large Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods, SIAM (1998)Google Scholar
  13. 13.
    Maschhoff, K.J., Sorensen, D.C. In: Wasniewski, J., Dongarra, J., Madsen, K., D. Olesen (eds.) : Applied Parallel Computing in Industrial Problems and Optimization, volume 1184 of Lecture Notes in Computer Science. Springer-Verlag, Berlin (1996)Google Scholar
  14. 14.
    Morgan, R.B.: On Restarting the Arnoldi Method for Large Nonsymmetric Eigenvalue Problem. Math. Comp. 65, 1213–1230 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Moriya, K., Nodera, T.: The DEFLATED-GMRES(m, k) method with switching the restart frequency dynamically. Numer. Linear Algebra Appl. 7, 569–584 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Parlett, B.N.: The Symmetric Eigenvalue Problem. Prentice-Hall, Inc. (1998)Google Scholar
  17. 17.
    Saad, Y.: Numerical Methods for Large Eigenvalue Problems. Manchester University Press (1993)Google Scholar
  18. 18.
    Saad, Y.: Chebyshev Acceleration Techniques for Solving Nonsymmetric Eigenvalue Problems. Math. Com. 42, 567–588 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Saad, Y.: Variations on Arnoldi’s Method for Computing Eigenelements of Large Unsymmetric Matrices. Linear Algebra Applications 34, 269–295 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Saad, Y.: Numerical Solution of Large Nonsymetric Eigen Problem. Comput. Phys. Commun. 53, 71–90 (1989). MR 90f:65064MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sorensen, D.C.: Implicit Application of Polynomial Filters in a k-step Arnoldi Method. SIAM Journal on Matrix Analysis and Applications 13, 357–385 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sorensen, D.C.: Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations, (invited paper). Kluwer (1995)Google Scholar
  23. 23.
    Sorensen, D.C.: Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations. In: Keyes, D. E., Sameh, A., Venkatakrishnan, V. (eds.) Parallel Numerical Algorithms, pp. 119–166. Kluwer, Dordrecht (1997)Google Scholar
  24. 24.
    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 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lehoucq, R.B.: Implicitly Restarted Arnoldi Methods and Subspace Iteration. SIAM J. Matrix Anal. Appl. 23, 551–562 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
  27. 27.

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • S. A. Shahzadeh Fazeli
    • 1
  • Nahid Emad
    • 2
    Email author
  • Zifan Liu
    • 2
  1. 1.Faculty of MathematicsYazd UniversityYazdIran
  2. 2.Maison de la simulation and PRiSM LaboratoryUniversity of VersaillesVersaillesFrance

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