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A key to choose subspace size in implicitly restarted Arnoldi method

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Abstract

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.

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References

  1. Arnoldi, W.E.: The Principle of Minimized Iteration in the Solution of Matrix Eigenvalue Problems. Quart. J. Appl. Math. 9, 17–29 (1951)

    MathSciNet  MATH  Google Scholar 

  2. Bai, Z., Day, D., Demmel, J., Dongarra, J.J.: A Test Matrix Collection for Non-Hermitian Eigenvalue Problems, http://math.nist.gov/MatrixMarket

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chatelain, F.: Eigenvalues of matrices. Wiley (1993)

  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)

  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)

    MathSciNet  Google Scholar 

  7. Duff, I.S., Scott, J.A.: Computing selected eigenvalues of large sparse unsymmetric matrices using subspace iteration 19, 137–159 (1993)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Embree, M.: TheTortoise and the Hare Restart GMRES. SIAM Rev. 45(2), 259–266 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lehoucq, R.B.: Analysis and Implementation of an Implicitly Restarted Iteration, PhD thesis, Rice University, Houston, Texas (1995)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  14. Morgan, R.B.: On Restarting the Arnoldi Method for Large Nonsymmetric Eigenvalue Problem. Math. Comp. 65, 1213–1230 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Parlett, B.N.: The Symmetric Eigenvalue Problem. Prentice-Hall, Inc. (1998)

  17. Saad, Y.: Numerical Methods for Large Eigenvalue Problems. Manchester University Press (1993)

  18. Saad, Y.: Chebyshev Acceleration Techniques for Solving Nonsymmetric Eigenvalue Problems. Math. Com. 42, 567–588 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  19. Saad, Y.: Variations on Arnoldi’s Method for Computing Eigenelements of Large Unsymmetric Matrices. Linear Algebra Applications 34, 269–295 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Saad, Y.: Numerical Solution of Large Nonsymetric Eigen Problem. Comput. Phys. Commun. 53, 71–90 (1989). MR 90f:65064

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sorensen, D.C.: Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations, (invited paper). Kluwer (1995)

  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. 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)

    Article  MathSciNet  Google Scholar 

  25. Lehoucq, R.B.: Implicitly Restarted Arnoldi Methods and Subspace Iteration. SIAM J. Matrix Anal. Appl. 23, 551–562 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. http://snap.stanford.edu/data/

  27. https://forge.prism.uvsq.fr/projects/miram

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Authors and Affiliations

  1. Faculty of Mathematics, Yazd University, Yazd, Iran

    S. A. Shahzadeh Fazeli

  2. Maison de la simulation and PRiSM Laboratory, University of Versailles, 45 avenue des Etats-Unis, 78035, Versailles, France

    Nahid Emad & Zifan Liu

Authors
  1. S. A. Shahzadeh Fazeli
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  2. Nahid Emad
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  3. Zifan Liu
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Correspondence to Nahid Emad.

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Shahzadeh Fazeli, S.A., Emad, N. & Liu, Z. A key to choose subspace size in implicitly restarted Arnoldi method. Numer Algor 70, 407–426 (2015). https://doi.org/10.1007/s11075-014-9954-5

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  • DOI: https://doi.org/10.1007/s11075-014-9954-5

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