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A refined Arnoldi type method for large scale eigenvalue problems

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Abstract

We present a refined Arnoldi-type method for extracting partial eigenpairs of large matrices. The approximate eigenvalues are the Ritz values of (A−τ I)−1 with respect to a shifted Krylov subspace. The approximate eigenvectors are derived by satisfying certain optimal properties, and they can be computed cheaply by a small sized singular value problem. Theoretical analysis show that the approximate eigenpairs computed by the new method converges as the approximate subspace expands. Finally, numerical results are reported to show the efficiency of the new method.

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References

  1. Arnoldi W.E.: The principle of minimized iteration in the solution of the matrix eigenvalue problem. Quart. Appl. Math. 9, 17–29 (1951)

    MathSciNet  MATH  Google Scholar 

  2. Bai, Z., Barret, R., Day, D., Demmel, J., Dongarra, J.: Test matrix collection for non-Hermitian eigenvalue problems, Technical Report CS-97-355, University of Tennessee, 1997. Available at http://math.nist.gov/MatrixMarket

  3. Chen G., Jia Z.: An analogue of the results of Saad and Stewart for harmonic Ritz vectors. J. Comput. Appl. Math. 167, 493–498 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen G., Jia Z.: A refined harmonic Rayleigh–Ritz procedure and an explicitly restarted refined harmonic Arnoldi algorithm. Math. Comput. Model. 41, 615–627 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Duff I.S., Grimes R.G., Lewis J.G.: Sparse matrix test problems. ACM Trans. Math. Soft. 15, 1–14 (1989)

    Article  MATH  Google Scholar 

  6. Eiermann M., Ernst O.G., Schneider O.: Analysis of acceleration stategies for restarted minimal residual methods. J. Comput. Appl. Math. 123, 261–292 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jia Z.: Refined iterative algorithms based on Arnoldi’s process for large unsymmetric eigenpriblem. Linear Algebra Appl. 259, 1–23 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jia Z., Stewart G.W.: An analysis of the Rayleigh–Ritz method for approximating eigenspaces. Math. Comput. 70, 637–647 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Jia, Z., Stewart, G.W.: On the convergence of the Ritz values, Ritz vectors and refined Ritz vectors, Technical Report 99-08, Institute of Advanced Computer Studies, Technical Report 3896

  10. Mittelman H., Law C.C., Jankowski D.F., Neitzel G.P.: A large, sparse, and indefinite generalized eigenvalue problem from fluid mechanics. SIAM J. Sci. Comput. 13, 411–424 (1992)

    Article  Google Scholar 

  11. Morgan R.B.: Computing interioreigenvalues of large matrices. Linear Algebra Appl. 154(156), 289–309 (1991)

    Article  MathSciNet  Google Scholar 

  12. Morgan R.B., Zeng M.: Harmonic projection methods for large nonsymmetric eigenvalue problems. Numer. Linear Algebra Appl. 5, 33–55 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Niu Q., Lu L.-Z.: An invert-free Arnoldi method for computing interior eigenvalues of large matrices. Int. J. Comput. Math. 84, 477–490 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Paige C.C., Parlett B.N., van der Vorst H.A.: Approximate solutions and eigenvalue bounds from Krylov subspace. Numer. Linear. Algebra Appl. 2, 115–133 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Saad Y.: Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices. Linear Algebra Appl. 34, 269–295 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  16. Saad Y.: Numerical Methods for Large Eigenvalue Problems. Manchester University Press, UK (1992)

    MATH  Google Scholar 

  17. Scott D.S.: The advantages of the inverted operators in Rayleigh–Ritz approximations. SIAM J. Sci. Statist. Comput. 3, 68–75 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Scott J.A.: An Arnoldi Code for Computing Selected Eigenvalues of Sparse, Real, Unsymmetric Matrices. ACM Trans. Math. Softw. 214, 432–475 (1995)

    Article  Google Scholar 

  19. Sleijpen G.L.G., Vander Vorst H.A.: A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 401–425 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Steffen B.: Subspace methods for large sparse interior eigenvalue problems. Int. J. Differ Appl. 3, 339–351 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Stewart G.W.: Matrix Algorithms, vol. II. Eigensystems SIAM, Philadelphia (2001)

    Book  Google Scholar 

  22. Wu G.: A dynamic thick restarted semi-refined ABLE algorithm for computing a few selected eigentriplets of large nonsymmetric matrices. Linear. Algebra Appl. 416, 313–335 (2001)

    Article  Google Scholar 

  23. Jiang W., Wu G.: A thick-restarted block Arnoldi algorithm with modified Ritz vectors for large eigenproblems. Comput. Math. Appl. 60, 873–889 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Van der Vorst, H.A.: Computational methods for large eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds) Handbook of Numerical Analysis, vol. VIII, North-Holland, pp. 3–179. Elsevier, Amsterdam, (2002)

  25. Walker H.F., Zhou L.: A simpler GMRES. Numer. Linear Algebra Appl. 1, 571–581 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Xiang Wang.

Additional information

This work is supported by NNSF of China No. 11101204, supported by Young Scientists of Jiangxi province, NSF of Jiangxi, China No. 20114BAB201004, Science Funds of The Education Department of Jiangxi Province No.GJJ12011, and the Scientific Research Foundation of Graduate School of Nanchang University with No.YC2011-S007.

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Wang, X., Niu, Q. & Lu, Lz. A refined Arnoldi type method for large scale eigenvalue problems. Japan J. Indust. Appl. Math. 30, 129–143 (2013). https://doi.org/10.1007/s13160-012-0090-0

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  • DOI: https://doi.org/10.1007/s13160-012-0090-0

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