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Papers on Krylov Subspace Methods for the Eigenproblem

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G.W. Stewart

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References

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

    MathSciNet  MATH  Google Scholar 

  2. Z. Bai and J. W. Demmel, On swapping diagonal blocks in real Schur form, Linear Algebra Appl., 186 (1993), pp. 73–95.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. R. Fokkema, G. L. G. Sleijpen, and H. A. van der Vorst, Jacobi – Davidson style QR and QZ algorithms for the reduction of matrix pencils, SIAM J. Sci. Comput., 20 (1998), pp. 94–125.

    Article  MathSciNet  Google Scholar 

  4. G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed., Johns Hopkins University Press, Baltimore, MD, 1989.

    MATH  Google Scholar 

  5. Z. Jia, Refined iterative algorithm based on Arnoldi ’ s process for large unsymmetric eigenproblems, Linear Algebra Appl., 259 (1997), pp. 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards, 45 (1950), pp. 255–282.

    Article  MathSciNet  Google Scholar 

  7. R. B. Lehoucq, private communication.

    Google Scholar 

  8. R. B. Lehoucq and D. C. Sorensen, Deflation techniques for an implicitly restarted Arnoldi iteration, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 789–821.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users ’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, Philadelphia, 1998.

    Book  Google Scholar 

  10. R. B. Morgan, Computing interior eigenvalues of large matrices, Linear Algebra Appl., 154–156 (1991), pp. 289–309.

    Article  Google Scholar 

  11. R. B. Morgan, On restarting the Arnoldi method for large nonsymmetric eigenvalue problems, Math. Comp., 65 (1996), pp. 1213–1230.

    Article  MathSciNet  MATH  Google Scholar 

  12. R. B. Morgan, GMRES with Deflated Restarting, Department of Mathematics, Baylor University, Waco, TX, 1999.

    Google Scholar 

  13. J. A. Scott, An Arnoldi code for computing selected eigenvalues of sparse, real, unsymmetric matrices, ACM Trans. Math. Software, 21 (1995), pp. 432–475.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357–385.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Stathopoulos, Y. Saad, and K. Wu, Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods, SIAM J. Sci. Comput., 19 (1998), pp. 227–245.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. W. Stewart, Matrix Algorithms I: Basic Decompositions, SIAM, Philadelphia, 1998.

    Book  MATH  Google Scholar 

  17. D. S. Watkins, Forward stability and transmission of shifts in the QR algorithm, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 469–487.

    Article  MathSciNet  MATH  Google Scholar 

  18. K. Wu and H. Simon, Thick-restart Lanczos method for large symmetric eigenvalue problems, SIAM J. Matrix Anal. Appl., 22 (2000), pp. 602–616.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Genseberger and G. L. G. Sleijpen, Alternative correction equations in the Jacobi-Davidson method, Numer. Linear Algebra Appl., 6 (1999), pp. 235–253.

    Article  MathSciNet  MATH  Google Scholar 

  20. Z. Jia and G. W. Stewart, An analysis of the Rayleigh – Ritz method for approximating eigenspaces, Math. Comp., 70 (2001), pp. 637–647.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. B. Morgan, Computing interior eigenvalues of large matrices, Linear Algebra Appl., 154–156 (1991), pp. 289–309.

    Article  Google Scholar 

  22. G. W. Stewart, A Krylou – Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 601–614.

    Article  MathSciNet  MATH  Google Scholar 

  23. G. W. Stewart, Matrix Algorithms Volume II: Eigensystems, SIAM, Philadelphia, 2001.

    Book  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM, Philadelphia, PA, 1997.

    Book  MATH  Google Scholar 

  26. R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.

    Book  MATH  Google Scholar 

  27. Y. Saad, Numerical Methods for Large Eigenvalue Problems: Theory and Algorithms, Wiley, New York, 1992.

    MATH  Google Scholar 

  28. G.W. Stewart, A Krylov–Schur algorithm for large eigenproblems. Technical Report TR-4127, Department of Computer Science, University of Maryland, College Park, 2000, SIAM J. Matrix Anal. Appl. (to appear).

    Google Scholar 

  29. G.W. Stewart, Matrix Algorithms II: Eigensystems. SIAM, Philadelphia, PA, 2001 (to appear). Pre-print available at http://www.cs.umd.edu/ ∼stewart.

  30. G.W. Stewart, J.-G. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990.

    MATH  Google Scholar 

  31. J. Grcar, Analyses of the Lanczos Algorithm and of the Approximation Problem in Richardson ’ s Method, Ph.D. thesis, University of Illinois at Urbana–Champaign, 1981.

    Google Scholar 

  32. Z. Jia and G. W. Stewart, An analysis of the Rayleigh – Ritz method for approximating eigenspaces, Math. Comp., 70 (2001), pp. 637–647.

    Article  MathSciNet  MATH  Google Scholar 

  33. Y. Saad, Numerical Methods for Large Eigenvalue Problems: Theory and Algorithms, John Wiley, New York, 1992.

    MATH  Google Scholar 

  34. H. D. Simon, Analysis of the symmetric Lanczos algorithm with reorthogonalization methods, Linear Algebra Appl., 61 (1984), pp. 101–132.

    Article  MathSciNet  MATH  Google Scholar 

  35. H. D. Simon, The Lanczos algorithm with partial reorthogonalization, Math. Comp., 42 (1984), pp. 115–142.

    Article  MathSciNet  MATH  Google Scholar 

  36. G. W. Stewart, A Krylou – Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 601–614.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Tufts University, Medford, MA, 02155, USA

    Misha E. Kilmer

  2. Computer Science Department and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, 20742, USA

    Dianne P. O’Leary

Authors
  1. Misha E. Kilmer
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  2. Dianne P. O’Leary
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Correspondence to Misha E. Kilmer .

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Editors and Affiliations

  1. Department of Mathematics, Tufts University, Medford, 02155, MA, USA

    Misha E. Kilmer

  2. Computer Science Department and Institute for Advanced Computer Studies, University of Maryland, College Park, 20742, MD, USA

    Dianne P. O’Leary

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Kilmer, M.E., O’Leary, D.P. (2010). Papers on Krylov Subspace Methods for the Eigenproblem. In: Kilmer, M.E., O’Leary, D.P. (eds) G.W. Stewart. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4968-5_18

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