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Extracting partial canonical structure for large scale eigenvalue problems

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Abstract

We present methods for computing a nearby partial Jordan-Schur form of a given matrix and a nearby partial Weierstrass-Schur form of a matrix pencil. The focus is on the use and the interplay of the algorithmic building blocks – the implicitly restarted Arnoldi method with prescribed restarts for computing an invariant subspace associated with the dominant eigenvalue, the clustering method for grouping computed eigenvalues into numerically multiple eigenvalues and the staircase algorithm for computing the structure revealing form of the projected problem. For matrix pencils, we present generalizations of these methods. We introduce a new and more accurate clustering heuristic for both matrices and matrix pencils. Particular emphasis is placed on reliability of the partial Jordan-Schur and Weierstrass-Schur methods with respect to the choice of deflation parameters connecting the steps of the algorithm such that the errors are controlled. Finally, successful results from computational experiments conducted on problems with known canonical structure and varying ill-conditioning are presented.

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References

  1. Z. Bai and G.W. Stewart, Algorithm 776: SRRIT: A Fortran Subroutine to calculate the dominant invariant subspace of a nonsymmetric matrix, ACM Trans. Math. Software 23(4) (1997) 494-523.

    Article  MATH  MathSciNet  Google Scholar 

  2. T. Beelen and P. Van Dooren, An improved algorithm for the computation of Kronecker's canonical form of a singular pencil, Linear Algebra Appl. 105 (1988) 9-65.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Bennani and T. Braconnier, Stopping criteria for eigensolvers, Technical Report TR/PA/94/22, CERFACS (1994).

  4. T. Braconnier, V. Frayss´e and J.-C. Rioual, ARNCHEB users' guide: Solution of large non symmetric or non Hermitian eigenvalue problems by the Arnoldi-Tchebycheff method, Technical Report TR/PA/97/50, CERFACS (1997).

  5. F. Chatelin, Eigenvalues of Matrices (Wiley, New York, 1993).

    Google Scholar 

  6. J. Demmel and B. Kå gström, The generalized Schur decomposition of an arbitrary pencil A-λB: Robust software with error bounds and applications. Part I: Theory and algorithms, ACM Trans. Math. Software 19(2) (1993) 160-174.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Demmel and B. Kå gström, The generalized Schur decomposition of an arbitrary pencil A-λB: Robust software with error bounds and applications. Part II: Software and applications, ACM Trans. Math. Software 19(2) (1993) 175-201.

    Article  MATH  MathSciNet  Google Scholar 

  8. I.S. Duff and J.A. Scott, Computing selected eigenvalues of sparse unsymmetric matrices using subspace iteration, ACM Trans. Math. Software 19(2) (1993) 137-159.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Edelman, E. Elmroth and B. Kå gström, A geometric approach to perturbation theory of matrices and matrix pencils. Part I: Versal deformations, SIAM J. Matrix Anal. Appl. 18(3) (1997) 653-692.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Edelman, E. Elmroth and B. Kå gström, A geometric approach to perturbation theory of matrices and matrix pencils. Part II: A stratification-enhanced staircase algorithm, SIAM J. Matrix Anal. Appl. 20(3) (1999) 667-699.

    Article  MATH  MathSciNet  Google Scholar 

  11. E. Elmroth and B. Kå gström, The set of 2-by-3 matrix pencils-Kronecker structures and their transitions under perturbations, SIAM J. Matrix Anal. Appl. 17(1) (1996) 1-34.

    Article  MATH  MathSciNet  Google Scholar 

  12. G.L. Fokkema, G.L.G. Sleijpen and H.A. Van der Vorst, A Jacobi-Davidson style QR and QZ algorithm for partial reduction of matrix pencils, Technical Report 941, Department of Mathematics, University of Utrecht (1996).

  13. F.R. Gantmacher, The Theory of Matrices, Vol. II (Chelsea, New York, 1959).

    Google Scholar 

  14. G. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the computation of the Jordan canonical form, SIAM Review 18(4) (1976) 578-619.

    Article  MATH  MathSciNet  Google Scholar 

  15. G.H. Golub and C.F. Van Loan, Matrix Computations, 2nd ed. (Johns Hopkins Univ. Press, Baltimore, MD, 1989).

    Google Scholar 

  16. B. Hartley and T.O. Hawkes, Rings, Modules and Linear Algebra (Chapman & Hall, London, 1970).

    Google Scholar 

  17. B. Kå gström, How to compute the Jordan normal form-the choice between similarity transformations and methods using the chain relations, Technical Report UMINF-91.81, Department of Numerical Analysis, Institute of Information Processing, University of Umeå (1981).

  18. B. Kå gström, RGSVD-an algorithm for computing the Kronecker canonical form and reducing subspaces of singular matrix pencils A-λB, SIAM J. Sci. Statist. Comput. 7(1) (1986) 185-211.

    Article  MathSciNet  Google Scholar 

  19. B. Kå gström, A perturbation analysis of the generalized Sylvester equation (AR_LB,DR_LE) = (C,F), SIAM J. Matrix Anal. Appl. 15(4) (1994) 1045-1060.

    Article  MathSciNet  Google Scholar 

  20. B. Kå gström and P. Poromaa, Computing eigenspaces with specified eigenvalues of a regular matrix pair (A,B) and condition estimation: Theory, algorithms and software, Numer. Algorithms 12 (1996) 369-407.

    Article  MathSciNet  Google Scholar 

  21. B. Kå gström and A. Ruhe, An algorithm for the numerical computation of the Jordan normal form of a complex matrix, ACM Trans. Math. Software 6(3) (1980) 389-419.

    Google Scholar 

  22. B. Kå gström and A. Ruhe, ALGORITHM 560: An algorithm for the numerical computation of the Jordan normal form of a complex matrix [F2], ACM Trans. Math. Software 6(3) (1980) 437-443.

    Article  Google Scholar 

  23. W. Kahan, B.N. Parlett and E. Jiang, Residual bounds on approximate eigensystems of nonnormal matrices, SIAM J. Num. Anal. 19(3) (1982) 470-484.

    Article  MATH  MathSciNet  Google Scholar 

  24. T. Kato, Perturbation Theory for Linear Operators, 2nd ed. (Springer, Berlin, 1980).

    Google Scholar 

  25. V.N. Kublanovskaya, On a method of solving the complete eigenvalue problem for a degenerate matrix (in Russian), Zh. Vychisl. Mat. i Mat. Fiz. 6 (1966) 611-620. (USSR Comput. Math. Math. Phys. 6(4) (1968) 1-16.)

    MATH  Google Scholar 

  26. V. Kublanovskaya, AB-algorithm and its modifications for the spectral problem of linear pencils of matrices, Numer. Math. 43 (1984) 329-342.

    Article  MATH  MathSciNet  Google Scholar 

  27. R.B. Lehoucq and D.C. Sorensen, Deflation techniques for an implicitly restarted Arnoldi iteration, SIAM J. Matrix Anal. Appl. 17(2) (1996) 789-821.

    Article  MATH  MathSciNet  Google Scholar 

  28. 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, PA, 1998).

    Google Scholar 

  29. K. Meerbergen and D. Roose, The restarted Arnoldi method applied to iterative linear system solvers for the computation of rightmost eigenvalus, SIAM J. Matrix Anal. Appl. 18(1) (1997) 1-20.

    Article  MATH  MathSciNet  Google Scholar 

  30. A. Ruhe, An algorithm for numerical determination of the structure of a general matrix, BIT 10 (1970) 196-216.

    Article  MATH  MathSciNet  Google Scholar 

  31. A. Ruhe, Perturbation bound for means of eigenvalues and invariant subspaces, BIT (1970).

  32. A. Ruhe, Rational Krylov algorithms for nonsymmetric eigenvalue problems. II. Matrix pairs, Linear Algebra Appl. 197/198 (1994) 283-295.

    Article  MathSciNet  Google Scholar 

  33. Y. Saad, Numerical Methods for Large Eigenvalue Problems (Halsted Press, New York, 1992).

    Google Scholar 

  34. J.A. Scott, An Arnoldi code for computing selected eigenvalues of sparse, real, unsymmetric matrices, ACM Trans. Math. Software 21(4) (1995) 432-475.

    Article  MATH  MathSciNet  Google Scholar 

  35. G.E. Shilov, Linear Algebra (Dover, New York, 1977).

    Google Scholar 

  36. G.L.G. Sleijpen and H.A. van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl. 17(2) (1996) 401-425.

    Article  MATH  MathSciNet  Google Scholar 

  37. D.C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13(2) (1992) 357-385.

    Article  MATH  MathSciNet  Google Scholar 

  38. D.C. Sorensen, Truncated QZ methods for large scale generalized eigenvalue problems, Electron. Trans. Numer. Anal. 7 (1998) 141-168.

    MATH  MathSciNet  Google Scholar 

  39. D.C. Sorensen and C. Yang, A truncated RQ-iteration for large scale eigenvalue calculations, SIAM J. Matrix Anal. Appl. 19(4) (1998) 1045-1073.

    Article  MATH  MathSciNet  Google Scholar 

  40. G.W. Stewart, Matrix Algorithms: Basic Decompositions (SIAM, Philadelphia, PA, 1998).

    Google Scholar 

  41. G.W. Stewart and J.-G. Sun, Matrix Perturbation Theory (Academic Press, London, 1990).

    Google Scholar 

  42. P. Van Dooren, The computation of Kronecker's canonical form of a singular pencil, Linear Algebra Appl. 27 (1979) 103-141.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computing Science, Umeå University, SE-901 87, Umeå, Sweden

    Bo Kågström & Petter Wiberg

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  1. Bo Kågström
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  2. Petter Wiberg
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Kågström, B., Wiberg, P. Extracting partial canonical structure for large scale eigenvalue problems. Numerical Algorithms 24, 195–237 (2000). https://doi.org/10.1023/A:1019153528915

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