Numerical Algorithms

, Volume 39, Issue 4, pp 437–462 | Cite as

On theoretical and numerical aspects of symplectic Gram–Schmidt-like algorithms

Article

Abstract

Gram–Schmidt-like orthogonalization process with respect to a given skew-symmetric scalar product is a key step in model reduction methods, structure-preserving, for large sparse Hamiltonian eigenvalue problem. Theoretical as well as numerical aspects of this step do not benefit of enough attention, compared to the one allowed to the classical Gram–Schmidt algorithm and its modified version. The aim of this paper is to revisit the symplectic Gram–Schmidt algorithms, to built some modified versions and to deal with their theoretical and numerical features.

Keywords

skew-symmetric scalar product symplectic Gram–Schmidt algorithms modified versions symplectic geometry large sparse Hamiltonian eigenvalue problem 

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References

  1. [1]
    G. Ammar, P. Benner and V. Mehrmann, A multishift algorithm for the numerical solution of the algebraic Riccati equations, Electron. Trans. Numer. Anal. 1 (1993) 33–48. Google Scholar
  2. [2]
    G. Ammar, C. Mehl and V. Mehrmann, Schur-like forms for matrix Lie groups, Lie algebras and Jordan algebras, Linear Algebra Appl. 287 (1999) 11–39. Google Scholar
  3. [3]
    G. Ammar and V. Mehrmann, On Hamiltonian and symplectic Hessenberg forms, Linear Algebra Appl. 149 (1991) 55–72. Google Scholar
  4. [4]
    E. Artin, Geometric Algebra (Interscience, New York, 1957). Google Scholar
  5. [5]
    P. Benner, R. Byers, H. Faßbender, V. Mehrmann and D. Watkins, Cholesky-like factorisations of skew-symmetric matrices, Electron. Trans. Numer. Anal. 11 (2000) 85–93. Google Scholar
  6. [6]
    P. Benner and H. Faßbender, An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem, Linear Algebra Appl. 263 (1997) 75–111. Google Scholar
  7. [7]
    Å. Björck, Solving linear least squares problem by Gram–Schmidt orthogonalization, BIT 7 (1967) 1–21. Google Scholar
  8. [8]
    Å. Björck, Numerics of Gram–Schmidt orthogonalization, Linear Algebra Appl. 197/198 (1994) 297–316. Google Scholar
  9. [9]
    C. Brezinski, Computational Aspects of Linear Control (Kluwer Academic Publishers, Dordrecht/Hardbound, 2002). Google Scholar
  10. [10]
    A. Bunse-Gerstner, An analysis of the HR algorithm for computing the eigenvalues of a matrix, Linear Algebra Appl. 35 (1981) 155–173. Google Scholar
  11. [11]
    A. Bunse-Gerstner, Matrix factorizations for symplectic QR-like methods, Linear Algebra Appl. 83 (1986) 49–77. Google Scholar
  12. [12]
    A. Bunse-Gerstner and V. Mehrmann, A symplectic QR-like algorithm for the solution of the real algebraic Riccati equation, IEEE Trans. Automat. Control 31 (1986) 1104–1113. Google Scholar
  13. [13]
    R. Byers, A Hamiltonian QR algorithm, SIAM J. Sci. Statist. Comput. 7 (1986) 212–229. Google Scholar
  14. [14]
    J. Della-Dora, Numerical linear algorithms and group theory, Linear Algebra Appl. 10 (1975) 267–283. Google Scholar
  15. [15]
    R. Freund, Lanczos-type algorithms for structured non-Hermitian eigenvalue problems, in: Proc. of the Cornelius Lanczos Internat. Centenary Conference, eds. J. Brown, M. Chu, D. Ellison and R. Plemmons (SIAM, Philadelphia, PA, 1994) pp. 243–245. Google Scholar
  16. [16]
    R. Freund and N. Nachtigal, Software for simplified lanczos and QMR algorithms, Appl. Numer. Math. 19 (1995) 319–341. Google Scholar
  17. [17]
    G. Golub and C. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins Univ. Press, Baltimore, MD, 1996). Google Scholar
  18. [18]
    E. Grimme, D. Sorensen and P. Van Dooren, Model reduction of state space systems via an implicitly restarted Lanczos method, Numer. Algorithms 12 (1996) 1–31. Google Scholar
  19. [19]
    N.J. Higham, Accuracy and Stability of Numerical Algorithms (SIAM, Philadelphia, PA, 1996). Google Scholar
  20. [20]
    H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley-Interscience, New York, 1972). Google Scholar
  21. [21]
    P. Lancaster and L. Rodman, The Algebraic Riccati Equation (Oxford Univ. Press, Oxford, 1995). Google Scholar
  22. [22]
    C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Standards 45 (1950) 255–282. Google Scholar
  23. [23]
    A. Laub and K. Meyer, Canonical forms for symplectic and Hamiltonian matrices, Celest. Mech. 9 (1974) 213–238. Google Scholar
  24. [24]
    D.S. Mackey, N. Mackey and F. Tisseur, G-reflectors: Analogues of Householder transformations in scalar product spaces, Linear Algebra Appl. 385 (2004) 187–213. Google Scholar
  25. [25]
    V. Mehrmann, Der SR-algorithms zur Bestimmung der Eigenwerte einer Matrix, Diplomarbeit, Universität Bielefeld (1979). Google Scholar
  26. [26]
    V. Mehrmann and D. Watkins, Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils, SIAM J. Sci. Comput. 22 (2001) 1905–1925. Google Scholar
  27. [27]
    G. Mei, A new method for solving the algebraic Riccati equation, Master’s thesis, Nanjing Aeronautical Institute, Nanjing, P.R. China (1986). Google Scholar
  28. [28]
    C. Paige and C. Van Loan, A Schur decomposition for Hamiltonian matrices, Linear Algebra Appl. 41 (1981) 11–32. Google Scholar
  29. [29]
    Y. Saad, Iterative Methods for Sparse Linear Systems (PWS, Boston, 1996). Google Scholar
  30. [30]
    C. Van Loan, A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix, Linear Algebra Appl. 61 (1984) 233–251. Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Pures et AppliquéesUniversité du Littoral–Côte d’OpaleCalais CedexFrance

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