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A periodic Krylov-Schur algorithm for large matrix products

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Abstract

Stewart's recently introduced Krylov-Schur algorithm is a modification of the implicitly restarted Arnoldi algorithm which employs reordered Schur decompositions to perform restarts and deflations in a numerically reliable manner. This paper describes a variant of the Krylov-Schur algorithm suitable for addressing eigenvalue problems associated with products of large and sparse matrices. It performs restarts and deflations via reordered periodic Schur decompositions and, by taking the product structure into account, it is capable of achieving qualitatively better approximations to eigenvalues of small magnitude.

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References

  1. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia, PA, 2005

  2. Antoulas, A.C., Sorensen, D.C., Gugercin, S.: A survey of model reduction methods for large-scale systems. In: Structured matrices in mathematics, computer science, and engineering, I (Boulder, CO, 1999), Vol. 280 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2001, pp. 193–219

  3. Beattie, C.A., Embree, M., Sorensen, D.C.: Convergence of polynomial restart Krylov methods for eigenvalue computations. SIAM Rev. 47(3), 492–515 (2005)

    Article  MathSciNet  Google Scholar 

  4. Bhatia, R.: Matrix Analysis. Springer-Verlag, New York, 1997

  5. Bojanczyk, A., Golub, G.H., Van Dooren, P.: The periodic Schur decomposition; algorithm and applications. In: Proc. SPIE Conference, Vol. 1770, pp. 31–42, 1992

  6. Bojanczyk, A., Van Dooren, P.: On propagating orthogonal transformations in a product of 2× 2 triangular matrices. In: (Reichel, L. Ruttan, A. and R. S. Varga, eds.), Numerical Linear Algebra (Kent, OH, 1992), pp. 1–9. de Gruyter, 1993

  7. Bomhof, W.: Iterative and parallel methods for linear systems, with applications in circuit simulation. PhD thesis, Universiteit Utrecht, Faculteit der Wiskunde en Informatica, Utrecht, Netherlands, 2001

  8. Bomhof, W., van der Vorst, H.: A parallelizable GMRES-type method for p-cyclic matrices, with applications in circuit simulation. In: U. Van Rienen, M. Gunther, and D. Hecht, eds.), Scientific computing in electrical engineering: proceedings of the 3rd international workshop, August 20–23, 2000, Warnemünde, Germany, volume 18 of Lecture Notes in Computational Science and Engineering, Berlin, Heidelberg, London, 2001. Springer Verlag, pp. 293–300

  9. Bonhoure, F., Dallery, Y., Stewart, W.J.: On the use of periodicity properties for the efficient numerical solution of certain Markov chains. Numer. Linear Algebra Appl. 1(3), 265–286 (1994)

    Article  MathSciNet  Google Scholar 

  10. Bru, R., Cantó, R., Ricarte, B.: Modelling nitrogen dynamics in citrus trees. Math and Comp. Mod 38, 975–987 (2003)

    Article  MATH  Google Scholar 

  11. Bunch, J.R.: The weak and strong stability of algorithms in numerical linear algebra. Linear Algebra Appl. 88/89, 49–66 (1987)

    Google Scholar 

  12. Chahlaoui, Y., Van Dooren, P.: Benchmark examples for model reduction of linear time-invariant dynamical systems. In: (P. Benner, V. Mehrmann, and D. C. Sorensen eds.), Dimension Reduction of Large-Scale Systems, Vol. 45 of Lecture Notes in Computational Science and Engineering, Springer, Heidelberg, 2005, pp. 379–392

  13. Daniel, J., Gragg, W.B., Kaufman, L., Stewart, G.W.: Reorthogonalization and stable algorithms for updating the Gram–Schmidt QR factorization. Math. Comput. 30, 772–795 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ernst, O.G.: Equivalent iterative methods for p-cyclic matrices. Numer. Algorithms 25(1–4), 161–180 (2000)

    Google Scholar 

  15. Freund, R.W., Golub, G.H., Nachtigal, N.M.: Recent advances in Lanczos-based iterative methods for nonsymmetric linear systems. In: Algorithmic trends in computational fluid dynamics (1991), ICASE/NASA LaRC Ser., Springer, New York, 1993, pp. 137–162

  16. Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore, MD, third edition, 1996

  17. Granat, R., Kågström, B.: Direct eigenvalue reordering in a product of matrices in extended periodic Schur form. Report UMINF-05.05, Department of Computing Science, Umeå University, Umeå , Sweden, 2005

  18. Hammarling, S.J.: Numerical solution of the stable, non-negative definite Lyapunov equation. IMA J. Numer. Anal. 2, 303–323 (1982)

    MATH  MathSciNet  Google Scholar 

  19. Hench, J.J., Laub, A.J.: Numerical solution of the discrete-time periodic Riccati equation. IEEE Trans. Automat. Control 39(6), 1197–1210 (1994)

    Article  MathSciNet  Google Scholar 

  20. Jaimoukha, I.M., Kasenally, E.M.: Krylov subspace methods for solving large Lyapunov equations. SIAM J. Numer. Anal. 31, 227–251 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kressner, D.: The periodic QR algorithm is a disguised QR algorithm, 2003. To appear in Linear Algebra Appl.

  22. Kressner, D.: Numerical Methods and Software for General and Structured Eigenvalue Problems. PhD thesis, TU Berlin, Institut für Mathematik, Berlin, Germany, 2004

  23. Lehoucq, R.B.: Analysis and Implementation of an Implicitly Restarted Arnoldi Iteration. PhD thesis, Rice U., 1995

  24. Lehoucq, R.B., Sorensen, D.C.: Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Anal. Appl. 17(4), 789–821 (1996)

    Article  MathSciNet  Google Scholar 

  25. Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK users' guide. SIAM, Philadelphia, PA, 1998. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods

  26. Lin, W.-W., Van Dooren, P., Xu, Q.-F.: Periodic invariant subspaces in control. In: Proc. of IFAC Workshop on Periodic Control Systems, Como, Italy, 2001

  27. Lust, K.: Numerical Bifurcation Analysis of Periodic Solutions of Partial Differential Equations. PhD thesis, Department of Computer Science, KU Leuven, Belgium, 1997

  28. Lust, K.: Improved numerical Floquet multipliers. Internat. J. Bifur. Chaos Appl. Sci. Eng. 11(9), 2389–2410 (2001)

    Article  MathSciNet  Google Scholar 

  29. Parlett, B.N., Le, J.: Forward instability of tridiagonal QR. SIAM J. Matrix Anal. Appl. 14(1), 279–316 (1993)

    Article  MathSciNet  Google Scholar 

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

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Sorensen, D.C.: Implicitly restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations. In: Parallel numerical algorithms (Hampton, VA, 1994), Vol. 4 of ICASE/LaRC Interdiscip. Ser. Sci. Eng. Kluwer Acad. Publ., Dordrecht, 1997, pp. 119–165

  33. Sreedhar, J., Van Dooren, P.: Pole placement via the periodic Schur decomposition. In: Proceedings Amer. Contr. Conf. pp. 1563–1567, 1993

  34. Sreedhar, J., Van Dooren, P.: A Schur approach for solving some periodic matrix equations. In: (U. Helmke, R. Mennicken, and J. Saurer, eds.), Systems and Networks : Mathematical Theory and Applications, volume 77, Akademie Verlag, Berlin, 1994, pp. 339–362

  35. Stewart, G.W.: Matrix Algorithms. Vol. II. SIAM, Philadelphia, PA, 2001. Eigensystems

  36. Stewart, G.W.: A Krylov-Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23(3), 601–614 (2001/02)

    Article  Google Scholar 

  37. Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton, NJ, 1994

  38. Van Loan, C.F.: A general matrix eigenvalue algorithm. SIAM J. Numer. Anal. 12(6), 819–834 (1975)

    Article  Google Scholar 

  39. Varga, A.: Balancing related methods for minimal realization of periodic systems. Systems Control Lett. 36(5), 339–349 (1999)

    Article  Google Scholar 

  40. Varga, A., Van Dooren, P.: Computational methods for periodic systems - an overview. In: Proc. of IFAC Workshop on Periodic Control Systems, Como, Italy, 2001, pp. 171–176

  41. Walker, H.F.: Implementation of the GMRES method using Householder transformations. SIAM J. Sci. Stat. Comp. 9, 152–163 (1988)

    Article  MATH  Google Scholar 

  42. Watkins, D.S.: Product eigenvalue problems. SIAM Rev. 47, 3–40 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics,  , Bijenička 30, 10000, Zagreb, Croatia

    Daniel Kressner

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Correspondence to Daniel Kressner.

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Supported by DFG Research Center Matheon, Mathematics for key technologies, in Berlin.

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Kressner, D. A periodic Krylov-Schur algorithm for large matrix products. Numer. Math. 103, 461–483 (2006). https://doi.org/10.1007/s00211-006-0682-1

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  • DOI: https://doi.org/10.1007/s00211-006-0682-1

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