Skip to main content
SpringerLink
Log in

Solving Stable Sylvester Equations via Rational Iterative Schemes

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We investigate the numerical solution of stable Sylvester equations via iterative schemes proposed for computing the sign function of a matrix. In particular, we discuss how the rational iterations for the matrix sign function can efficiently be adapted to the special structure implied by the Sylvester equation. For Sylvester equations with factored constant term as those arising in model reduction or image restoration, we derive an algorithm that computes the solution in factored form directly. We also suggest convergence criteria for the resulting iterations and compare the accuracy and performance of the resulting methods with existing Sylvester solvers. The algorithms proposed here are easy to parallelize. We report on the parallelization of those algorithms and demonstrate their high efficiency and scalability using experimental results obtained on a cluster of Intel Pentium Xeon processors

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abels J., and Benner, P. (1999). CAREX—a collection of benchmark examples for continuous-time algebraic Riccati equations (version 2.0). SLICOT Working Note 1999-14. Available from http://www.win.tue.nl/niconet/NIC2/reports.html

  2. Aldhaheri R. (1991). Model order reduction via real Schur-form decomposition. Int. J. Control 53(3):709–716

    Article  MathSciNet  MATH  Google Scholar 

  3. Anderson E., Bai Z., Bischof C., Demmel J., Dongarra J., Du Croz J., Greenbaum A., Hammarling S., McKenney A., Sorensen D. (1999). LAPACK Users’ Guide 3rd ed. SIAM, Philadelphia, PA

    Google Scholar 

  4. Antoulas A. (2005). pproximation of Large-Scale Dynamical Systems. SIAM Publications, Philadelphia, PA

    Google Scholar 

  5. Bai, Z., and Demmel, J. (1993). Design of a parallel nonsymmetric eigenroutine toolbox, Part I. In R. S. et al., (eds.), Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, SIAM, Philadelphia, PA, 391–398, See also: Tech. Report CSD-92-718, Computer Science Division, University of California, Berkeley, CA.

  6. Bartels R., Stewart G. (1972). Solution of the matrix equation AX+XB=C: Algorithm 432. Comm. ACM 15:820–826

    Article  Google Scholar 

  7. Beavers A.N., Denman E.D. (1975). A new solution method for the Lyapunov matrix equations. SIAM J. Appl. Math. 29:416–42

    Article  MathSciNet  MATH  Google Scholar 

  8. Benner P. (2004). Factorized solution of Sylvester equations with applications in control. In Proc. Intl. Symp. Math. Theory Networks and Syst. MTNS 2004, http://www.mtns2004.be

  9. Benner P., Claver v., Quintana-Ortí E. (1999). Parallel distributed solvers for large stable generalized Lyapunov equations. Parallel Processing Lett. 9(1):147–158

    Article  Google Scholar 

  10. Benner P., Quintana-Ortí E. (1999). Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Algorithms 20(1):75–100

    Article  MathSciNet  MATH  Google Scholar 

  11. Benner P., Quintana-Ortí E., Quintana-Ortí G. (2002). Numerical solution of discrete stable linear matrix equations on multicomputers. Parallel Algorithms and Appl. 17(1):127–146

    MathSciNet  MATH  Google Scholar 

  12. Blackford L., Choi J., Cleary A., D’Azevedo E., Demmel J., Dhillon I., Dongarra J., Hammarling S., Henry G., Petitet A., Stanley K., Walker D., Whaley R. (1997). ScaLAPACK Users’ Guide. SIAM, Philadelphia, PA

    MATH  Google Scholar 

  13. Byers R. (1987). Solving the algebraic Riccati equation with the matrix sign function. Linear Algebra Appl. 85:267–279

    Article  MathSciNet  MATH  Google Scholar 

  14. Calvetti D., Reichel L. (1996). Application of ADI iterative methods to the restoration of noisy images. SIAM J. Matrix Anal. Appl. 17:165–186

    Article  MathSciNet  MATH  Google Scholar 

  15. Chan T. (1987). Rank revealing QR factorizations. Linear Algebra Appl. 88/89:67–82

    Article  Google Scholar 

  16. Choi C., Laub A. (1990). Efficient matrix-valued algorithms for solving stiff Riccati differential equations. IEEE Trans. Automat. Control 35:770–776

    Article  MathSciNet  MATH  Google Scholar 

  17. Datta B. (2003). Numerical Methods for Linear Control Systems Design and Analysis. Elsevier Press, Amsterdam

    Google Scholar 

  18. Dieci L., Osborne M., Russell R. (1988). A. Riccati, transformation method for solving linear BVPs I: Theoretical aspects. SIAM J. Numer. Anal. 25(5):1055–1073

    Article  MathSciNet  MATH  Google Scholar 

  19. Dongarra J., Croz J.D., Duff I., Hammarling S. (1990). A set of level 3 Basic linear algebra subprograms. ACM Trans. Math. Soft. 16:1–17

    Article  MATH  Google Scholar 

  20. Dongarra J., Duff I.S., Sorensen D.C., van der Vorst H.A. (1991). Solving Linear Systems on Vector and Shared Memory computers. SIAM, Philadelphia, PA

    Google Scholar 

  21. Enright W. (1978). Improving the efficiency of matrix operations in the numerical solution of stiff ordinary differential equations. ACM Trans. Math. Soft. 4:127–136

    Article  MathSciNet  MATH  Google Scholar 

  22. Epton M. (1980). Methods for the solution of AXD−BXC=E and its application in the numerical solution of implicit ordinary differential equations. BIT 20:341–345

    Article  MathSciNet  MATH  Google Scholar 

  23. Fernando K., Nicholson H. (1984). On a fundamental property of the cross-Gramian matrix. IEEE Trans. Circuits Syst. CAS- 31(5): 504–505

    Article  Google Scholar 

  24. Gajić Z., Qureshi M. (1995). Lyapunov Matrix Equation in System Stability and Control. Math. in Science and Engineering. Academic Press, San Diego, CA

    Google Scholar 

  25. Gardiner J., Laub A. (1991). Parallel algorithms for algebraic Riccati equations. Int. J. Control 54(6):1317–1333

    Article  MathSciNet  MATH  Google Scholar 

  26. Gardiner J., Laub A., Amato J., Moler C. (1992). Solution of the Sylvester matrix equation AXB+CXD=E. ACM Trans. Math. Soft 18:223–231

    Article  MathSciNet  MATH  Google Scholar 

  27. Golub G., Van Loan C. (1996). Matrix Computations 3rd ed. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  28. Golub G.H., Nash S., Van Loan C.F. (1979). A Hessenberg–Schur method for the problem AX+XB=C. IEEE Trans. Automat. Control. AC- 24:909–913

    Article  MathSciNet  Google Scholar 

  29. Goto K., van de Geijn R. (2002). On reducing TLB misses in matrix multiplication. FLAME Working Note 9, Department of Computer Sciences, The University of Texas at Austin, http://www.cs.utexas.edu/users/flame

  30. Grasedyck L. (2004). Existence of a low rank or H-matrix approximant to the solution of a Sylvester equation. Numer. Lin. Alg. Appl. 11:371–389

    Article  MathSciNet  MATH  Google Scholar 

  31. Grasedyck L., Hackbusch W. (2004). A multigrid method to solve large scale Sylvester equations. Preprint 48, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig Germany

    Google Scholar 

  32. Gropp W., Lusk E., Skjellum A. (1994). Using MPI: Portable Parallel Programming with the Message-Passing Interface. MIT Press, Cambridge, MA

    Google Scholar 

  33. Halley E. (1694). Methodus nova, accurata & facilis inveniendi radices aequationum quarumcumque generaliter, sine praevia reductione. Philos. Trans. R. Soc. Lond 18:136–148

    Article  Google Scholar 

  34. Henry G., van de Geijn R. (1997). Parallelizing the QR algorithm for the unsymmetric algebraic eigenvalue problem: myths and reality. SIAM J. Sci. Comput. 17:870–883

    Article  Google Scholar 

  35. Higham N. (1986). Computing the polar decomposition—with applications. SIAM J. Sci. Statist. Comput. 7:1160–1174

    Article  MathSciNet  MATH  Google Scholar 

  36. Hoskins W., Meek D., Walton D. (1977). The numerical solution of the matrix equation X A+A Y=F. BIT 17:184–190

    Article  MathSciNet  MATH  Google Scholar 

  37. Hu D., Reichel L. (1992). Krylov-subspace methods for the Sylvester equation. Linear Algebra Appl. 172:283–313

    Article  MathSciNet  MATH  Google Scholar 

  38. Kågström B., Poromaa P. (1996). Lapack-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs. ACM Trans. Math. Soft 22(1):78–103

    Article  MATH  Google Scholar 

  39. Kenney C., and Laub A. (1991). Rational iterative methods for the matrix sign function. SIAM J. Matrix Anal. Appl. 12: 273–291

    Article  MathSciNet  MATH  Google Scholar 

  40. Kenney C., Laub A. (1995). The matrix sign function. IEEE Trans Automat Control 40(8):1330–1348

    Article  MathSciNet  MATH  Google Scholar 

  41. Quintana-Ortí E., Quintana-Ortí G., Sun X., van de Geijn R. (2001). A note on parallel matrix inversion. SIAM J. Sci. Comput. 22:1762–1771

    Article  Google Scholar 

  42. Roberts, J. (1980). Linear model reduction and solution of the algebraic Riccati equation by use of the sign function. Int. J. Control, 32, 677–687, (Reprint of Technical Report No. TR-13, CUED/B-Control, Cambridge University, Engineering Department, 1971).

  43. Schulz G. (1993). Iterative berechnung der reziproken matrix. Z. Angew. Math. Mech. 13:57–59 In German

    Article  Google Scholar 

  44. Sima V. (1996). Algorithms for Linear-Quadratic Optimization. Pure and Applied Mathematics vol 200. Marcel Dekker Inc., New York NY

    Google Scholar 

  45. Slowik, M., Benner, P., and Sima, V. (2004). Evaluation of the linear matrix equation solvers in SLICOT. SLICOT Working Note 2004–1 Available from http://www.win.tue.nl/niconet/NIC2/reports.html

  46. van de Geijn R. (1997). Using PLAPACK: Parallel Linear Algebra Package. MIT Press, Cambridge MA

    Google Scholar 

  47. Wachspress E. (1988). Iterative solution of the Lyapunov matrix equation. Appl. Math. Lett. 107:87–90

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Peter Benner

  2. Depto. de Ingeniería y Ciencia de Computadores, Universidad Jaume I, 12.071, Castellón, Spain

    Enrique S. Quintana-Ortí & Gregorio Quintana-Ortí

Authors
  1. Peter Benner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Enrique S. Quintana-Ortí
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gregorio Quintana-Ortí
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Benner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Benner, P., Quintana-Ortí, E.S. & Quintana-Ortí, G. Solving Stable Sylvester Equations via Rational Iterative Schemes. J Sci Comput 28, 51–83 (2006). https://doi.org/10.1007/s10915-005-9007-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-005-9007-2

Keywords

Search

Navigation