Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2014, Article ID 578102, 9 pages
http://dx.doi.org/10.1155/2014/578102
Research Article

A Generalized HSS Iteration Method for Continuous Sylvester Equations

1School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, China
2Department of Mathematics, Federal University of Paraná, Centro Politécnico, CP 19.081, 81531-980 Curitiba, PR, Brazil

Received 20 August 2013; Accepted 13 December 2013; Published 12 January 2014

Academic Editor: Qing-Wen Wang

Copyright © 2014 Xu Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. P. Lancaster, “Explicit solutions of linear matrix equations,” SIAM Review, vol. 12, no. 4, pp. 544–566, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. P. Lancaster and M. Tismenetsky, The Theory of Matrices, Computer Science and Applied Mathematics, Academic Press, Orlando, Fla, USA, 2nd edition, 1985. View at MathSciNet
  3. Q.-W. Wang and Z.-H. He, “Solvability conditions and general solution for mixed Sylvester equations,” Automatica, vol. 49, no. 9, pp. 2713–2719, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  4. Z.-Z. Bai, “On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations,” Journal of Computational Mathematics, vol. 29, no. 2, pp. 185–198, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. Z.-Z. Bai, Y.-M. Huang, and M. K. Ng, “On preconditioned iterative methods for Burgers equations,” SIAM Journal on Scientific Computing, vol. 29, no. 1, pp. 415–439, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. Z.-Z. Bai and M. K. Ng, “Preconditioners for nonsymmetric block toeplitz-like-plus-diagonal linear systems,” Numerische Mathematik, vol. 96, no. 2, pp. 197–220, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford Science Publications, The Clarendon Press, New York, NY, USA, 1995. View at MathSciNet
  8. G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, Md, USA, 3rd edition, 1996. View at MathSciNet
  9. A. Halanay and V. Răsvan, Applications of Liapunov Methods in Stability, vol. 245 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  10. A. N. P. Liao, Z.-Z. Bai, and Y. Lei, “Best approximate solution of matrix equation AXB+CYD=E,” SIAM Journal on Matrix Analysis and Applications, vol. 27, no. 3, pp. 675–688, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. Z.-Z. Bai, X.-X. Guo, and S.-F. Xu, “Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations,” Numerical Linear Algebra with Applications, vol. 13, no. 8, pp. 655–674, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. X.-X. Guo and Z.-Z. Bai, “On the minimal nonnegative solution of nonsymmetric algebraic Riccati equation,” Journal of Computational Mathematics, vol. 23, no. 3, pp. 305–320, 2005. View at Google Scholar · View at MathSciNet
  13. G.-P. Xu, M.-S. Wei, and D.-S. Zheng, “On solutions of matrix equation AXB+CYD=F,” Linear Algebra and Its Applications, vol. 279, no. 1–3, pp. 93–109, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. J.-T. Zhou, R.-R. Wang, and Q. Niu, “A preconditioned iteration method for solving Sylvester equations,” Journal of Applied Mathematics, vol. 2012, Article ID 401059, 12 pages, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. F. Yin and G.-X. Huang, “An iterative algorithm for the generalized reflexive solutions of the generalized coupled Sylvester matrix equations,” Journal of Applied Mathematics, vol. 2012, Article ID 152805, 28 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. H. Bartels and G. W. Stewart, “Solution of the matrix equation AX+XB=C[F4],” Communications of the ACM, vol. 15, no. 9, pp. 820–826, 1972. View at Publisher · View at Google Scholar
  17. G. H. Golub, S. Nash, and C. Van Loan, “A Hessenberg-Schur method for the problem AX+XB=C,” IEEE Transactions on Automatic Control, vol. 24, no. 6, pp. 909–913, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. R. A. Smith, “Matrix equation XA+BX=C,” SIAM Journal on Applied Mathematics, vol. 16, no. 1, pp. 198–201, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. D. Calvetti and L. Reichel, “Application of ADI iterative methods to the restoration of noisy images,” SIAM Journal on Matrix Analysis and Applications, vol. 17, no. 1, pp. 165–186, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. D. Y. Hu and L. Reichel, “Krylov-subspace methods for the Sylvester equation,” Linear Algebra and Its Applications, vol. 172, pp. 283–313, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. N. Levenberg and L. Reichel, “A generalized ADI iterative method,” Numerische Mathematik, vol. 66, no. 1, pp. 215–233, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. E. L. Wachspress, “Iterative solution of the Lyapunov matrix equation,” Applied Mathematics Letters, vol. 1, no. 1, pp. 87–90, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. G. Starke and W. Niethammer, “SOR for AX-XB=C,” Linear Algebra and Its Applications, vol. 154–156, pp. 355–375, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  24. D. J. Evans and E. Galligani, “A parallel additive preconditioner for conjugate gradient method for AX+XB=C,” Parallel Computing, vol. 20, no. 7, pp. 1055–1064, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. L. Jódar, “An algorithm for solving generalized algebraic Lyapunov equations in Hilbert space, applications to boundary value problems,” Proceedings of the Edinburgh Mathematical Society: Series 2, vol. 31, no. 1, pp. 99–105, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. C.-Q. Gu and H.-Y. Xue, “A shift-splitting hierarchical identification method for solving Lyapunov matrix equations,” Linear Algebra and Its Applications, vol. 430, no. 5-6, pp. 1517–1530, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. Z.-Z. Bai, G. H. Golub, and M. K. Ng, “Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems,” SIAM Journal on Matrix Analysis and Applications, vol. 24, no. 3, pp. 603–626, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  28. Z.-Z. Bai, “Splitting iteration methods for non-Hermitian positive definite systems of linear equations,” Hokkaido Mathematical Journal, vol. 36, no. 4, pp. 801–814, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. Z.-Z. Bai, G. H. Golub, L.-Z. Lu, and J.-F. Yin, “Block triangular and skew-Hermitian splitting methods for positive-definite linear systems,” SIAM Journal on Scientific Computing, vol. 26, no. 3, pp. 844–863, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  30. Z.-Z. Bai, G. H. Golub, and M. K. Ng, “On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations,” Numerical Linear Algebra with Applications, vol. 14, no. 4, pp. 319–335, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  31. Z.-Z. Bai, G. H. Golub, and M. K. Ng, “On inexact hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems,” Linear Algebra and Its Applications, vol. 428, no. 2-3, pp. 413–440, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  32. Z.-Z. Bai, G. H. Golub, and J.-Y. Pan, “Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems,” Numerische Mathematik, vol. 98, no. 1, pp. 1–32, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. M. Benzi, “A Generalization of the he rmitian and skew-hermitian splitting iteration,” SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 2, pp. 360–374, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. L. Li, T.-Z. Huang, and X.-P. Liu, “Asymmetric Hermitian and skew-Hermitian splitting methods for positive definite linear systems,” Computers and Mathematics with Applications, vol. 54, no. 1, pp. 147–159, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  35. A.-L. Yang, J. An, and Y.-J. Wu, “A generalized preconditioned HSS method for non-Hermitian positive definite linear systems,” Applied Mathematics and Computation, vol. 216, no. 6, pp. 1715–1722, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  36. J.-F. Yin and Q.-Y. Dou, “Generalized preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems,” Journal of Computational Mathematics, vol. 30, no. 4, pp. 404–417, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  37. Z.-Z. Bai and G. H. Golub, “Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems,” IMA Journal of Numerical Analysis, vol. 27, no. 1, pp. 1–23, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  38. X. Li, A.-L. Yang, and Y.-J. Wu, “Parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method for saddle-point problems,” International Journal of Computer Mathematics, 2013. View at Publisher · View at Google Scholar
  39. W. Li, Y.-P. Liu, and X.-F. Peng, “The generalized HSS method for solving singular linear systems,” Journal of Computational and Applied Mathematics, vol. 236, no. 9, pp. 2338–2353, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. O. Axelsson, Z.-Z. Bai, and S.-X. Qiu, “A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part,” Numerical Algorithms, vol. 35, no. 2-4, pp. 351–372, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  41. Z.-Z. Bai, “A class of two-stage iterative methods for systems of weakly nonlinear equations,” Numerical Algorithms, vol. 14, no. 4, pp. 295–319, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus