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Abstract and Applied Analysis
Volume 2012, Article ID 857284, 18 pages
http://dx.doi.org/10.1155/2012/857284
Research Article

An Iterative Algorithm for the Least Squares Generalized Reflexive Solutions of the Matrix Equations 𝐴 𝑋 𝐡 = 𝐸 , 𝐢 𝑋 𝐷 = 𝐹

1School of Science, Sichuan University of Science and Engineering, Zigong 643000, China
2Geomathematics Key Laboratory of Sichuan Province, College of Mathematics, Chengdu University of Technology, Chengdu 610059, China

Received 10 October 2011; Accepted 9 December 2011

Academic Editor: Zhenya Yan

Copyright © 2012 Feng Yin and Guang-Xin Huang. 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. F. Li, X. Hu, and L. Zhang, β€œThe generalized reflexive solution for a class of matrix equations AX=B; XC=D,” Acta Mathematica Scientia Series B, vol. 28, no. 1, pp. 185–193, 2008. View at Publisher Β· View at Google Scholar
  2. J.-C. Zhang, X.-Y. Hu, and L. Zhang, β€œThe (P,Q) generalized reflexive and anti-reflexive solutions of the matrix equation AX=B,” Applied Mathematics and Computation, vol. 209, no. 2, pp. 254–258, 2009. View at Publisher Β· View at Google Scholar
  3. B. Zhou, Z.-Y. Li, G.-R. Duan, and Y. Wang, β€œWeighted least squares solutions to general coupled Sylvester matrix equations,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 759–776, 2009. View at Publisher Β· View at Google Scholar
  4. A.-P. Liao and Y. Lei, β€œLeast-squares solution with the minimum-norm for the matrix equation (AXB,GXH)=(C,D),” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 539–549, 2005. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  5. F. Ding, P. X. Liu, and J. Ding, β€œIterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 41–50, 2008. View at Publisher Β· View at Google Scholar
  6. F. Ding and T. Chen, β€œOn iterative solutions of general coupled matrix equations,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 2269–2284, 2006. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  7. L. Xie, J. Ding, and F. Ding, β€œGradient based iterative solutions for general linear matrix equations,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1441–1448, 2009. View at Publisher Β· View at Google Scholar
  8. F. Ding and T. Chen, β€œGradient based iterative algorithms for solving a class of matrix equations,” IEEE Transactions on Automatic Control, vol. 50, no. 8, pp. 1216–1221, 2005. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  9. J. Ding, Y. Liu, and F. Ding, β€œIterative solutions to matrix equations of the form AiXBi=Fi,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3500–3507, 2010. View at Publisher Β· View at Google Scholar
  10. F. Ding and T. Chen, β€œIterative least-squares solutions of coupled Sylvester matrix equations,” Systems & Control Letters, vol. 54, no. 2, pp. 95–107, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  11. F. Ding and T. Chen, β€œHierarchical gradient-based identification of multivariable discrete-time systems,” Automatica, vol. 41, no. 2, pp. 315–325, 2005. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  12. F. Ding and T. Chen, β€œHierarchical least squares identification methods for multivariable systems,” IEEE Transactions on Automatic Control, vol. 50, no. 3, pp. 397–402, 2005. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  13. F. Ding and T. Chen, β€œHierarchical identification of lifted state-space models for general dual-rate systems,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 6, pp. 1179–1187, 2005. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  14. Y.-X. Peng, X.-Y. Hu, and L. Zhang, β€œAn iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 1127–1137, 2006. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  15. X. Sheng and G. Chen, β€œA finite iterative method for solving a pair of linear matrix equations (AXB,CXD)=(E,F),” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1350–1358, 2007. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  16. Z.-H. Peng, X.-Y. Hu, and L. Zhang, β€œAn efficient algorithm for the least-squares reflexive solution of the matrix equation A1XB1=C1, A2XB2=C2,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 988–999, 2006. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  17. M. Dehghan and M. Hajarian, β€œAn iterative algorithm for solving a pair of matrix equations AYB=E, CYD=F over generalized centro-symmetric matrices,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3246–3260, 2008. View at Publisher Β· View at Google Scholar
  18. J. Cai and G. Chen, β€œAn iterative algorithm for the least squares bisymmetric solutions of the matrix equations A1XB1=C1; A2XB2=C2,” Mathematical and Computer Modelling, vol. 50, no. 7-8, pp. 1237–1244, 2009. View at Publisher Β· View at Google Scholar
  19. A. Kılıçman and Z. A. A. A. Zhour, β€œVector least-squares solutions for coupled singular matrix equations,” Journal of Computational and Applied Mathematics, vol. 206, no. 2, pp. 1051–1069, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  20. M. Dehghan and M. Hajarian, β€œAn iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices,” Applied Mathematical Modelling, vol. 34, no. 3, pp. 639–654, 2010. View at Publisher Β· View at Google Scholar
  21. A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, β€œIterative solutions to coupled Sylvester-conjugate matrix equations,” Computers & Mathematics with Applications, vol. 60, no. 1, pp. 54–66, 2010. View at Publisher Β· View at Google Scholar
  22. A.-G. Wu, B. Li, Y. Zhang, and G.-R. Duan, β€œFinite iterative solutions to coupled Sylvester-conjugate matrix equations,” Applied Mathematical Modelling, vol. 35, no. 3, pp. 1065–1080, 2011. View at Publisher Β· View at Google Scholar
  23. A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, β€œFinite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1463–1478, 2010. View at Publisher Β· View at Google Scholar
  24. I. Jonsson and B. Kågström, β€œRecursive blocked algorithm for solving triangular systems. I. One-sided and coupled Sylvester-type matrix equations,” ACM Transactions on Mathematical Software, vol. 28, no. 4, pp. 392–415, 2002. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  25. I. Jonsson and B. Kågström, β€œRecursive blocked algorithm for solving triangular systems. II. Two-sided and generalized Sylvester and Lyapunov matrix equations,” ACM Transactions on Mathematical Software, vol. 28, no. 4, pp. 416–435, 2002. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  26. G. X. Huang, N. Wu, F. Yin, Z. L. Zhou, and K. Guo, β€œFinite iterative algorithms for solving generalized coupled Sylvester systems—part I: one-sided and generalized coupled Sylvester matrix equations over generalized reflexive solutions,” Applied Mathematical Modelling, vol. 36, no. 4, pp. 1589–1603, 2012. View at Publisher Β· View at Google Scholar
  27. F. Yin, G. X. Huang, and D. Q. Chen, β€œFinite iterative algorithms for solving generalized coupled Sylvester systems-Part II: two-sided and generalized coupled Sylvester matrix equations over reflexive solutions,” Applied Mathematical Modelling, vol. 36, no. 4, pp. 1604–1614, 2012. View at Publisher Β· View at Google Scholar
  28. M. Dehghan and M. Hajarian, β€œThe general coupled matrix equations over generalized bisymmetric matrices,” Linear Algebra and Its Applications, vol. 432, no. 6, pp. 1531–1552, 2010. View at Publisher Β· View at Google Scholar
  29. M. Dehghan and M. Hajarian, β€œAnalysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations,” Applied Mathematical Modelling, vol. 35, no. 7, pp. 3285–3300, 2011. View at Publisher Β· View at Google Scholar
  30. G.-X. Huang, F. Yin, and K. Guo, β€œAn iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C,” Journal of Computational and Applied Mathematics, vol. 212, no. 2, pp. 231–244, 2008. View at Publisher Β· View at Google Scholar
  31. Z.-Y. Peng, β€œNew matrix iterative methods for constraint solutions of the matrix equation AXB=C,” Journal of Computational and Applied Mathematics, vol. 235, no. 3, pp. 726–735, 2010. View at Publisher Β· View at Google Scholar
  32. Z.-Y. Peng and X.-Y. Hu, β€œThe reflexive and anti-reflexive solutions of the matrix equation AX=B,” Linear Algebra and Its Applications, vol. 375, pp. 147–155, 2003. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  33. Z.-h. Peng, X.-Y. Hu, and L. Zhang, β€œAn efficient algorithm for the least-squares reflexive solution of the matrix equation A1XB1=C1, A2XB2=C2,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 988–999, 2006. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  34. X. Sheng and G. Chen, β€œAn iterative method for the symmetric and skew symmetric solutions of a linear matrix equation AXB+CYD=E,” Journal of Computational and Applied Mathematics, vol. 233, no. 11, pp. 3030–3040, 2010. View at Publisher Β· View at Google Scholar
  35. Q.-W. Wang, J.-H. Sun, and S.-Z. Li, β€œConsistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra,” Linear Algebra and Its Applications, vol. 353, pp. 169–182, 2002. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  36. Q.-W. Wang, β€œA system of matrix equations and a linear matrix equation over arbitrary regular rings with identity,” Linear Algebra and Its Applications, vol. 384, pp. 43–54, 2004. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  37. Q.-W. Wang, β€œBisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 641–650, 2005. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  38. A.-G. Wu, G.-R. Duan, and Y. Xue, β€œKronecker maps and Sylvester-polynomial matrix equations,” IEEE Transactions on Automatic Control, vol. 52, no. 5, pp. 905–910, 2007. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  39. A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, β€œClosed-form solutions to Sylvester-conjugate matrix equations,” Computers & Mathematics with Applications, vol. 60, no. 1, pp. 95–111, 2010. View at Publisher Β· View at Google Scholar
  40. Y. Yuan and H. Dai, β€œGeneralized reflexive solutions of the matrix equation AXB=D and an associated optimal approximation problem,” Computers & Mathematics with Applications, vol. 56, no. 6, pp. 1643–1649, 2008. View at Publisher Β· View at Google Scholar
  41. A. Antoniou and W.-S. Lu, Practical Optimization: Algorithm and Engineering Applications, Springer, New York, NY, USA, 2007.