Abstract and Applied Analysis

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

Feng Yin and Guang-Xin Huang

Full-text: Open access

Abstract

The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: ( A X B C X D ) - ( E F ) = min over generalized reflexive matrix X . For any initial generalized reflexive matrix X 1 , by the iterative algorithm, the generalized reflexive solution X {\ast} can be obtained within finite iterative steps in the absence of round-off errors, and the unique least-norm generalized reflexive solution X {\ast} can also be derived when an appropriate initial iterative matrix is chosen. Furthermore, the unique optimal approximate solution X ^ to a given matrix X 0 in Frobenius norm can be derived by finding the least-norm generalized reflexive solution X ~ {\ast} of a new corresponding minimum Frobenius norm residual problem: min ( A X ~ B C X ~ D ) - ( E ~ F ~ ) with E ~ = E - A X 0 B , F ~ = F - C X 0 D . Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 857284, 18 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495658

Digital Object Identifier
doi:10.1155/2012/857284

Mathematical Reviews number (MathSciNet)
MR2898044

Zentralblatt MATH identifier
1242.65085

Citation

Yin, Feng; Huang, Guang-Xin. An Iterative Algorithm for the Least Squares Generalized Reflexive Solutions of the Matrix Equations $AXB=E,$ $CXD=F$. Abstr. Appl. Anal. 2012 (2012), Article ID 857284, 18 pages. doi:10.1155/2012/857284. https://projecteuclid.org/euclid.aaa/1355495658


Export citation

References

  • F. Li, X. Hu, and L. Zhang, “The 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.
  • J.-C. Zhang, X.-Y. Hu, and L. Zhang, “The $(P,Q)$ generalized reflexive and anti-reflexive solutions of the matrix equation $AX=B$,” Applied Mathematics and Computation, vol. 209, no. 2, pp. 254–258, 2009.
  • B. Zhou, Z.-Y. Li, G.-R. Duan, and Y. Wang, “Weighted least squares solutions to general coupled Sylvester matrix equations,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 759–776, 2009.
  • A.-P. Liao and Y. Lei, “Least-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.
  • F. Ding, P. X. Liu, and J. Ding, “Iterative 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.
  • F. Ding and T. Chen, “On iterative solutions of general coupled matrix equations,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 2269–2284, 2006.
  • L. Xie, J. Ding, and F. Ding, “Gradient based iterative solutions for general linear matrix equations,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1441–1448, 2009.
  • F. Ding and T. Chen, “Gradient based iterative algorithms for solving a class of matrix equations,” IEEE Transactions on Automatic Control, vol. 50, no. 8, pp. 1216–1221, 2005.
  • J. Ding, Y. Liu, and F. Ding, “Iterative solutions to matrix equations of the form ${A}_{i}X{B}_{i}={F}_{i}$,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3500–3507, 2010.
  • F. Ding and T. Chen, “Iterative least-squares solutions of coupled Sylvester matrix equations,” Systems & Control Letters, vol. 54, no. 2, pp. 95–107, 2005.
  • F. Ding and T. Chen, “Hierarchical gradient-based identification of multivariable discrete-time systems,” Automatica, vol. 41, no. 2, pp. 315–325, 2005.
  • F. Ding and T. Chen, “Hierarchical least squares identification methods for multivariable systems,” IEEE Transactions on Automatic Control, vol. 50, no. 3, pp. 397–402, 2005.
  • F. Ding and T. Chen, “Hierarchical 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.
  • Y.-X. Peng, X.-Y. Hu, and L. Zhang, “An 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.
  • X. Sheng and G. Chen, “A 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.
  • Z.-H. Peng, X.-Y. Hu, and L. Zhang, “An efficient algorithm for the least-squares reflexive solution of the matrix equation ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 988–999, 2006.
  • M. Dehghan and M. Hajarian, “An 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.
  • J. Cai and G. Chen, “An iterative algorithm for the least squares bisymmetric solutions of the matrix equations ${A}_{1}X{B}_{1}={C}_{1}$; ${A}_{2}X{B}_{2}={C}_{2}$,” Mathematical and Computer Modelling, vol. 50, no. 7-8, pp. 1237–1244, 2009.
  • A. K\il\içman and Z. A. A. A. Zhour, “Vector least-squares solutions for coupled singular matrix equations,” Journal of Computational and Applied Mathematics, vol. 206, no. 2, pp. 1051–1069, 2007.
  • M. Dehghan and M. Hajarian, “An 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.
  • A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, “Iterative solutions to coupled Sylvester-conjugate matrix equations,” Computers & Mathematics with Applications, vol. 60, no. 1, pp. 54–66, 2010.
  • A.-G. Wu, B. Li, Y. Zhang, and G.-R. Duan, “Finite iterative solutions to coupled Sylvester-conjugate matrix equations,” Applied Mathematical Modelling, vol. 35, no. 3, pp. 1065–1080, 2011.
  • A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, “Finite 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.
  • I. Jonsson and B. Kågström, “Recursive 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.
  • I. Jonsson and B. Kågström, “Recursive 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.
  • G. X. Huang, N. Wu, F. Yin, Z. L. Zhou, and K. Guo, “Finite iterative algorithms for solving generalized coupled Sylvester systems–-part I: one-sided and čommentComment on ref. [40?]: Please update the information of these references [40, 41?], if possible.generalized coupled Sylvester matrix equations over generalized reflexive solutions,” Applied Mathematical Modelling, vol. 36, no. 4, pp. 1589–1603, 2012.
  • F. Yin, G. X. Huang, and D. Q. Chen, “Finite 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.
  • M. Dehghan and M. Hajarian, “The general coupled matrix equations over generalized bisymmetric matrices,” Linear Algebra and Its Applications, vol. 432, no. 6, pp. 1531–1552, 2010.
  • M. Dehghan and M. Hajarian, “Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations,” Applied Mathematical Modelling, vol. 35, no. 7, pp. 3285–3300, 2011.
  • G.-X. Huang, F. Yin, and K. Guo, “An 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.
  • Z.-Y. Peng, “New 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.
  • Z.-Y. Peng and X.-Y. Hu, “The reflexive and anti-reflexive solutions of the matrix equation $AX=B$,” Linear Algebra and Its Applications, vol. 375, pp. 147–155, 2003.
  • Z.-h. Peng, X.-Y. Hu, and L. Zhang, “An efficient algorithm for the least-squares reflexive solution of the matrix equation ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 988–999, 2006.
  • X. Sheng and G. Chen, “An 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.
  • Q.-W. Wang, J.-H. Sun, and S.-Z. Li, “Consistency 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.
  • Q.-W. Wang, “A 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.
  • Q.-W. Wang, “Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 641–650, 2005.
  • A.-G. Wu, G.-R. Duan, and Y. Xue, “Kronecker maps and Sylvester-polynomial matrix equations,” IEEE Transactions on Automatic Control, vol. 52, no. 5, pp. 905–910, 2007.
  • A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, “Closed-form solutions to Sylvester-conjugate matrix equations,” Computers & Mathematics with Applications, vol. 60, no. 1, pp. 95–111, 2010.
  • Y. Yuan and H. Dai, “Generalized 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.
  • A. Antoniou and W.-S. Lu, Practical Optimization: Algorithm and Engineering Applications, Springer, New York, NY, USA, 2007.