Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2015, Special Issue (2015), Article ID 562529, 12 pages.

LSMR Iterative Method for General Coupled Matrix Equations

F. Toutounian, D. Khojasteh Salkuyeh, and M. Mojarrab

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Abstract

By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations k = 1 q A i k X k B i k = C i , i = 1,2 , , p , (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups ( X 1 , X 2 , , X q ) , such as symmetric, generalized bisymmetric, and ( R , S ) -symmetric matrix groups. By this iterative method, for any initial matrix group ( X 1 ( 0 ) , X 2 ( 0 ) , , X q ( 0 ) ) , a solution group ( X 1 * , X 2 * , , X q * ) can be obtained within finite iteration steps in absence of round-off errors, and the minimum Frobenius norm solution or the minimum Frobenius norm least-squares solution group can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation solution group to a given matrix group ( X ¯ 1 , X ¯ 2 , , X ¯ q ) in the Frobenius norm can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the presented method.

Article information

Source
J. Appl. Math., Volume 2015, Special Issue (2015), Article ID 562529, 12 pages.

Dates
First available in Project Euclid: 15 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1429105060

Digital Object Identifier
doi:10.1155/2015/562529

Mathematical Reviews number (MathSciNet)
MR3332127

Citation

Toutounian, F.; Khojasteh Salkuyeh, D.; Mojarrab, M. LSMR Iterative Method for General Coupled Matrix Equations. J. Appl. Math. 2015, Special Issue (2015), Article ID 562529, 12 pages. doi:10.1155/2015/562529. https://projecteuclid.org/euclid.jam/1429105060


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References

  • T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems, Springer, London, UK, 1995.
  • T. Chen and L. Qiu, “${H}_{\infty }$ design of general multirate sampled-data control systems,” Automatica, vol. 30, no. 7, pp. 1139–1152, 1994.
  • L. P. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method, Academic Press, New York, NY, USA, 1961.
  • Z. Y. Li, Y. Wang, B. Zhou, and G. R. Duan, “Least squares solution with the minimum-norm to general matrix equations via iteration,” Applied Mathematics and Computation, vol. 215, no. 10, pp. 3547–3562, 2010.
  • B. C. Moore, “Principal component analysis in linear systems: controllability, observability, and model reduction,” IEEE Transactions on Automatic Control, vol. 26, no. 1, pp. 17–32, 1981.
  • L. Qiu and T. Chen, “Multirate sampled-data systems: all ${H}_{\infty }$ suboptimal controllers and the minimum entropy controller,” IEEE Transactions on Automatic Control, vol. 44, no. 3, pp. 537–550, 1999.
  • F. Toutounian and S. Karimi, “Global least squares method (Gl-LSQR) for solving general linear systems with several right-hand sides,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 452–460, 2006.
  • M. Dehghan and M. Hajarian, “An efficient algorithm for solving general coupled matrix equations and its application,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1118–1134, 2010.
  • 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.
  • 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, “On iterative solutions of general coupled matrix equations,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 2269–2284, 2006.
  • 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 generalized coupled SYLvester matrix equations over generalized reflexive solutions,” Applied Mathematical Modelling, vol. 36, no. 4, pp. 1589–1603, 2012.
  • I. Jonsson and B. Kågström, “Recursive blocked algorithm for solving triangular systems. I. One-sided and coupled Sylvester-type matrix equations,” Association for Computing Machinery. Transactions on Mathematical Software, vol. 28, no. 4, pp. 392–415, 2002.
  • I. Jonsson and B. Kågström, “Recursive blocked algorithms for solving triangular systems-Part II: two-sided and generalized Sylvester and Lyapunov matrix eqations,” ACM Transactions on Mathematical Software, vol. 28, no. 4, pp. 416–435, 2002.
  • Z. H. Peng, X. Y. Hu, and L. Zhang, “The bisymmetric solutions of the matrix equation image and its optimal approximation,” Linear Algebra and Its Applications, vol. 426, no. 2-3, pp. 583–595, 2007.
  • G. Starke and W. Niethammer, “SOR for $AX-XB=C$,” Linear Algebra and Its Applications, vol. 154–156, no. 3, pp. 355–375, 1991.
  • C. Tsui, “New approach to robust observer design,” International Journal of Control, vol. 47, no. 3, pp. 745–751, 1988.
  • F. Yin, G. Huang, and D. 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.
  • B. Zhou, G. Duan, and Z. Li, “Gradient based iterative algorithm for solving coupled matrix equations,” Systems & Control Letters, vol. 58, no. 5, pp. 327–333, 2009.
  • Z. Chen and L. Lu, “A gradient based iterative solutions for Sylvester tensor equations,” Mathematical Problems in Engineering, vol. 2013, Article ID 819479, 7 pages, 2013.
  • D. Chen, F. Yin, and G. X. Huang, “An iterative algorithm for the generalized reflexive solution of the matrix equations $AXB=E$, $CXD=F$,” Journal of Applied Mathematics, vol. 2012, Article ID 492951, 20 pages, 2012.
  • F. Ding, “Combined state and least squares parameter estimation algorithms for dynamic systems,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 38, no. 1, pp. 403–412, 2014.
  • F. Ding, Y. Liu, and B. Bao, “Gradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systems,” Proceedings of the Institution of Mechanical Engineers. Part I. Journal of Systems and Control Engineering, vol. 226, no. 1, pp. 43–55, 2012.
  • K. Wang, Z. Liu, and C. Xu, “A modified gradient based algorithm for solving matrix equations $AXB+C{X}^{T}D=F$,” Journal of Applied Mathematics, vol. 2014, Article ID 954523, 6 pages, 2014.
  • H. Yin and H. Zhang, “Least squares based iterative algorithm for the coupled sylvester matrix equations,” Mathematical Problems in Engineering, vol. 2014, Article ID 831321, 8 pages, 2014.
  • H. Zhang and F. Ding, “A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations,” Journal of the Franklin Institute, vol. 351, no. 1, pp. 340–357, 2014.
  • J. Zhou, 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.
  • 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, X. Liu, H. Chen, and G. Yao, “Hierarchical gradient based and hierarchical least squares based iterative parameter identification for CARARMA systems,” Signal Processing, vol. 97, pp. 31–39, 2014.
  • 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.
  • Y. Liu, F. Ding, and Y. Shi, “An efficient hierarchical identification method for general dual-rate sampled-data systems,” Automatica, vol. 50, no. 3, pp. 962–973, 2014.
  • B. Zhou and G. R. Duan, “On the generalized Sylvester mapping and matrix equations,” Systems & Control Letters, vol. 57, no. 3, pp. 200–208, 2008.
  • M. Dehghan and M. Hajarian, “Efficient iterative method for solving the second-order Sylvester matrix equation ${EVF}^{2}-AVF-CV=BW$,” IET Control Theory & Applications, vol. 3, no. 10, pp. 1401–1408, 2009.
  • 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. Hajarian and M. Dehghan, “The generalized centro-symmetric and least squares generalized centro-symmetric solutions of the matrix equation $AYB+{CY}^{T}D=E$,” Mathematical Methods in the Applied Sciences, vol. 34, no. 13, pp. 1562–1579, 2011.
  • S. Li and T. Huang, “LSQR iterative method for generalized coupled Sylvester matrix equations,” Applied Mathematical Modelling, vol. 36, no. 8, pp. 3545–3554, 2012.
  • M. Hajarian, “The generalized QMRCGSTAB algorithm for solving Sylvester-transpose matrix equations,” Applied Mathematics Letters, vol. 26, no. 10, pp. 1013–1017, 2013.
  • Y. Lin and V. Simoncini, “Minimal residual methods for large scale Lyapunov equations,” Applied Numerical Mathematics, vol. 72, pp. 52–71, 2013.
  • D. C. Fong and M. Saunders, “LSMR: an iterative algorithm for sparse least-squares problems,” SIAM Journal on Scientific Computing, vol. 33, no. 5, pp. 2950–2971, 2011.
  • M. Dehghan and M. Hajarian, “An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 571–588, 2008.
  • M. L. Liang, L. F. Dai, and S. F. Wang, “An iterative method for $\left(R,S\right)$-symmetric solution of matrix equation $AXB=C$,” Scientia Magna, vol. 4, no. 3, pp. 60–70, 2008.
  • 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.
  • G. Golub and W. Kahan, “Calculating the singular values and pseudo-inverse of a matrix,” Journal of the Society for Industrial and Applied Mathematics B: Numerical Analysis, vol. 2, no. 2, pp. 205–224, 1965.
  • A. Bouhamidi and K. Jbilou, “A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications,” Applied Mathematics and Computation, vol. 206, no. 2, pp. 687–694, 2008. \endinput