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Generalized conjugate direction algorithm for solving the general coupled matrix equations over symmetric matrices

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Abstract

Symmetric solutions of the linear matrix equations have wide applications in both mechanical and electrical engineering. In this work, an analytic study of the generalized conjugate direction (CD) algorithm for finding the symmetric solution group (X 1,X 2,...,X m ) of the general coupled matrix equations

$$\sum\limits_{j=1}^{m}A_{ij}X_{j}B_{ij}=C_{i},\qquad i=1,2,...,n, $$

is performed. We show that the generalized CD algorithm can compute the (least Frobenius norm) symmetric solution group of the general coupled matrix equations for any (special) initial symmetric matrix group within a finite number of iterations in the absence of round-off errors. In order to illustrate the effectiveness of the generalized CD algorithm, two numerical examples are finally given.

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References

  1. Liao, A., Bai, Z., Lei, Y.: Best approximate solution of matrix equation A X B + C Y D = E. SIAM J. Matrix Anal. Appl. 27, 675–688 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Yuan, Y., Liu, H.: An iterative updating method for undamped structural systems. Meccanica 47, 699–706 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Yuan, Y., Liu, H.: An iterative updating method for damped structural systems using symmetric eigenstructure assignment. J. Comput. Appl. Math. 256, 268–277 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jiang, J., Yuan, Y.: A new model updating method for damped structural systems. Math. Meth. Appl. Sci. 32, 2138–2147 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kȧgström, B., van Dooren, P.: A generalized state-space approach for the additive decomposition of a transfer matrix. J. Numer. Linear Algebra Appl. 1, 165–181 (1992)

    MathSciNet  Google Scholar 

  6. Kȧgström, B., Westin, L.: Generalized Schur methods with condition estimators for solving the generalized Sylvester equation. IEEE Trans. Autom. Contr. 34, 745–751 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kȧgström, B., Poromaa, P.: Distributed and shared memory block algorithms for the triangular Sylvester equation with sep−1 estimators. SIAM J. Matrix Anal. Appl. 13, 90–101 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kȧgström, B., Poromaa, P.: LAPACK-Style algorithms and software for solving the generalized sylvester equation and estimating the separation between regular matrix pairs. ACM Trans. Math. Softw. 22, 78–103 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Varga, A.: Robust pole assignment via Sylvester equation based state feedback parametrization. In: Proceedings of the 2000 IEEE International Symposium on Computer-Aided Control System Design, Alsaka, USA, pp 13–18 (2000)

  10. Zhang, Y.N., Jiang, D.C., Wang, J.: A recurrent neural network for solving Sylvester equation with time-varying coefficients. IEEE Trans. Neural Netw. 13, 1053–1063 (2002)

    Article  Google Scholar 

  11. Shahzad, A., Jones, B.L., Kerrigan, E.C., Constantinides, G.A.: An efficient algorithm for the solution of a coupled Sylvester equation appearing in descriptor systems. Automatica 47, 244–248 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wang, X., Li, Y., Dai, L.: On Hermitian and skew-Hermitian splitting iteration methods for the linear matrix equation A X B = C. Comput. Math. Appl. 65, 657–664 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, H., Gao, Z., Zhao, D.: Least squares solutions of the matrix equation A X B + C Y D = E with the least norm for symmetric arrowhead matrices. Appl. Math. Comput. 226, 719–724 (2014)

    MathSciNet  Google Scholar 

  14. Wang, Q.W., He, Z.H.: Systems of coupled generalized Sylvester matrix equations. Automatica 50, 2840–2844 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, Q.W., He, Z.H.: Solvability conditions and general solution for the mixed Sylvester equations. Automatica 49, 2713–2719 (2013)

    Article  MathSciNet  Google Scholar 

  16. Lee, S.G., Vu, Q.P.: Simultaneous solutions of matrix equations and simultaneous equivalence of matrices. Linear Algebra Appl. 437, 2325–2339 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wimmer, H.K.: Consistency of a pair of generalized Sylvester equations. IEEE Trans. Autom. Control 39, 1014–1016 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huang, N., Ma, C.: The modified conjugate gradient methods for solving a class of generalized coupled Sylvester-transpose matrix equations. Comput. Math. Appl. 67, 1545–1558 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ding, F., Chen, T.: Iterative least squares solutions of coupled Sylvester matrix equations. Syst. Control Lett. 54, 95–107 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ding, F., Chen, T.: On iterative solutions of general coupled matrix equations. SIAM J. Control. Optim. 44, 2269–2284 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ding, F., Chen, T.: Gradient based iterative algorithms for solving a class of matrix equations. IEEE Trans. Autom. Control 50, 1216–1221 (2005)

    Article  MathSciNet  Google Scholar 

  22. Hajarian, M.: Developing BiCOR and CORS methods for coupled Sylvester-transpose and periodic Sylvester matrix equations. Appl. Math. Model. 39, 6073–6084 (2015)

    Article  MathSciNet  Google Scholar 

  23. Hajarian, M.: Matrix GPBiCG algorithms for solving the general coupled matrix equations. IET Control Theory Appl. 9, 74–81 (2015)

    Article  MathSciNet  Google Scholar 

  24. Hajarian, M.: Matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations. J. Franklin Inst. 350, 3328–3341 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hajarian, M.: Matrix form of the CGS method for solving general coupled matrix equations. Appl. Math. Lett. 34, 37–42 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hajarian, M.: The generalized QMRCGSTAB algorithm for solving Sylvester-transpose matrix equations. Appl. Math. Lett. 26, 1013–1017 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Huang, G.X., Wu, N., Yin, F., Zhou, Z.L., Guo, K.: Finite iterative algorithms for solving generalized coupled Sylvester systems-Part I: One-sided and generalized coupled Sylvester matrix equations over generalized reflexive solutions. Appl. Math. Model. 36, 1589–1603 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yin, F., Huang, G.X., Chen, D.Q.: Finite iterative algorithms for solving generalized coupled Sylvester systems-Part II: Two-sided and generalized coupled Sylvester matrix equations over reflexive solutions. Appl. Math. Model. 36, 1604–1614 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dehghan, M., Hajarian, M.: Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations. Appl. Math. Model. 35, 3285–3300 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Dehghan, M., Hajarian, M.: On the generalized reflexive and anti-reflexive solutions to a system of matrix equations. Linear Algebra Appl. 437, 2793–2812 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hegedűs, C.J.: Generating conjugate directions for arbitrary matrices by matrix equations I. Comput. Math. Appl. 21, 71–85 (1991)

    Article  MathSciNet  Google Scholar 

  32. Hegedűs, C.J.: Generating conjugate directions for arbitrary matrices by matrix equations II. Comput. Math. Appl. 21, 87–94 (1991)

    Article  MathSciNet  Google Scholar 

  33. Dehghan, M., Hajarian, M.: The general coupled matrix equations over generalized bisymmetric matrices. Linear Algebra Appl. 432, 1531–1552 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hajarian, M.: Developing the CGLS algorithm for the least squares solutions of the general coupled matrix equations. Math. Methods Appl. Sci. 37, 2782–2798 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Peng, Z.: The reflexive least squares solutions of the matrix equation A 1 X 1 B 1 + A 2 X 2 B 2+… + A l X l B l = C with a submatrix constraint. Numerical Algorithms 64, 455–480 (2013)

    Article  MathSciNet  Google Scholar 

  36. Peng, Z., Xin, H.: The reflexive least squares solutions of the general coupled matrix equations with a submatrix constraint. Appl. Math. Comput. 225, 425–445 (2013)

    MathSciNet  MATH  Google Scholar 

  37. Peng, Z.: The (R,S)-symmetric least squares solutions of the general coupled matrix equations. Linear and Multilinear Algebra 63, 1086–1105 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Sheng, X., Chen, G.: An iterative method for the symmetric and skew symmetric solutions of a linear matrix equation A X B + C Y D = e. J. Comput. Appl. Math. 233, 3030–3040 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Huang, G.X., Yin, F., Guo, K.: An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation A X B = c. J. Comput. Appl. Math. 212, 231–244 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ding, F., Zhang, H.: Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems. IET Control Theory Appl. 8, 1588–1595 (2014)

    Article  MathSciNet  Google Scholar 

  41. Zhang, H., Ding, F.: A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations. J. Franklin Inst. 351, 340–357 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ding, F., Wang, Y., Ding, J.: Recursive least squares parameter identification algorithms for systems with colored noise using the filtering technique and the auxilary model. Digital Signal Process. 37, 100–108 (2015)

    Article  Google Scholar 

  43. Ding, F., Wang, X., Chen, Q., Xiao, Y.: Recursive least squares parameter estimation for a class of output nonlinear systems based on the model decomposition. Circuits, Systems and Signal Processing. doi:10.1007/s00034-015-0190-6

  44. Guennouni, A.E., Jbilou, K., Riquet, A.J.: Block Krylov subspace methods for solving large Sylvester equations. Numer. Algo. 29, 75–96 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lin, Y., Simoncini, V.: Minimal residual methods for large scale Lyapunov equations. Appl. Numer. Math. 72, 52–71 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  46. Shank, S.D., Simoncini, V.: Krylov subspace methods for large scale constrained Sylvester equations. SIAM J. Matrix Anal. Appl. 34, 1448–1463 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kressner, D., Sirković, P.: Truncated low-rank methods for solving general linear matrix equations. Numerical Linear Algebra with Applications 22, 564–583 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  48. Dehghan, M., Hajarian, M.: An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation. Appl. Math. Comput. 202, 571–588 (2008)

    MathSciNet  MATH  Google Scholar 

  49. Dehghan, M., Hajarian, M.: An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices. Appl. Math. Model. 34, 639–654 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  50. Dehghan, M., Hajarian, M.: An iterative algorithm for solving a pair of matrix equations A Y B = E, C Y D = F over generalized centro-symmetric matrices. Computers & Mathematics with Applications 56, 3246–3260 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  51. Dehghan, M., Hajarian, M.: Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 = C. Math. Comput. Model. 49, 1937–1959 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. Dehghan, M., Hajarian, M.: The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations. Rocky Mountain Journal of Mathematics 40, 825–848 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, General Campus, Evin, Tehran, 19839, Iran

    Masoud Hajarian

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  1. Masoud Hajarian
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Correspondence to Masoud Hajarian.

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This work was supported by Iran National Science Foundation (INSF) and the author thanks Iran National Science Foundation (INSF) for this support.

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Hajarian, M. Generalized conjugate direction algorithm for solving the general coupled matrix equations over symmetric matrices. Numer Algor 73, 591–609 (2016). https://doi.org/10.1007/s11075-016-0109-8

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  • DOI: https://doi.org/10.1007/s11075-016-0109-8

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