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LSMR Iterative Method for General Coupled Matrix Equations

Author

Listed:
  • F. Toutounian
  • D. Khojasteh Salkuyeh
  • M. Mojarrab

Abstract

By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations ∑k=1qAikXkBik=Ci, i = 1,2, …, p, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups (X1, X2, …, Xq), such as symmetric, generalized bisymmetric, and (R, S)‐symmetric matrix groups. By this iterative method, for any initial matrix group (X1(0),X2(0),…,Xq(0)), a solution group (X1*,X2*,…,Xq*) 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.

Suggested Citation

  • F. Toutounian & D. Khojasteh Salkuyeh & M. Mojarrab, 2015. "LSMR Iterative Method for General Coupled Matrix Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2015(1).
    • Handle: RePEc:wly:jnljam:v:2015:y:2015:i:1:n:562529
    DOI: 10.1155/2015/562529
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    References listed on IDEAS

    as
    1. Jituan Zhou & Ruirui Wang & Qiang Niu, 2012. "A Preconditioned Iteration Method for Solving Sylvester Equations," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-12, July.
    2. Kanmin Wang & Zhibing Liu & Chengfeng Xu, 2014. "A Modified Gradient Based Algorithm for Solving Matrix Equations," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-6, February.
    3. Feng Yin & Guang-Xin Huang, 2012. "An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    4. Kanmin Wang & Zhibing Liu & Chengfeng Xu, 2014. "A Modified Gradient Based Algorithm for Solving Matrix Equations AXB + CXTD = F," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    5. Deqin Chen & Feng Yin & Guang-Xin Huang, 2012. "An Iterative Algorithm for the Generalized Reflexive Solution of the Matrix Equations AXB = E, CXD = F," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    6. Feng Yin & Guang-Xin Huang, 2012. "An Iterative Algorithm for the Least Squares Generalized Reflexive Solutions of the Matrix Equations AXB = E,CXD = F," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    7. Feng Yin & Guang-Xin Huang, 2012. "An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-28, August.
    8. Jituan Zhou & Ruirui Wang & Qiang Niu, 2012. "A Preconditioned Iteration Method for Solving Sylvester Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
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