Advertisement

Numerical Algorithms

, Volume 64, Issue 3, pp 455–480 | Cite as

The reflexive least squares solutions of the matrix equation \(A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l=C\)with a submatrix constraint

  • Zhuohua Peng
Original Paper

Abstract

In this paper, an efficient algorithm is presented for minimizing \(\|A_1X_1B_1 + A_2X_2B_2+\cdots +A_lX_lB_l-C\|\) where \(\|\cdot \|\) is the Frobenius norm, \(X_i\in R^{n_i \times n_i}(i=1,2,\cdots ,l)\) is a reflexive matrix with a specified central principal submatrix \([x_{ij}]_{r\leq i,j\leq n_i-r}\). The algorithm produces suitable \([X_1,X_2,\cdots ,X_l]\) such that \(\|A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l-C\|=\min \) within finite iteration steps in the absence of roundoff errors. We show that the algorithm is stable any case. The algorithm requires little storage capacity. Given numerical examples show that the algorithm is efficient.

Keywords

Matrix equation Central principal submatrix Reflexive solutions Submatrix constraint Least squares solution 

Mathematics Subject Classifications (2010)

65F10 65F30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baksalary, J.K., Kala, R.: The matrix equation \(AXB+CYD=E\). Linear Algebra Appl. 30, 141–147 (1980).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Xu, G., Wei, M., Zheng, D.: On solutions of matrix equation \(AXB+CYD=F\). Linear Algebra Appl. 279, 93–109 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chu, K.E.: Singular value and generalized value decompositions and the solution of linear matrx equations. Linear Algebra Appl. 87, 83–98 (1987)CrossRefGoogle Scholar
  4. 4.
    Shim, S.-Y., Chen, Y.: Least squares solution of matrix equation \(AXB^*+CYD^*=E\). SIAM J. Matrix Anal. Appl. 3, 8002–8008 (2003)MathSciNetGoogle Scholar
  5. 5.
    Wang, Q.W.: A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity. Linear Algebra Appl. 384, 43–54 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Wang, Q.W., Chang, H.X., Lin, C.Y.: \(P-\)(skew) symmetric common solutions to a pair of quaternion matrix equations. Appl. Math. Comput. 195, 721–732 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Wang, Q.W.,Wu, Z.C., Lin, C.Y.: Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications. Appl. Math. Comput. 182, 1755–1764 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Liao, A.P., Bai, Z.Z., Lei, Y.: Best approximate solution of matrix equation \(AXB+CYD=E\). SIAM J. Matrix Anal. Appl. 27, 675–688 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Liao, A.P., Lei, Y.: Least squares solutions of matrix inverse problem for bisymmetric matrices with a submatrix constraint. Numer. Linear Algebr. Appl. 14, 425–444 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chen, H.C.: Generalized reflexive matrices: special properties and applications. SIAM J. Matrix Anal. Appl. 19, 140–153 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chen, H.C., Sameh, A.: Numerical linear algebra algorithms on the ceder system. In: Noor, A.K. (ed.) Parallel Computations and their Impact on Mechanics, Vol. 86, pp. 101–125. The American Society of Mechanical Engineers, New York (1987)Google Scholar
  12. 12.
    Chen, H.C.: The SAS Domain Decomposition Method for Structural Analysis, CSRD Teach, report 754. Center for Supercomputing Research and Development, University of Illinois, Urbana, IL (1988)Google Scholar
  13. 13.
    Peng, Z.H., Liu, J.W.: An iterative method for the reflexive solutions of the matrix equation \(A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l=C\). In: Proceedings of the Third InternationalWorkshop on Matrix Analysis and Applications, Hangzhou, China, Vol. 2, pp. 198–201 (2009)Google Scholar
  14. 14.
    Bai, Z.J.: The inverse eigen problem of centrosymmetric matrices with a submatrix constraint and its approximation. SIAM J. Matrix Anal. Appl. 26, 1100–1114 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Yin, Q.X.: Construction of real antisymmetric and bi-antisymmetric matrices with prescribed spectrum data. Linear Algebra Appl. 389, 95–106 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Deift, P., Nanda, T.: On the determination of a tridiagonal matrix from its spectrum and a submatrix. Linear Algebra Appl. 60, 43–55 (1984)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gong, L.S., Hu, X.Y., Zhang, L.: The expansion problem of anti-symmetric matrix under a linear consraint and the optimal approximation. J. Comput. Appl. Math. 197, 44–52 (2006)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Zhao, L.J., Hu, X.Y., Zhang, L.: Least squares solutions to \(AX=B\) for bi-symmetric matrix under a central principal submatrix constraint and the optimal approximation. Linear Algebra Appl. 428, 871–880 (2008)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Wang, R.S.: Functional Analysis and Optimization Theory. Beijing Univ of Aeronautics Astronautics, Beijing (2003)Google Scholar
  20. 20.
    Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia, PA (1995)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.School of Mathematics and Computing ScienceHunan University of Science and TechnologyXiangtanPeople’s Republic of China

Personalised recommendations