An Efficient Method for Solving a Quaternionic Least-Squares Problem

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Abstract

The quaternions play a crucial role in various research areas such as computer graphic, integrated navigation systems, color image processing, signal processing, quantum physics etc. Consider the quaternion least-squares problem
$$\begin{aligned} \min \,\,\left\| {\sum \limits _{j = 1}^q { {\left( {{A_{ij}}{X_j}{B_{ij}} + {C_{ij}}X_j^{\eta H}{D_{ij}} - {E_i}} \right) } } } \right\| ,\qquad i=1,2,\ldots ,p, \end{aligned}$$
where \(A_{ij},\,B_{ij},\,C_{ij},\,D_{ij}\in {\mathbb {Q}}^{m\times m}\), \({\mathbb {Q}}\) is the division algebra of quaternions, \(X=(X_1,\ldots ,X_q)\) and \(X^{\eta H}=\eta X^H\eta \) where \(\eta =\{\mathbf{i},\mathbf{j},\mathbf{k}\}\) are quaternion units. In this paper, we propose an efficient method to solve the above least-squares problem. We present the convergence analysis in detail. The convergence analysis show that the proposed algorithm converges in a finite number of iterations. Finally, to show the efficiency of our method two numerical examples are presented.

Keywords

Quaternion matrices Matrix equation Least-squares 

Mathematics Subject Classification

Primary 15A24 Secondary 15B33 65F10 

Notes

Acknowledgements

The authors would like to express their heartfelt gratitude to two anonymous referees for constructive criticism, which have helped us to improve the presentation. The authors are also very indebted to Dr. Santanu Saha Ray (Editor in Chief) for managing the review process of this paper.

References

  1. 1.
    Ahmadi-Asl, S., Beik, F.P.A.: Iterative algorithms for least-squares solution of a quaternion matrix equation. J. Appl. Math. Comput. 53(1), 95–127 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ahmadi-Asl, S., Beik, F.P.A.: An efficient iterative algorithm for quaternionic least-squares problems over the generalized \(\eta \)-(anti-)bi-Hermitian matrices. Linear Multilinear Algebra (2016).  https://doi.org/10.1080/03081087.2016.1255172MATHGoogle Scholar
  3. 3.
    Beik, F.P.A., Ahmadi-Asl, S.: An iterative algorithm for \(\eta \)-(anti)-Hermitian least-squares solutions of quaternion matrix equation. Electron. J. Linear Algebra 30, 372–401 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bihan, N.L., Sangwine, S.J.: Color image decomposition using quaternion singular value decomposition. In: International Conference on Visual Information Engineering, 2003. VIE 2003. IET (2003)Google Scholar
  5. 5.
    Bihan, N.L., Mars, J.: Singular value decomposition of matrices: a new tool for vector-sensor signal processing. Signal Process. 84(7), 1177–1199 (2004)CrossRefMATHGoogle Scholar
  6. 6.
    Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)CrossRefMATHGoogle Scholar
  7. 7.
    Conway, J.H., Smith, D.A.: On Quaternions and Octonions. A K Peters Ltd, Natick (2003)MATHGoogle Scholar
  8. 8.
    Leo, S.D., Scolarici, G.: Right eigenvalue equation in quaternionic quantum mechanics. J. Phys. A 33, 2971–2995 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Li, N., Wang, Q.W.: Iterative algorithm for solving a class of quaternion matrix equation over the generalized \((P,Q)\)-reflexive matrices. Abstr. Appl. Anal. (2013).  https://doi.org/10.1155/2013/831656
  10. 10.
    Ling, S., Wang, M., Wei, M.: Hermitian tridiagonal solution with the least norm to quaternionic least squares problem. Comput. Phys. Commun. 181(3), 481–488 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Rogers, R.M.: Applied Mathematics in Integrated Navigation Systems. American Institute of Aeronautics and Astronautics, Reston (2003)Google Scholar
  12. 12.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)CrossRefMATHGoogle Scholar
  13. 13.
    University of Florida Sparse Matrix Collection. http://www.cise.ufl.edu/research/sparse/matrices
  14. 14.
    Vince, J.: Quaternions for Computer Graphics. Springer, Berlin (2011)CrossRefMATHGoogle Scholar
  15. 15.
    Vince, J.: Rotation Transforms for Computer Graphics. Springer, Berlin (2011)CrossRefMATHGoogle Scholar
  16. 16.
    Wang, R.S.: Functional Analysis and Optimization Theory. Beijing University of Aeronautics Astronautics, Beijing (2003)Google Scholar

Copyright information

© Springer (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceImam Hossein Comprehensive UniversityTehranIran

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