An Efficient Method for Solving a Quaternionic Least-Squares Problem

  • Alireza Shojaei-Fard
  • Ali Nakhaei Amroudi
Original Paper


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.


Quaternion matrices Matrix equation Least-squares 

Mathematics Subject Classification

Primary 15A24 Secondary 15B33 65F10 



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.


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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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