Journal of Applied Mathematics and Computing

, Volume 53, Issue 1–2, pp 95–127 | Cite as

Iterative algorithms for least-squares solutions of a quaternion matrix equation

Original Research


This paper deals with developing four efficient algorithms (including the conjugate gradient least-squares, least-squares with QR factorization, least-squares minimal residual and Paige algorithms) to numerically find the (least-squares) solutions of the following (in-) consistent quaternion matrix equation
$$\begin{aligned} {A_1}X + {\left( {{A_1}X} \right) ^{\eta H}} + {B_1}YB_1^{\eta H} + {C_1}ZC_1^{\eta H} = {D_1}, \end{aligned}$$
in which the coefficient matrices are large and sparse. More precisely, we construct four efficient iterative algorithms for determining triple least-squares solutions (XYZ) such that X may have a special assumed structure, Y and Z can be either \(\eta \)-Hermitian or \(\eta \)-anti-Hermitian matrices. In order to speed up the convergence of the offered algorithms for the case that the coefficient matrices are possibly ill-conditioned, a preconditioned technique is employed. Some numerical test problems are examined to illustrate the effectiveness and feasibility of presented algorithms.


Quaternion matrix equations \(\eta \)-(anti)-Hermitian matrix Iterative algorithm Preconditioner Convergence 

Mathematics Subject Classification

Primary 15A24 Secondary 65F10 15B33 



The authors would like to express their heartfelt thanks to anonymous referees for their valuable suggestions and constructive comments which have improved the quality of the paper.


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Copyright information

© Korean Society for Computational and Applied Mathematics 2015

Authors and Affiliations

  1. 1.Department of MathematicsVali-e-Asr University of RafsanjanRafsanjanIran

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