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

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• 作者：Salman Ahmadi-Asl ; Fatemeh Panjeh Ali Beik
• 关键词：Quaternion matrix equations ; \(\eta \) ; (anti) ; Hermitian matrix ; Iterative algorithm ; Preconditioner ; Convergence
• 刊名：Journal of Applied Mathematics and Computing
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：53
• 期：1-2
• 页码：95-127
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Computational Mathematics and Numerical Analysis; Appl.Mathematics/Computational Methods of Engineering; Theory of Computation; Mathematics of Computing;
• 出版者：Springer Berlin Heidelberg
• ISSN：1865-2085
• 卷排序：53

文摘

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 (X, Y, Z) 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. 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