An iterative algorithm for η-(Anti)-Hermitian least-squares solutions of quaternion matrix equations

Fatemeh Panjeh Ali Beik, Salman Ahmadi-Asl

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)

Abstract

Recently, some research has been devoted to finding the explicit forms of the η- Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient leastsquares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

Original languageEnglish
Article number26
Pages (from-to)372-401
Number of pages30
JournalElectronic Journal of Linear Algebra
Volume30
Publication statusPublished - 2015
Externally publishedYes

Keywords

  • Convergence
  • Iterative algorithm
  • Preconditioner
  • Quaternion matrix equation
  • η- Hermitian matrix
  • η-Anti-Hermitian matrix

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