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

Salman Ahmadi-Asl, Fatemeh Panjeh Ali Beik

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


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 A1X+(A1X)ηH+B1YB1ηH+C1ZC1ηH=D1,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 η-Hermitian or η-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.

Original languageEnglish
Pages (from-to)95-127
Number of pages33
JournalJournal of Applied Mathematics and Computing
Issue number1-2
Publication statusPublished - 1 Feb 2017
Externally publishedYes


  • Convergence
  • Iterative algorithm
  • Preconditioner
  • Quaternion matrix equations
  • η-(anti)-Hermitian matrix


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