## Abstract

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 language | English |
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Pages (from-to) | 95-127 |

Number of pages | 33 |

Journal | Journal of Applied Mathematics and Computing |

Volume | 53 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 1 Feb 2017 |

Externally published | Yes |

## Keywords

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