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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2026, Vol. 43 ›› Issue (1): 15-37.doi: 10.3969/j.issn.1005-3085.2026.01.002cstr: 32411.14.cjem.CN61-1269/O1.2026.01.002

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一类四元数Sylvester共轭张量方程的CGLS算法及其应用

胡晶晶1,2,  柯艺芬2,  马昌凤3   

  1. 1. 华北水利水电大学数学与统计学院，郑州 450046

    2. 福建师范大学数学与统计学院，福州 350117

    3. 厦门工学院人工智能学院，厦门 361021

  • 收稿日期:2023-04-13 接受日期:2023-07-13 出版日期:2026-02-15 发布日期:2026-04-15
  • 通讯作者: 柯艺芬 E-mail: keyifen@fjnu.edu.cn
  • 基金资助:
    国家自然科学基金 (11901098)；福建省自然科学基金 (2020J05034).

CGLS Algorithm for a Class of Quaternion Sylvester Conjugate Tensor Equations and Its Application

HU Jingjing1,2,  KE Yifen2,  MA Changfeng3   

  1. 1. School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046
    2. School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117\
    3. School of Artificial Intelligence, Xiamen Institute of Technology, Xiamen 361021
  • Received:2023-04-13 Accepted:2023-07-13 Online:2026-02-15 Published:2026-04-15
  • Contact: Y. Ke. E-mail address: keyifen@fjnu.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (11901098); the Natural Science Foundation of Fujian Province (2020J05034).

摘要：

提出张量形式的共轭梯度最小二乘算法求解一类四元数Sylvester共轭张量方程。证明在不计舍入误差的情况下，所提方法可在有限迭代步内获得张量方程组的最小二乘解。进一步，通过选择特殊类型的初始张量，可获得方程组的唯一极小Frobenius范数最小二乘解。数值实验验证了该算法在彩色视频恢复中的可行性和有效性。

关键词: 四元数Sylvester张量方程, CGLS算法, 彩色视频

Abstract:

In this paper, the conjugate gradient least squares (CGLS) algorithm based on tensor form is proposed to solve a class of quaternion Sylvester conjugate tensor equations. It is proved that the proposed algorithm can obtain the least squares solution in a finite iterative steps with the absence of round-off errors. Furthermore, by selecting the special initial tensors, the unique minimum Frobenius norm least squares solution can be obtained. Numerical experiments demonstrate the feasibility and effectiveness of the proposed algorithm in color video restoration.

Key words: quaternion Sylvester tensor equation, CGLS algorithm, color video

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