JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (12): 66-74.doi: 10.6040/j.issn.1671-9352.0.2023.456

Previous Articles     Next Articles

Global quasi-minimal residual method for solving quaternion matrix equation AX+XB=C

WU Yuling1, ZHENG Jiali1, KE Yifen1,2,3*, XU Xiaofang1   

  1. 1. School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, Fujian, China;
    2. Key Laboratory of Analytical Mathematical and Applications(Ministry of Education), Fujian Normal University, Fuzhou 350117, Fujian, China;
    3. Center for Applied Mathematics of Fujian province(FJNU), Fuzhou 350117, Fujian, China
  • Published:2025-12-10

PDF (PC)

31

Abstract: In this paper, the quaternion nonsymmetric Lanczos method is proposed. Based on this method, the global quaternion quasi-minimum residual method is established for solving the Sylvester quaternion matrix equation AX+XB=C, which can effectively reduce the storage space during algorithm execution. Further, some numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed method.

Key words: quaternion, Sylvester matrix equation, nonsymmetric Lanczos method, global quaternion quasi-minimal residual method

CLC Number: 

  • O241
[1] SUN Y F, CHEN S Y, YIN B C. Color face recognition based on quaternion matrix representation[J]. Pattern Recognition Letters, 2011, 32(4):597-605.
[2] ZOU C M, KOU K I, WANG Y L. Quaternion collaborative and sparse representation with application to color face recognition[J]. IEEE Transactions on Image Processing, 2016, 25(7):3287-3302.
[3] JIANG T S. An algorithm for quaternionic linear equations in quaternionic quantum theory[J]. Journal of Mathematical Physics, 2004, 45(11):4218-4222.
[4] CHEN Y Y, XIAO X L, ZHOU Y C. Low-rank quaternion approximation for color image processing[J]. IEEE Transactions on Image Processing, 2019, 29:1426-1439.
[5] LANCZOS C. Solution of systems of linear equations by minimized iterations[J]. Journal of Research of the National Bureau of Standards, 1952, 49(1):33-53.
[6] SAAD Y. Practical use of some Krylov subspace methods for solving indefinite and nonsymmetric linear systems[J]. SIAM Journal on Scientific and Statistical Computing, 1984, 5(1):203-228.
[7] SAAD Y, SCHULTZ M H. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems[J]. SIAM Journal on Scientific and Statistical Computing, 1986, 7:856-869.
[8] FREUND R W, NACHTIGAL N M. QMR: a quasi-minimal residual method for non-Hermitian linear systems[J]. Numerische Mathematik, 1991, 60(1):315-339.
[9] JBILOU K, MESSAOUDI A, SADOK H. Global FOM and GMRES algorithms for matrix equations[J]. Applied Numerical Mathematics, 1999, 31(1):49-63.
[10] LI S K, HUANG T Z. A shifted complex global Lanczos method and the quasi-minimal residual variant for the Stein-conjugate matrix equation X+A(-overX)B=C[J]. Journal of Computational and Applied Mathematics, 2019, 357:1-11.
[11] JIA Z G, NG M K. Structure preserving quaternion generalized minimal residual method[J]. SIAM Journal on Matrix Analysis and Applications, 2021, 42(2):616-634.
[12] LI T, WANG Q W. Structure preserving quaternion full orthogonalization method with applications[J]. Numerical Linear Algebra with Applications, 2023:e2495.
[13] LI Tao, WANG Qingwen, ZHANG Xinfang. Gl-QFOM and Gl-QGMRES: two efficient algorithms for quaternion linear systems with multiple right-hand sides[EB/OL].(2023-07-24)[2024-07-11]. http: //doi.org/10.48550/arXiv.2308.13214.
[14] RODMAN L. Topics in quaternion linear algebra[M]. Princeton: Princeton University Press, 2014:37.
[15] GHILONI R, MORETTI V, PEROTTI A. Continuous slice functional calculus in quaternionic Hilbert spaces[J]. Reviews in Mathematical Physics, 2013, 25(4):1350006.
[16] BOUYOULI R, JBILOU K, SADAKA R, et al. Convergence properties of some block Krylov subspace methods for multiple linear systems[J]. Journal of Computational Applied Mathematics, 2006, 196(2):498-511.
[17] University of Florida Sparse Matrix Collection web page[EB/OL].(2011-03-11)[2024-07-11]. https://www.cise.ufl.edu/research/sparse/matrices/list_by_id.html.
[1] FAN Xue-ling, LI Ying, ZHAO Jian-li, LIU Zhi-hong. A new method for solving quaternion linear system [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(4): 55-64.
[2] LIAO Qing-qing, GUO Ji-dong, ZHANG Liang. Number of homomorphisms between intra-commutative p-groups and quaternion groups [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(2): 45-49.
[3] DING Wen-xu, LI Ying, WANG Dong, ZHAO Jian-li. Solutions of the quaternion matrix equation based on semi-tensor product of matrices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(6): 103-110.
[4] CAO Mei-hong, ZHANG Jian-hua. Jordan and Lie centralizers on quaternion rings [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(12): 67-71.
[5] LI Feng-jiao, GAO Bai-jun. Number of homomorphisms between two classes of non-abelian finite groups [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(12): 25-29.
[6] YANG Liu, MA Jing. Armendariz property for a group ring over the Hamiltons quaternion ring [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(1): 1-4.
[7] LING Si-Chao, CHENG Wue-Han, WEI Mu-Sheng. On Hermitian solutions to general linear quaternionic matrix equations [J]. J4, 2008, 43(12): 1-4.
[8] ZHAO Jian-li,LI Ying and ZHANG Li-mei . QR decomposition and the equality constrained least squares problem over a quaternion field [J]. J4, 2007, 42(6): 65-68 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] LIU Ting-ting, CHEN Zhi-yong, LI Xiao-qin*, YANG Wen-zhi. The Berry-Esseen bound for the sequence of #br# negatively associated random variables#br#[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 101 -106 .
[2] LI Shi-long,ZHANG Yun-feng . Error analysis of the rational interpolation based on arithmetic average difference quotient[J]. J4, 2007, 42(10): 106 -110 .
[3] LI Min1,2, LI Qi-qiang1. Observer-based sliding mode control of uncertain singular time-delay systems#br#[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 37 -42 .
[4] ZHANG Xiao, ZHANG Yu-zhen, LIU Jun-jun, LIU Hong-lei*. Differential expression of aromatic hydrocarbon degradation related proteins in Pseudomonas putida F1 grown on succinate, toluene or ethylbenzene[J]. J4, 2013, 48(05): 14 -19 .
[5] WANG Wen-ang . Maximal general Armendariz subrings of matrix rings[J]. J4, 2007, 42(8): 74 -78 .
[6] PANG Guan-song, ZHANG Li-sha, JIANG Sheng-yi*, KUANG Li-min, WU Mei-ling. A multi-level clustering approach based on noun phrases for search results[J]. J4, 2010, 45(7): 39 -44 .
[7] WANG Dun-Xin. Some conditions for finite groups to be solvable[J]. J4, 2009, 44(8): 35 -38 .
[8] DONG Xin-mei . The average order and mean value estimates ofthe error term of function δrk(n)[J]. J4, 2006, 41(5): 91 -94 .
[9] PAN Zhen-kuan,WEI Wei-bo,ZHANG Hai-tao . Variational models for image diffusion based on gradient and Laplacian[J]. J4, 2008, 43(11): 11 -16 .
[10] YANG Bi-Cheng. A Hilbert-type integral inequality with the homogeneous kernel of degree zero[J]. J4, 2010, 45(2): 103 -106 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd