This paper focuses on the solution of the reduced biquaternion equation $ AX+XB = C $ using the E-extra iteration method. By utilizing the complex decomposition of a reduced biquaternion matrix, we transform the equation into a complex matrix equation. Subsequently, we analyze the convergence of this method and provide guidelines for selecting optimal parameters. Finally, numerical examples are presented to demonstrate the efficacy of our algorithm.
Citation: Jiaxin Lan, Jingpin Huang, Yun Wang. An E-extra iteration method for solving reduced biquaternion matrix equation $ AX+XB = C $[J]. AIMS Mathematics, 2024, 9(7): 17578-17589. doi: 10.3934/math.2024854
This paper focuses on the solution of the reduced biquaternion equation $ AX+XB = C $ using the E-extra iteration method. By utilizing the complex decomposition of a reduced biquaternion matrix, we transform the equation into a complex matrix equation. Subsequently, we analyze the convergence of this method and provide guidelines for selecting optimal parameters. Finally, numerical examples are presented to demonstrate the efficacy of our algorithm.
| [1] | F. R. Gantmacher, J. L. Brenner, Applications of the Theory of Matrices, New York: Dover Publications, 2005. |
| [2] | R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge: Cambridge University Press, 2012. http://dx.doi.org/10.1017/CBO9781139020411 |
| [3] |
J. S. Respondek, Matrix black box algorithms-a survey, Bull. Pol. Acad. Sci. Tech. Sci., 70 (2022), e140535. http://dx.doi.org/10.24425/bpasts.2022.140535 doi: 10.24425/bpasts.2022.140535
|
| [4] |
K. F. Wang, Universal thermal balance matrix equation of thermodynamic system with external steam cooler, Proc. CSEE, 39 (2019), 7280–7285. http://dx.doi.org/10.13334/j.0258-8013.pcsee.182450 doi: 10.13334/j.0258-8013.pcsee.182450
|
| [5] |
V. B. Larin, Solutions of matrix equations in problems of mechanics and control, Int. Appl. Mechan., 45 (2009), 847–872. http://dx.doi.org/10.1007/s10778-009-0232-5 doi: 10.1007/s10778-009-0232-5
|
| [6] |
S. Zhou, L. Wang, M. H. Han, Equation and its application to inverse problem of vibration theory, Appl. Math. Mechan., 34 (2013), 306–317. http://dx.doi.org/10.3879/j.issn.1000-0887.2013.03.010 doi: 10.3879/j.issn.1000-0887.2013.03.010
|
| [7] |
D. Wang, Y. Li, W. X. Ding, The least squares Bisymmetric solution of quaternion matrix equation $AXB = C$, AIMS Math., 6 (2021), 13247–13257. http://dx.doi.org/10.3934/math.2021766 doi: 10.3934/math.2021766
|
| [8] |
Q. X. He, L. S. Hou, J. Y. Zhou, On the solution of fuzzy Sylvester matrix equation, Soft Comput., 22 (2018), 6515–6523. http://dx.doi.org/10.1007/s00500-017-2702-8 doi: 10.1007/s00500-017-2702-8
|
| [9] |
S. K. Mitra, Common solutions to a pair of linear matrix equations $A_1XB_1$ = $C_1$ and $A_2XB_2$ = $C_2$, Math. Proc. Cambridge Philos. Soc., 74 (1973), 213–216. http://dx.doi.org/10.1017/S030500410004799X doi: 10.1017/S030500410004799X
|
| [10] |
A. B. Özgüler, N. Akar, A common solution to a pair of linear matrix equations over a principal domain, Linear Algebra Appl., 144 (1991), 85–99. http://dx.doi.org/10.1016/0024-3795(91)90063-3 doi: 10.1016/0024-3795(91)90063-3
|
| [11] |
[ 10.1016/j.jfranklin.2014.08.003] S. F. Yuan, A. P. Liao, Least squares Hermitian solution of the complex matrix equation $AXB$+$CXD$ = $E$ with the least norm, J. Franklin Inst., 351 (2014), 4978–4997. http://dx.doi.org/10.1016/j.jfranklin.2014.08.003 doi: 10.1016/j.jfranklin.2014.08.003
|
| [12] |
G. J. Song, S. W. Yu, The solution of a generalized Sylvester quaternion matrix equation and its application, Adv. Appl. Clifford Algebras, 27 (2017), 2473–2492. http://dx.doi.org/10.1007/s00006-017-0782-2 doi: 10.1007/s00006-017-0782-2
|
| [13] |
F. X. Zhang, W. S. Mu, Y. Li, J. L. Zhao, Special least squares solutions of the quaternion matrix equation $AXB$+$CXD$ = $E$, Comput. Math. Appl., 72 (2016), 1426–1435. http://dx.doi.org/10.1016/j.camwa.2016.07.019 doi: 10.1016/j.camwa.2016.07.019
|
| [14] |
Z. H. He, Q. W. Wang, The general solutions to some systems of matrix equations, Linear Multil. Algebra, 63 (2015), 2017–2032. http://dx.doi.org/10.1080/03081087.2014.896361 doi: 10.1080/03081087.2014.896361
|
| [15] |
F. X. Zhang, Y. Li, J. L. Zhao, The semi-tensor product method for special least squares solutions of the complex generalized Sylvester matrix equation, AIMS Math., 8 (2023), 5200–5215. http://dx.doi.org/10.3934/math.2023261 doi: 10.3934/math.2023261
|
| [16] |
H. L. Zhou, An iterative algorithm for solutions of the system of matrix equations $A_1XB_1$ = $C_1$, $A_2XB_2$ = $C_2$ over linear subspace, Math. Numer. Sin., 39 (2017), 213–228. http://dx.doi.org/10.12286/jssx.2017.2.213 doi: 10.12286/jssx.2017.2.213
|
| [17] |
Z. Z. Bai, On Hermitian and skew-Hermitian spliting iteration methods for continuous Sylvester equations, J. Comput. Math., 29 (2011), 185–198. http://dx.doi.org/10.4208/jcm.1009-m3152 doi: 10.4208/jcm.1009-m3152
|
| [18] |
D. M. Zhou, G. L. Chen, Q. Y. Cai, On modified HSS iteration methods for continuous Sylvester equations, Appl. Math. Comput., 263 (2015), 84–93. http://dx.doi.org/10.1016/j.amc.2015.04.020 doi: 10.1016/j.amc.2015.04.020
|
| [19] |
M. Benzi, A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl., 31 (2009), 360–374. http://dx.doi.org/10.1137/080723181 doi: 10.1137/080723181
|
| [20] |
Y. Cao, W. W. Tan, M. Q. Jiang, A generalization of the positive-definite and skew-Hermitian splitting iteration, Numer. Algebra Control Optim., 2 (2012), 811–821. http://dx.doi.org/10.3934/naco.2012.2.811 doi: 10.3934/naco.2012.2.811
|
| [21] |
Y. Li, A splitting iterative algorithm for solving continuous Sylvester matrix equations, Appl. Math. Mechan., 41 (2020), 115–124. http://dx.doi.org/10.21656/1000-0887.400133 doi: 10.21656/1000-0887.400133
|
| [22] |
S. S. Zhang, J. P. Huang, H. Xiong, On NPSS iteration and extrapolation method of the sub-positive definite quaternion matrix equation, J. Chongqing Normal Uni. (Natural Science), 39 (2022), 96–102. http://dx.doi.org/10.11721/cqnuj20220202 doi: 10.11721/cqnuj20220202
|
| [23] |
C. F. Ma, An E-extra iteration method for solving continuous Sylvester equations, Math. Appl., 36 (2023), 220–229. http://dx.doi.org/10.13642/j.cnki.42-1184/o1.2023.01.017 doi: 10.13642/j.cnki.42-1184/o1.2023.01.017
|
| [24] |
M. Erdogdu, M. Ozdemir, On exponetial of split quaternionic matrices, Appl. Math. Comput., 315 (2017), 468–476. http://dx.doi.org/10.1016/j.amc.2017.08.007 doi: 10.1016/j.amc.2017.08.007
|
| [25] |
S. Gai, M. H. Wan, L. Wang, C. H. Yang, Reduced quaternion matrix for color texture classification, Neural Comput. Appl., 25 (2014), 945–954. http://dx.doi.org/10.1007/s00521-014-1578-0 doi: 10.1007/s00521-014-1578-0
|
| [26] |
S. C. Pei, J. H. Chang, J. J. Ding, Commutative reduced biquaternions and their Fourier transform for signal and image processing applications, IEEE Trans. Signal Process., 52 (2004), 2012–2031. http://dx.doi.org/10.1109/TSP.2004.828901 doi: 10.1109/TSP.2004.828901
|
| [27] |
S. Gai, X. Huang, Reduced biquaternion convolutional neural network for color image processing, IEEE Trans. Circ. Syst. Vid. Tech., 32 (2022), 1061–1075. http://dx.doi.org/10.1109/TCSVT.2021.3073363 doi: 10.1109/TCSVT.2021.3073363
|
| [28] |
S. F. Yuan, Y. Tian, M. Z. Li, On Hermitian solutions of the reduced biquaternion matrix equation $(AXB, CXD)$ = $(E, G)$, Linear Multil. Algebra, 68 (2020), 1355–1373. http://dx.doi.org/10.1080/03081087.2018.1543383 doi: 10.1080/03081087.2018.1543383
|
Figures(1) / Tables(1)
Jiaxin Lan, Jingpin Huang, Yun Wang. An E-extra iteration method for solving reduced biquaternion matrix equation $ AX+XB = C $[J]. AIMS Mathematics, 2024, 9(7): 17578-17589. doi: 10.3934/math.2024854
DownLoad: