| Citation: |
Jun Yu Luo, Zhen Chuan Ng. HERMITIAN POSITIVE SEMI-DEFINITE SOLUTION OF THE QUATERNION MATRIX EQUATION $ X^2+XB^*X+BX+Q=0 $[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 744-766. doi: 10.11948/20250173
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HERMITIAN POSITIVE SEMI-DEFINITE SOLUTION OF THE QUATERNION MATRIX EQUATION $ X^2+XB^*X+BX+Q=0 $
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Abstract
This paper discusses necessary and sufficient conditions for the existence of Hermitian positive semi-definite solutions of the quadratic matrix equation $ X^2+XB^*X+BX+Q=0\, (Q\geq0) $ on the quaternion field and its iterative solution method. Some necessary and sufficient conditions for the existence of Hermitian positive semi-definite solutions of this equation are proved mainly for the characteristics of coefficient matrices by introducing appropriate parameters to establish matrix inequalities and by using the theory of fixed points on convex sets. On this basis, three convergent iteration formats are constructed for different conditions and solution existence intervals, and the selection of the initial matrix is given according to each iteration property. The algorithm is solved in the MATLAB environment by using the complex operator of the quaternion matrix. At the same time, a perturbation analysis is carried out on the solution of the equation, and two perturbation error bounds are obtained by considering the Frobenius norm. Three numerical examples are used to test the effectiveness and feasibility of the given method.
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References
[1] M. S. I. Adam, J. Ding, Q. Huang and L. Zhu, Solving a class of quadratic matrix equations, Appl. Math. Lett., 2018, 82, 58–63. doi: 10.1016/j.aml.2018.02.017 [2] S. C. Chacha, Fixed point theorems and sensitivity analysis in solving nonlinear matrix equations: A study involving partially ordered sets, Tanzan. J. Sci., 2024, 50(4), 813–820. doi: 10.4314/tjs.v50i4.8 [3] C. Chen, R. -C. Li and C. Ma, Highly accurate doubling algorithm for quadratic matrix equation from quasi-birth-and-death process, Linear Algebra Appl., 2019, 583, 1–45. [4] L. Chen, N. Yang, Y. Zhao and Z. Ma, Fixed point theorems for set-valued G-contractions in a graphical convex metric space with applications, J. Fixed Point Theory Appl., 2020, 22(4), Paper No. 88, 23. [5] X. Ding, L. Guo and P. M. Frank, Parameterization of linear observers and its application to observer design, IEEE Trans. Automat. Control, 1994, 39(8), 1648–1652. doi: 10.1109/9.310042 [6] X. Duan, Z. Jiang and A. Liao, Low rank approximation solution of a class of generalized Lyapunov equation, J. Comput. Math., 2016, 34(4), 407–420. doi: 10.4208/jcm.1601-m2015-0388 [7] R. Erfanifar, K. Sayevand and M. Hajarian, Solving system of nonlinear matrix equations over Hermitian positive definite matrices, Linear Multilinear Algebra, 2023, 71(4), 597–630. doi: 10.1080/03081087.2022.2038058 [8] J. Guan, R. Song and A. Zubair, New iteration method for a quadratic matrix equation associated with an M-matrix, Math. Appl. (Wuhan), 2021, 34(1), 1–7. [9] B. Huang and C. Ma, Some iterative algorithms for positive definite solution to nonlinear matrix equations, J. Appl. Anal. Comput., 2019, 9(2), 526–546. [10] J. Huang, S. Zhang and H. Xiong, Iterative methods of the positive definite solution to a quadratic quaternion system, Math. Appl. (Wuhan), 2021, 34(2), 357–364. [11] J. P. Huang, Parameter iterative method for solving a mixed-type Lyapunov matrix equation, Math. Numer. Sin., 2007, 29(3), 285–292. [12] J. P. Huang and Y. Yu, Complex representation of quaternion matrix equations and a structure-preserving algorithm, Pure Appl. Math. (Xi'an), 2008, 24(2), 321–326. [13] M. Jleli and B. Samet, Lyapunov-type inequalities for fractional boundary-value problems, Electron. J. Differential Equations, 2015, No. 88, 11. [14] A. Kassymov and B. T. Torebek, Lyapunov-type inequalities for a nonlinear fractional boundary value problem, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 2021, 115(1), Paper No. 15, 10. [15] W. L. Li, Quaternion Matrix, National University of Defense Technology Press, Changsha, 2002. [16] J. Liu, J. Zhang and Q. Li, Upper and lower eigenvalue summation bounds of the Lyapunov matrix differential equation and the application in a class time-varying nonlinear system, Internat. J. Control, 2020, 93(5), 1115–1126. doi: 10.1080/00207179.2018.1494389 [17] J. Liu, J. Zhang and Q. Li, Upper and lower eigenvalue summation bounds of the Lyapunov matrix differential equation and the application in a class time-varying nonlinear system, Internat. J. Control, 2020, 93(5), 1115–1126. doi: 10.1080/00207179.2018.1494389 [18] L. Lu, Z. Ahmed and J. Guan, Numerical methods for a quadratic matrix equation with a nonsingular M-matrix, Appl. Math. Lett., 2016, 52, 46–52. doi: 10.1016/j.aml.2015.08.006 [19] M. Masoudi and A. Salemi, On Hermitian positive definite solutions of a nonlinear matrix equation, J. Fixed Point Theory Appl., 2021, 23(2), Paper No. 30, 19. [20] M. Masoudi and A. Salemi, On Hermitian positive definite solutions of a nonlinear matrix equation, J. Fixed Point Theory Appl., 2021, 23(2), Paper No. 30, 19. [21] V. Simoncini, Computational methods for linear matrix equations, SIAM Rev., 2016, 58(3), 377–441. doi: 10.1137/130912839 [22] Y. O. Vorontsov and K. D. Ikramov, Numerical algorithm for solving quadratic matrix equations of a certain class, Comput. Math. Math. Phys., 2014, 54(11), 1643–1646. doi: 10.1134/S0965542514110128 [23] L. W. Yi, Numeric Analysis, Tsinghua University Press, Peking, 2012. [24] S. Zhang and L. Hu, The existence of solutions and generalized Lyapunov-type inequalities to boundary value problems of differential equations of variable order, AIMS Math., 2020, 5(4), 2923–2943. doi: 10.3934/math.2020189 -
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Jun Yu Luo, Zhen Chuan Ng. HERMITIAN POSITIVE SEMI-DEFINITE SOLUTION OF THE QUATERNION MATRIX EQUATION $ X^2+XB^*X+BX+Q=0 $[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 744-766. doi: 10.11948/20250173
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