SOME ITERATIVE ALGORITHMS FOR POSITIVE DEFINITE SOLUTION TO NONLINEAR MATRIX EQUATIONS

Baohua Huang; Changfeng Ma

doi:10.11948/2156-907X.20170324

2019 Volume 9 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2019 > 9(2): 526-546

Next Article Previous Article

Baohua Huang, Changfeng Ma. SOME ITERATIVE ALGORITHMS FOR POSITIVE DEFINITE SOLUTION TO NONLINEAR MATRIX EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 526-546. doi: 10.11948/2156-907X.20170324

Citation:

Baohua Huang, Changfeng Ma. SOME ITERATIVE ALGORITHMS FOR POSITIVE DEFINITE SOLUTION TO NONLINEAR MATRIX EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 526-546. doi: 10.11948/2156-907X.20170324

SOME ITERATIVE ALGORITHMS FOR POSITIVE DEFINITE SOLUTION TO NONLINEAR MATRIX EQUATIONS

Baohua Huang^1,,
Changfeng Ma^1, ,

1.
College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Fuzhou, 350117, China

Corresponding authors: Email address: baohuahuang@126.com(B.-H. Huang); Email address: baohuahuang@126.com(B.-H. Huang)
Fund Project: The authors were supported by National Key Research and Development Program of China(No. 2018YFC0603500) and the CAS/CAFEA international partnership Program for creative research teams(No. KZZD-EW-TZ-19 and KZZD-EW-TZ-15)

Abstract

Abstract

This paper is concerned with the unique positive definite solution to a system of nonlinear matrix equations $X-A^*\bar{Y}^{-1}A=I_n$ and $Y-B^*\bar{X}^{-1}B=I_n$, where $A, B\in\mathbb{C}^{n\times n}$ are given matrices. Based on the special structure of the system of nonlinear matrix equations, the system can be equivalently reformulated as $V-C^*\bar{V}^{-1}C=I_{2n}$. Moreover, by means of Sherman-Moorison-Woodbury formula, we derive the relationship between the solutions of $V-C^*\bar{V}^{-1}C =I_{2n}$ and the well studied standard nonlinear matrix equation $Z+D^*Z^{-1}D=Q$, where $D$, $Q$ are uniquely determined by $C$. Then, we present a structure-preserving doubling algorithm and two modified structure-preserving doubling algorithms to compute the positive definite solution of the system. Furthermore, cyclic reduction algorithm and two modified cyclic reduction algorithms for the positive definite solution of the system are proposed. Finally, some numerical examples are presented to illustrate the efficiency of the theoretical results and the behavior of the considered algorithms.
- Nonlinear matrix equation /
- structure-preserving algorithm /
- cyclic reduction algorithm /
- positive definite solution /
- convergence theory.
MSC: 65F10, 65F30, 65H10, 15A24

FullText(HTML)

References

[1]	A. M. Al-Dubiban, Iterative algorithm for solving a system of nonlinear matrix equations, J. Appl. Math., 2012. DOI:10. 1155/2012/461407. CrossRef Google Scholar
[2]	A. M. Al-Dubiban, On the iterative method for the system of nonlinear matrix equations, Abst. Appl. Anal., 2013. DOI:10. 1155/2013/685753. CrossRef Google Scholar
[3]	W. N. Anderson, Jr., T. D. Morley, G. E. Trapp, Positive solutions to $X=A-BX.^{-1}B.*$, Linear Algebra Appl., 1990, 134, 53-62. doi: 10.1016/0024-3795(90)90005-W CrossRef Google Scholar
[4]	J. H. Bevis, F. J. Hall and R. E. Hartwing, Consimilarity and the matrix equation $A\bar{X}-XB = C$, Current Trends in Matrix Theory, North-Holland, New York, 1987, 51-64. Google Scholar
[5]	J. H. Bevis, F. J. Hall and R. E. Hartwing, The matrix equation $A\bar{X}-XB=C$and its special cases, SIAM J. Matrix Anal. Appl., 1998, 9, 348-359. Google Scholar
[6]	J. Cai and G. Chen, On the Hermitian positive definite solutions of nonlinear matrix equation $X^s+A^*X^{-t}A = Q$, Appl. Math. Comput., 2010, 217, 117-123. Google Scholar
[7]	C. Y. Chiang, Eric K. W. Chu and W. W. Lin, On the H-Sylvester equation $AX\pm X^B^=C$, Appl. Math. Comput., 2012, 218, 8393-8407. Google Scholar
[8]	F. Ding and T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim., 2006, 44, 2269-2284. doi: 10.1137/S0363012904441350 CrossRef Google Scholar
[9]	F. Ding, P. X. Liu and J. Ding, Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Appl. Math. Comput., 2008, 197, 41-50. Google Scholar
[10]	F. Ding, System Identification-new Theory and Methods, Science Press, Beijing, 2013. Google Scholar
[11]	X. F. Duan, C. Li and A. P. Liao, Solutions and perturbation analysis for the nonlinear matrix equation $X+\sum\limits_{i=1}^mA^*_iX^{-1}A_i=I$, Appl. Math. Comput., 2011, 218, 4458-4466. Google Scholar
[12]	S. M. El-sayed, Two iteration processes for computing positive definite solutions of the equation $X- A^*X^{-n}A = Q$, Comput. Math. Appl., 2001, 41, 579-588. doi: 10.1016/S0898-1221(00)00301-1 CrossRef Google Scholar
[13]	S. M. El-sayed and A. M. Al-Dbiban, A new inversion free iteration for solving the equation $X+A^TX^{-1}A = Q$, J. Comput. Appl. Math., 2005, 181, 148-156. doi: 10.1016/j.cam.2004.11.025 CrossRef Google Scholar
[14]	J. C. Engwerda, A. C. M. Ran and A. L. Rijkeboer, Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation $X +A^*X^{-1}A = Q$, Linear Algebra Appl., 1993, 186, 255-275. doi: 10.1016/0024-3795(93)90295-Y CrossRef Google Scholar
[15]	A. Ferrante and B. C. Levy, Hermitian solution of the equation $X = Q-NX^{-1}N^*$, Linear Algebra Appl., 1996, 247, 359-373. doi: 10.1016/0024-3795(95)00121-2 CrossRef Google Scholar
[16]	C. H. Guo and W. W. Lin, The matrix equation $X+A^TX^{-1}A = Q$ and its application in nano research, SIAM J. Sci. Comput., 2010, 32, 3020-3038. doi: 10.1137/090758209 CrossRef Google Scholar
[17]	C. H. Guo, Y. C. Kuo and W. W. Lin, Complex symmetric stabilizing solution of the matrix equation $X + A^TX^{-1}A = Q$, Linear Algebra Appl., 2011, 435, 1187-1192. doi: 10.1016/j.laa.2011.03.034 CrossRef Google Scholar
[18]	L. Guo, L. Liu, Y. Wu, Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions, Nonlinear Anal. Model. Control, 2015, 21, 635-650. Google Scholar
[19]	M. Han, L. Sheng, X. Zhang, Bifurcation theory for finitely smooth planar autonomous differential systems, J. Differ. Equations, 2018, 264, 3596-3618. doi: 10.1016/j.jde.2017.11.025 CrossRef Google Scholar
[20]	M. Han, X. Hou, L. Sheng, C. Wang, Theory of rotated equations and applications to a population model, Discrete Cont. Dyn. Syst. -A, 2018, 38, 2171-2185. doi: 10.3934/dcds.2018089 CrossRef Google Scholar
[21]	V. Hasanov and I. G. Ivanov, Solutions and perturbation estimates for the matrix equations $X \pm A^*X^{-n}A = Q$, Appl Math Comput., 2004, 156, 513-525. Google Scholar
[22]	N. Huang and C. F. Ma, The structure-preserving doubling algorithms for positive definite solution to a system of nonlinear matrix equations, Linear Multilinear Algebra, 2018, 66, 827-839. doi: 10.1080/03081087.2017.1329270 CrossRef Google Scholar
[23]	N. Huang and C. F. Ma, Two structure-preserving-doubling like algorithms for obtaining the positive definite solution to a class of nonlinear matrix equation, Comput. Math. Appl., 2015, 69, 494-502. doi: 10.1016/j.camwa.2015.01.008 CrossRef Google Scholar
[24]	I. G. Ivanov and S. M. El-sayed, Properties of positive definite solutions of the equation $X+A^*X^{-2}A =I$, Linear Algebra Appl., 1998, 279, 303-316. doi: 10.1016/S0024-3795(98)00023-8 CrossRef Google Scholar
[25]	T. S. Jiang, C. H. Cheng and L. Chen, An algebraic relation between consimilarity and similarity of complex matrices and its applications, Physica A, 2006, 38, 9215-9222. Google Scholar
[26]	T. S. Jiang and M. S. Wei, On solutions of the matrix equations $X -AXB = C$ and $X -A \bar{X}B = C$, Linear Algebra Appl., 2003, 367, 225-233. doi: 10.1016/S0024-3795(02)00633-X CrossRef Google Scholar
[27]	F. Li, G. Du, General energy decay for a degenerate viscoelastic Petrovsky-type plate equation with boundary feedback, J. Appl. Anal. Comput., 2018, 8, 390-401. Google Scholar
[28]	M. Li, J. Wang, Exploring delayed mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations, Appl. Math. Comput., 2018, 324, 254-265. Google Scholar
[29]	A. J. Liu, G. L. Chen, On the Hermitian positive definite solutions of nonlinear matrix equation $X^s+\sum\limits_{i=1}^{m}A_i^*X^{-t_i}A_i=Q$, Appl. Math. Comput., 2014, 243, 950-959. Google Scholar
[30]	A. J. Liu, G. L. Chen, X. Y. Zhang, A new method for the bisymmetric minimum norm solution of the consistent matrix equations $A_1XB_1=C_1, A_2XB_2=C_2$, J. Appl. Math., Vol. 2013, Article ID 125687, 6 pages. Google Scholar
[31]	Z. Y. Li, B. Zhou and J. Lam, Towards positive definite solutions of a class of nonlinear matrix equations, Appl. Math. Comput., 2014, 237, 546-559. Google Scholar
[32]	W. W. Lin and S. F. Xu, Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations, SIAM J. Matrix Anal. Appl., 2006, 29, 26-39. Google Scholar
[33]	B. Meini, Efficient computation of the extreme solutions of $X +A^X^{-1}A = Q$ and $X -A^X^{-1}A = Q$, Math. Comput., 2001, 71, 1189-1204. doi: 10.1090/S0025-5718-01-01368-0 CrossRef Google Scholar
[34]	M. Monsalve and M. Raydan, A new inversion-free method for a rational matrix equation, Linear Algebra Appl., 2010, 433, 64-71. doi: 10.1016/j.laa.2010.02.006 CrossRef Google Scholar
[35]	L. Ren, J. Xin, Almost global existence for the Neumann problem of quasilinear wave equations outside star-shaped domains in 3D, Electron J. Differ. Equations, 2018, 31, 1-22. Google Scholar
[36]	H. Tian, M. Han, Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems, J. Differ. Equations, 2017, 263, 7448-7474. doi: 10.1016/j.jde.2017.08.011 CrossRef Google Scholar
[37]	B. Wang, F. Meng, Y. Fang, Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations, Appl. Numer. Math., 2017, 119, 164-178. doi: 10.1016/j.apnum.2017.04.008 CrossRef Google Scholar
[38]	B. Wang, X. Wu, F. Meng, Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second order differential equations, J. Comput. Appl. Math., 2017, 313, 185-201. doi: 10.1016/j.cam.2016.09.017 CrossRef Google Scholar
[39]	B. Wang, Exponential Fourier collocation methods for solving first-order differential equations, J. Comput. Appl. Math., 2017, 35, 711-736. Google Scholar
[40]	S. F. Xu, Matrix Computation in Control Theory (in Chinese), Higher Education Press, Beijing, 2011. Google Scholar
[41]	X. Y. Yin and S. Y. Liu, Positive definite solutions of the matrix equations $X \pm A^*X^{-q}A = Q ~(q \geq1)$, Comput. Math. Appl., 2010, 59, 3727-3739. doi: 10.1016/j.camwa.2010.04.005 CrossRef Google Scholar
[42]	B. Zhou, G. B. Cai and J. Lam, Positive definite solutions of the nonlinear matrix equation $X +A^H\bar{X}^{-1}A = I$, Appl. Math. Comput., 2013, 219, 7377-7391. Google Scholar
[43]	B. Zhou, G. R. Duan and Z. Li, Gradient based iterative algorithm for solving coupled matrix equations, Syst. Control Lett., 2009, 58, 327-333. doi: 10.1016/j.sysconle.2008.12.004 CrossRef Google Scholar
[44]	B. Zhou, J. Lam and G. Duan, On Smith-type iterative algorithms for the Stein matrix equation, Appl. Math. Lett., 2009, 22, 1038-1044. doi: 10.1016/j.aml.2009.01.012 CrossRef Google Scholar
[45]	B. Zhou, J. Lam and G. Duan, Toward solution of matrix equation $X=Af (X)+B$, Linear Algebra Appl., 2011, 435, 1370-1398. doi: 10.1016/j.laa.2011.03.003 CrossRef Google Scholar