李旭, 李明翔

兰州理工大学应用数学系, 兰州 730050

收稿日期:2019-09-22出版日期:2021-08-15发布日期:2021-08-20
通讯作者:李旭,Email:lixu@lut.edu.cn.

基金资助:国家自然科学基金（11501272）资助.

GENERALIZED POSITIVE-DEFINITE AND SKEW-HERMITIAN SPLITTING ITERATION METHOD AND ITS SOR ACCELERATION FOR CONTINUOUS SYLVESTER EQUATIONS

Li Xu, Li Mingxiang

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China

Received:2019-09-22Online:2021-08-15Published:2021-08-20







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对于求解大型稀疏连续Sylvester方程，Bai提出了非常有效的Hermitian和反Hermitian分裂（HSS）迭代法.为了进一步提高求解这类方程的效率，本文建立一种广义正定和反Hermitian分裂（GPSS）迭代法，并且提出不精确GPSS（IGPSS）迭代法从而可以降低计算成本.对GPSS迭代法及其不精确变体的收敛性作了详细分析.另外，建立一种超松弛加速GPSS（AGPSS）方法并且讨论了收敛性.数值结果表明了方法的高效性和鲁棒性.
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[1] Lancaster P. Explicit solutions of linear matrix equations[J]. SIAM Rev., 1970, 12:544-566. [2] Lancaster P, Tismenetsky M. The theory of matrices[M]. Academic Press Inc., Orlando, FL, second edition, 1985. [3] Lancaster P, Rodman L. Algebraic Riccati equations[M]. The Clarendon Press, Oxford and New York, 1995. [4] Halanay A, Răsvan V. Applications of Liapunov methods in stability[M]. Kluwer Academic Publishers Group, Dordrecht, 1993. [5] Golub G H, Loan C F V. Matrix computations[M]. Johns Hopkins University Press, Baltimore, MD, third edition, 1996. [6] Anderson B D O, Agathoklis P, Jury E I, Mansour M. Stability and the matrix Lyapunov equation for discrete 2-dimensional systems[J]. IEEE Trans. Circuits and Systems, 1986, 33(3):261-267. [7] Calvetti D, Reichel L. Application of ADI iterative methods to the restoration of noisy images[J]. SIAM J. Matrix Anal. Appl., 1996, 17(1):165-186. [8] Haddad W M, Bernstein D S. The optimal projection equations for reduced-order, discrete-time state estimation for linear systems with multiplicative white noise[J]. Systems Control Lett., 1987, 8(4):381-388. [9] Bartels R H, Stewart G W. Solution of the matrix equation AX + XB=C[F4] [J]. Commun. ACM, 1972, 15(9):820-826. [10] Golub G H, Nash S, Loan C V. A Hessenberg-Schur method for the problem AX + XB=C[J]. IEEE Trans. Automat. Control, 1979, 24(6):909-913. [11] Smith R A. Matrix equation XA + BX=C[J]. SIAM J. Appl. Math., 1968, 16:198-201. [12] Niu Q, Wang X, Lu L Z. A relaxed gradient based algorithm for solving Sylvester equations[J]. Asian J. Control, 2011, 13(3):461-464. [13] Wang X, Dai L, Liao D. A modified gradient based algorithm for solving Sylvester equations[J]. Appl. Math. Comput., 2012, 218(9):5620-5628. [14] Hu D Y, Reichel L. Krylov-subspace methods for the Sylvester equation[J]. Linear Algebra Appl., 1992, 172:283-313. [15] Wachspress E L. Iterative solution of the Lyapunov matrix equation[J]. Appl. Math. Lett., 1988, 1(1):87-90. [16] Simoncini V. Computational methods for linear matrix equations[J]. SIAM Rev., 2016, 58(3):377-441. [17] Bai Z Z. On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations[J]. J. Comput. Math., 2011, 29(2):185-198. [18] Bai Z Z, Golub G H, Ng M K. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems[J]. SIAM J. Matrix Anal. Appl., 2003, 24(3):603-626. [19] Bai Z Z, Golub G H, Lu L Z, Yin J F. Block triangular and skew-Hermitian splitting methods for positive-definite linear systems[J]. SIAM J. Sci. Comput., 2005, 26(3):844-863. [20] Bai Z Z, Golub G H, Ng M K. On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations[J]. Numer. Linear Algebra Appl., 2007, 14(4):319-335. [21] Benzi M. A generalization of the Hermitian and skew-Hermitian splitting iteration[J]. SIAM J. Matrix Anal. Appl., 2009, 31(2):360-374. [22] Bai Z Z, Golub G H, Li C K. Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices[J]. Math. Comp., 2007, 76(257):287-298. [23] Yang A L, An J, Wu Y J. A generalized preconditioned HSS method for non-Hermitian positive definite linear systems[J]. Appl. Math. Comput., 2010, 216(6):1715-1722. [24] Yin J F, Dou Q Y. Generalized preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems[J]. J. Comput. Math., 2012, 30(4):404-417. [25] Wu Y J, Li X, Yuan J Y. A non-alternating preconditioned HSS iteration method for nonHermitian positive definite linear systems[J]. Comput. Appl. Math., 2017, 36(1):367-381. [26] Wang X, Li W W, Mao L Z. On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation AX + XB=C[J]. Comput. Math. Appl., 2013, 66(11):2352-2361. [27] Li X, Wu Y J, Yang A L, Yuan J Y. A generalized HSS iteration method for continuous Sylvester equations[J]. J. Appl. Math., 2014, pages Art. ID 578102, 9. [28] Zheng Q Q, Ma C F. On normal and skew-Hermitian splitting iteration methods for large sparse continuous Sylvester equations[J]. J. Comput. Appl. Math., 2014, 268:145-154. [29] Zhou D M, Chen G L, Cai Q Y. On modified HSS iteration methods for continuous Sylvester equations[J]. Appl. Math. Comput., 2015, 263:84-93. [30] Zhou R, Wang X, Tang X B. A generalization of the Hermitian and skew-Hermitian splitting iteration method for solving Sylvester equations[J]. Appl. Math. Comput., 2015, 271:609-617. [31] Zhou R, Wang X, Tang X B. Preconditioned positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equations AX + XB=C[J]. East Asian J. Appl. Math., 2017, 7(1):55-69. [32] Zak M K, Toutounian F. An iterative method for solving the continuous Sylvester equation by emphasizing on the skew-hermitian parts of the coefficient matrices[J]. Int. J. Comput. Math., 2017, 94(4):633-649. [33] Dong Y X, Gu C Q. On PMHSS iteration methods for continuous Sylvester equations[J]. J. Comput. Math., 2017, 35(5):600-619. [34] Li X, Huo H F, Yang A L. Preconditioned HSS iteration method and its non-alternating variant for continuous Sylvester equations[J]. Comput. Math. Appl., 2018, 75(4):1095-1106. [35] Wang X, Li Y, Dai L. On Hermitian and skew-Hermitian splitting iteration methods for the linear matrix equation AXB=C[J]. Comput. Math. Appl., 2013, 65(4):657-664. [36] Zhou R, Wang X, Zhou P. A modified HSS iteration method for solving the complex linear matrix equation AXB=C[J]. J. Comput. Math., 2016, 34(4):437-450. [37] Zhang W H, Yang A L, Wu Y J. Parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method for a class of linear matrix equations[J]. Comput. Math. Appl., 2015, 70(6):1357-1367. [38] Cao Y, Tan W W, Jiang M Q. A generalization of the positive-definite and skew-Hermitian splitting iteration[J]. Numer. Algebra Control Optim., 2012, 2(4):811-821. [39] Bai Z Z, Golub G H, Ng M K. On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems[J]. Linear Algebra Appl., 2008, 428(2-3):413-440. [40] Young D M. Iterative solution of large linear systems[M]. Academic Press, New York, 1971.

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