连续Sylvester方程的广义正定和反Hermitian分裂迭代法及其超松弛加速

doi:10.12286/jssx.j2019-0630

计算数学 ›› 2021, Vol. 43 ›› Issue (3): 354-366.DOI: 10.12286/jssx.j2019-0630

连续Sylvester方程的广义正定和反Hermitian分裂迭代法及其超松弛加速

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.

PDF ( 179 ) 引用

李旭, 李明翔. 连续Sylvester方程的广义正定和反Hermitian分裂迭代法及其超松弛加速[J]. 计算数学, 2021, 43(3): 354-366.

Li Xu, Li Mingxiang. GENERALIZED POSITIVE-DEFINITE AND SKEW-HERMITIAN SPLITTING ITERATION METHOD AND ITS SOR ACCELERATION FOR CONTINUOUS SYLVESTER EQUATIONS[J]. Mathematica Numerica Sinica, 2021, 43(3): 354-366.

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

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.

关键词： 连续Sylvester方程, GPSS迭代, 不精确迭代, SOR加速, 收敛性分析

Bai proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method for solving a broad class of large sparse continuous Sylvester equations. To further improve the efficiency, in this paper we present a generalized positive-definite and skew-Hermitian splitting (GPSS) iteration method for this matrix equation. Then we establish the inexact variant of the GPSS (IGPSS) iteration method which can reduce the computational cost. Convergence properties of the GPSS iteration method and its inexact variant are analyzed in detail. Moreover, an SOR accelerated GPSS (AGPSS) method is established and its convergence behavior is discussed. Numerical results illustrate the efficiency and robustness of our methods.

Key words: Continuous Sylvester equation, GPSS iteration, Inexact iteration, SOR acceleration, Convergence analysis

MR(2010)主题分类:

分享此文:

[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.

[1]	古振东. 非线性弱奇性Volterra积分方程的谱配置法[J]. 计算数学, 2021, 43(4): 426-443.
[2]	古振东, 孙丽英. 非线性第二类Volterra积分方程的Chebyshev谱配置法[J]. 计算数学, 2020, 42(4): 445-456.
[3]	王志强, 文立平, 朱珍民. 时间延迟扩散-波动分数阶微分方程有限差分方法[J]. 计算数学, 2019, 41(1): 82-90.
[4]	陈圣杰, 戴彧虹, 徐凤敏. 稀疏线性规划研究[J]. 计算数学, 2018, 40(4): 339-353.
[5]	古振东, 孙丽英. 一类弱奇性Volterra积分微分方程的级数展开数值解法[J]. 计算数学, 2017, 39(4): 351-362.
[6]	刘丽华, 马昌凤, 唐嘉. 求解广义鞍点问题的一个新的类SOR算法[J]. 计算数学, 2016, 38(1): 83-95.
[7]	黄娜, 马昌凤, 谢亚君. 求解非对称代数Riccati 方程几个新的预估-校正法[J]. 计算数学, 2013, 35(4): 401-418.
[8]	任志茹. 三阶线性常微分方程Sinc方程组的结构预处理方法[J]. 计算数学, 2013, 35(3): 305-322.
[9]	陈绍春, 梁冠男, 陈红如. Zienkiewicz元插值的非各向异性估计[J]. 计算数学, 2013, 35(3): 271-274.
[10]	张亚东, 石东洋. 各向异性网格下抛物方程一个新的非协调混合元收敛性分析[J]. 计算数学, 2013, 35(2): 171-180.
[11]	陈争, 马昌凤. 求解非线性互补问题一个新的 Jacobian 光滑化方法[J]. 计算数学, 2010, 32(4): 361-372.
[12]	来翔, 袁益让. 一类三维拟线性双曲型方程交替方向有限元法[J]. 计算数学, 2010, 32(1): 15-36.
[13]	蔚喜军. 非线性波动方程的交替显-隐差分方法[J]. 计算数学, 1998, 20(3): 225-238.

阅读次数

全文

摘要

连续Sylvester方程的广义正定和反Hermitian分裂迭代法及其超松弛加速

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

扫码分享