连续Sylvester矩阵方程求解的分裂迭代算法

李英

doi:10.21656/1000-0887.400133

连续Sylvester矩阵方程求解的分裂迭代算法

doi: 10.21656/1000-0887.400133

李英

商丘师范学院信息技术学院, 河南商丘 476000

详细信息

作者简介:
李英(1981—)，女，实验师，硕士(E-mail: sqsyjkx@126.com).

中图分类号: O246
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- 被引次数: 0
出版历程
- 收稿日期: 2019-04-04
- 修回日期: 2019-06-19
- 刊出日期: 2020-01-01

A Splitting Iterative Algorithm for Solving Continuous Sylvester Matrix Equations

LI Ying

School of Information Technology, Shangqiu Normal University,Shangqiu, Henan 476000, P.R.China

摘要

摘要: 有效求解连续的Sylvester矩阵方程对于科学和工程计算有着重要的应用价值，因此该文提出了一种可行的分裂迭代算法.该算法的核心思想是外迭代将连续Sylvester矩阵方程的系数矩阵分裂为对称矩阵和反对称矩阵，内迭代求解复对称矩阵方程.相较于传统的分裂算法，该文所提出的分裂迭代算法有效地避免了最优迭代参数的选取，并利用了复对称方程组高效求解的特点，进而提高了算法的易实现性、易操作性.此外，从理论层面进一步证明了该分裂迭代算法的收敛性.最后，通过数值算例表明分裂迭代算法具有良好的收敛性和鲁棒性，同时也证实了分裂迭代算法的收敛性很大程度依赖于内迭代格式的选取.
- Sylvester矩阵方程 /
- 复对称矩阵方程 /
- 分裂迭代算法 /
- 收敛性
Abstract: The solution of continuous Sylvester matrix equations has significant application value in scientific and engineering calculations, hence, a splitting iterative algorithm was proposed. The core idea of the algorithm is to split the coefficient matrix of the continuous Sylvester matrix equation into a symmetric matrix and an antisymmetric matrix with an outer iterative scheme, and to solve the complex symmetric matrix equation with the inner iterative scheme. Compared with the traditional splitting algorithms, the proposed splitting algorithm effectively avoids the selection of optimal iterative parameters and takes advantages of the efficient solution of complex symmetric equations, which improves the easy implementation and easy operation of the algorithm. In addition, the convergence of the splitting iterative algorithm was further proved theoretically. Numerical examples show that, the splitting iterative algorithm has good convergence and robustness, and the convergence of the splitting iterative algorithm depends on the selection of the inner iterative schemes.
- Sylvester matrix equation /
- complex symmetric matrix equation /
- splitting iterative algorithm /
- convergence

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参考文献(25)

[1]	LANCASTER P, RODMAN L. Algebraic Riccati Equations [M]. Oxford: The Clarendon Press, 1995.
[2]	CHIANG C Y. On the Sylvester-like matrix equation AX+f(X)B=C[J]. Journal of the Franklin Institute,2016,353: 1061-1074.
[3]	CALVETTI D, REICHEL L. Application of ADI iterative methods to the restoration of noisy images[J]. SIAM Journal on Matrix Analysis and Applications,1996,17(1): 165-186.
[4]	ANDERSON B, AGATHOKLIS P, JURY E, et al. Stability and the matrix Lyapunov equation for discrete 2-dimensional systems[J]. IEEE Transactions on Circuits and Systems,1986,33(3): 261-267.
[5]	BARTELS R H, STEWART G W. Solution of the matrix equation AX+XB=C[F4][J]. Communications of the ACM,1972,15(9): 820-826.
[6]	GOLUB G, NASH S, VAN LOAN C. A Hessenberg-Schur method for the problem AX+XB=C[J]. IEEE Transactions on Automatic Control,1979,24(6): 909-913.
[7]	BAI Z Z. On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations[J]. Journal of Computational Mathematics,2011,29(2): 185-198.
[8]	ZHOU D M, CHEN G L, CAI Q Y. On modified HSS iteration methods for continuous Sylvester equations[J]. Applied Mathematics and Computation,2015,263: 84-93.
[9]	XU L, HUO H F, YANG A L. Preconditioned HSS iteration method and its non-alternating variant for continuous Sylvester equations[J]. Computers and Mathematics With Applications,2018,75(4): 1095-1106.
[10]	DEHGHAN M, SHIRILORD A. A generalized modified Hermitian and skew-Hermitian splitting (GMHSS) method for solving complex Sylvester matrix equation[J]. Applied Mathematics and Computation,2019,348: 632-651.
[11]	ZHOU R, WANG X, TANG X B. A generalization of the Hermitian and skew-Hermitian splitting iteration method for solving Sylvester equations[J]. Applied Mathematics and Computation,2015,271: 609-617.
[12]	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 Journal on Applied Mathematics,2017,7(1): 55-69.
[13]	ZHENG Q Q, MA C F. On normal and skew-Hermitian splitting iteration methods for large sparse continuous Sylvester equations[J]. Journal of Computational and Applied Mathematics,2014,268: 145-154.
[14]	XIAO X Y, WANG X, YIN H W. Efficient single-step preconditioned HSS iteration methods for complex symmetric linear systems[J]. Computers and Mathematics With Applications,2017,74(10): 2269-2280.
[15]	XIAO X Y, WANG X, YIN H W. Efficient preconditioned NHSS iteration methods for solving complex symmetric linear systems[J]. Computers and Mathematics With Applications,2018,75(1): 235-247.
[16]	杨丽, 李军. Hilbert空间中分裂可行性问题的改进Halpern迭代和黏性逼近算法[J]. 应用数学和力学, 2017,38(9): 1072-1080.(YANG Li, LI Jun. Modified Halpern iteration and viscosity approximation methods for split feasibility problems in Hilbert spaces[J]. Applied Mathematics and Mechanics,2017,38(9): 1072-1080.(in Chinese))
[17]	LI C L, MA C F. On semi-convergence of parameterized SHSS method for a class of singular complex symmetric linear systems[J]. Computers and Mathematics With Applications,2019,77: 466-475.
[18]	HUANG Z G, WANG L G, XU Z, et al. The generalized double steps scale-SOR iteration method for solving complex symmetric linear systems[J].Journal of Computational and Applied Mathematics,2019,346: 284-306.
[19]	HUANG Z G, WANG L G, XU Z, et al. Preconditioned accelerated generalized successive overrelaxation method for solving complex symmetric linear systems[J]. Computers and Mathematics With Applications,2019,77(7): 1902-1916.
[20]	VAN DER VORST H A, MELISSEN J B M. A Petrov-Galerkin type method for solving Ax=b, where A is symmetric complex[J]. IEEE Transactions on Magnetics,1990,26(2): 706-708.
[21]	GU X M, CLEMENS M, HUANG T Z, et al. The SCBiCG class of algorithms for complex symmetric linear systems with applications in several electromagnetic model problems[J]. Computer Physics Communications,2015,191: 52-64.
[22]	SOGABE T, ZHANG S L. A COCR method for solving complex symmetric linear systems[J]. Journal of Computational and Applied Mathematics,2007,199(2): 297-303.
[23]	CLEMENS M, WEILAND T, VAN RIENEN U. Comparison of Krylov-type methods for complex linear systems applied to high-voltage problems[J]. IEEE Transactions on Magnetics,1998,34(5): 3335-3338.
[24]	ABE K, FUJINO S. Converting BiCR method for linear equations with complex symmetric matrices[J]. Applied Mathematics and Computation,2018,321: 564-576.
[25]	HAJARIAN M. Matrix form of the CGS method for solving general coupled matrix equations[J]. Applied Mathematics Letters,2014,34: 37-42.