求解一类复对称线性系统的广义AOR迭代法

doi:10.12288/szjs.s2021-0752

数值计算与计算机应用 ›› 2022, Vol. 43 ›› Issue (3): 295-306.DOI: 10.12288/szjs.s2021-0752

求解一类复对称线性系统的广义AOR迭代法

李旭, 李瑞丰

兰州理工大学应用数学系, 兰州 730050

收稿日期:2022-05-01 出版日期:2022-09-14 发布日期:2022-09-09
通讯作者: 李旭,Email:lixu@lut.edu.cn
基金资助:
国家自然科学基金（11501272）和甘肃省自然科学基金（20JR5RA464）资助.

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李旭, 李瑞丰. 求解一类复对称线性系统的广义AOR迭代法[J]. 数值计算与计算机应用, 2022, 43(3): 295-306.

Li Xu, Li Ruifeng. GENERALIZED AOR ITERATION METHODS FOR SOLVING A CLASS OF COMPLEX SYMMETRIC LINEAR SYSTEMS[J]. Journal on Numerica Methods and Computer Applications, 2022, 43(3): 295-306.

GENERALIZED AOR ITERATION METHODS FOR SOLVING A CLASS OF COMPLEX SYMMETRIC LINEAR SYSTEMS

Li Xu, Li Ruifeng

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

Received:2022-05-01 Online:2022-09-14 Published:2022-09-09

针对求解一类广义复对称线性系统，Salkuyeh等学者利用等价2×2块实值形式提出了一种广义SOR （GSOR）迭代法.为了进一步提高计算效率，本文建立一种含有两个参数的广义AOR （GAOR）迭代法.详细分析了该方法的收敛性，得到一个范围更广的收敛域.最后，通过两个数值算例验证了GAOR迭代法的可行性与高效性.

关键词： 复对称线性系统, GAOR迭代法, 收敛性分析, 特征值

For solving a broad class of complex symmetric linear systems, Salkuyeh et al. proposed the generalized SOR (GSOR) iteration method by using the equivalent block two-by-two real value forms. In order to further improve the computational efficiency, a generalized AOR (GAOR) iteration method with two parameters is established in this paper. The convergence properties of the method are analyzed in detail and a wider convergence domain is obtained. The feasibility and efficiency of the GAOR iteration method are verified by two numerical examples.

Key words: complex symmetric linear systems, GAOR iterative method, convergence analysis, eigenvalues

MR(2010)主题分类:

65F10
65F50

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[1] van Dijk W, Toyama F M. Accurate numerical solutions of the time-dependent Schrödinger equation[J]. Physical Review E, 2007, 75(3):036707.
[2] Arridge S R. Optical tomography in medical imaging[J]. Inverse Problems, 1999, 15(2):R41-R93.
[3] Feriani A, Perotti F, Simoncini V. Iterative system solvers for the frequency analysis of linear mechanical systems[J]. Comput. Methods Appl. Mech. Engrg., 2000, 190(13-14):1719-1739.
[4] Bertaccini D. Efficient preconditioning for sequences of parametric complex symmetric linear systems[J]. Electron. Trans. Numer. Anal., 2004, 18:49-64.
[5] Poirier B. Efficient preconditioning scheme for block partitioned matrices with structured sparsity[J]. Numer. Linear Algebra Appl., 2000, 7(7-8):715-726.
[6] 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.
[7] Bai Z Z, Benzi M, Chen F. Modified HSS iteration methods for a class of complex symmetric linear systems[J]. Computing, 2010, 87(3-4):93-111.
[8] Bai Z Z, Benzi M, Chen F. On preconditioned MHSS iteration methods for complex symmetric linear systems[J]. Numer. Algorithms, 2011, 56(2):297-317.
[9] Guo X X, Wang S. Modified HSS iteration methods for a class of non-Hermitian positive-definite linear systems[J]. Appl. Math. Comput., 2012, 218(20):10122-10128.
[10] Li X, Yang A L, Wu Y J. Lopsided PMHSS iteration method for a class of complex symmetric linear systems[J]. Numer. Algorithms, 2014, 66(3):555-568.
[11] Zeng M L, Ma C F. A parameterized SHSS iteration method for a class of complex symmetric system of linear equations[J]. Comput. Math. Appl., 2016, 71(10):2124-2131.
[12] Zheng Q Q, Ma C F. Accelerated PMHSS iteration methods for complex symmetric linear systems[J]. Numer. Algorithms, 2016, 73(2):501-516.
[13] Zheng Z, Huang F L, Peng Y C. Double-step scale splitting iteration method for a class of complex symmetric linear systems[J]. Appl. Math. Lett., 2017, 73:91-97.
[14] Xiao X Y, Wang X, Yin H W. Efficient single-step preconditioned HSS iteration methods for complex symmetric linear systems[J]. Comput. Math. Appl., 2017, 74(10):2269-2280.
[15] Xiao X Y, Wang X. A new single-step iteration method for solving complex symmetric linear systems[J]. Numer. Algorithms, 2018, 78(2):643-660.
[16] Axelsson O, Kucherov A. Real valued iterative methods for solving complex symmetric linear systems[J]. Numer. Linear Algebra Appl., 2000, 7(4):197-218.
[17] Day D, Heroux M A. Solving complex-valued linear systems via equivalent real formulations[J]. SIAM J. Sci. Comput., 2001, 23(2):480-498.
[18] Benzi M, Bertaccini D. Block preconditioning of real-valued iterative algorithms for complex linear systems[J]. IMA J. Numer. Anal., 2008, 28(3):598-618.
[19] Axelsson O, Neytcheva M, Ahmad B. A comparison of iterative methods to solve complex valued linear algebraic systems[J]. Numer. Algorithms, 2014, 66(4):811-841.
[20] Bai Z Z. On preconditioned iteration methods for complex linear systems[J]. J. Engrg. Math., 2015, 93:41-60.
[21] Bai Z Z, Benzi M, Chen F, Wang Z Q. Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems[J]. IMA J. Numer. Anal., 2013, 33(1):343-369.
[22] Zhang J, Wang Z, Zhao J. Double-step scale splitting real-valued iteration method for a class of complex symmetric linear systems[J]. Appl. Math. Comput., 2019, 353:338-346.
[23] Liu K, Gu G. Improved PMHSS iteration methods for complex symmetric linear systems[J]. J. Comput. Math., 2019, 37(2):278-296.
[24] Benzi M, Golub G H, Liesen J. Numerical solution of saddle point problems[J]. Acta Numer.,2005, 14:1-137.
[25] Benzi M, Golub G H. A preconditioner for generalized saddle point problems[J]. SIAM J. Matrix Anal. Appl., 2004, 26(1):20-41.
[26] Bai Z Z, Wang Z Q. On parameterized inexact Uzawa methods for generalized saddle point problems[J]. Linear Algebra Appl., 2008, 428(11-12):2900-2932.
[27] Zhang C H, Wang X, Tang X B. Generalized AOR method for solving a class of generalized saddle point problems[J]. J. Comput. Appl. Math., 2019, 350:69-79.
[28] Salkuyeh D K, Hezari D, Edalatpour V. Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations[J]. Int. J. Comput. Math., 2015, 92(4):802-815.
[29] van der Vorst H A. Iterative Krylov methods for large linear systems[M]. Cambridge University Press, Cambridge, 2003.
[30] Hezari D, Edalatpour V, Salkuyeh D K. Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations[J]. Numer. Linear Algebra Appl., 2015, 22(4):761-776.
[31] Chen C R, Ma C F. AOR-Uzawa iterative method for a class of complex symmetric linear system of equations[J]. Comput. Math. Appl., 2016, 72(9):2462-2472.
[32] Edalatpour V, Hezari D, Salkuyeh D K. Accelerated generalized SOR method for a class of complex systems of linear equations[J]. Math. Commun., 2015, 20(1):37-52.
[33] Huang Z G, Wang L G, Xu Z, Cui J J. Preconditioned accelerated generalized successive overrelaxation method for solving complex symmetric linear systems[J]. Comput. Math. Appl., 2019, 77(7):1902-1916.
[34] Liang Z Z, Zhang G F. On SSOR iteration method for a class of block two-by-two linear systems[J]. Numer. Algorithms, 2016, 71(3):655-671.
[35] Hadjidimos A. Accelerated overrelaxation method[J]. Math. Comp., 1978, 32(141):149-157.
[36] Young D M. Iterative solution of large linear systems[M]. Academic Press, New York, 1971.
[37] Golub G H, Loan C F V. Matrix computations[M]. Johns Hopkins University Press, Baltimore, MD, fourth edition, 2013.
[38] Saad Y. Iterative methods for sparse linear systems[M]. SIAM, Philadelphia, PA, second edition, 2003.

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摘要

求解一类复对称线性系统的广义AOR迭代法

GENERALIZED AOR ITERATION METHODS FOR SOLVING A CLASS OF COMPLEX SYMMETRIC LINEAR SYSTEMS

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