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《山东大学学报(理学版)》 ›› 2024, Vol. 59 ›› Issue (10): 1-9.doi: 10.6040/j.issn.1671-9352.0.2023.225

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一类复对称线性方程组的块三角分裂及其预处理迭代算法

王洋   

  1. 吉林师范大学数学与计算机学院, 吉林 四平 136100
  • 发布日期:2024-10-10
  • 作者简介:王洋(1982— ),女,副教授,硕士生导师,博士,研究方向为数值代数及其应用. E-mail:yyang3721@163.com
  • 基金资助:
    吉林省社会科学基金资助项目(2021B136)

Block triangular splitting and its preconditioning iterative algorithms for a class of complex symmetric linear systems

WANG Yang   

  1. College of Mathematics and Computer, Jilin Normal University, Siping 136100, Jilin, China
  • Published:2024-10-10

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摘要： 基于2×2块矩阵的块三角分裂,提出求解复对称线性方程组的块三角分裂(BTS)迭代方法及其预处理迭代方法(PBTS)。理论分析表明,在迭代参数α满足一定的条件下,BTS和PBTS迭代方法是收敛性的,给出两类方法迭代格式中理论最优参数的计算方法。数值实验结果证明BTS迭代方法和PBTS迭代方法的有效性和优越性。

关键词: 复对称线性方程组, 收敛性分析, 块三角分裂, 预处理

Abstract: Based on block triangular splitting for block 2×2 matrix, the block triangular splitting(BTS)iteration method and the preconditioned block triangular splitting(PBTS)iteration method for a class of complex symmetric linear system are proposed. Theoretical analysis shows that the BTS and PBTS methods converge under certain conditions. The optimal iteration parameters of these two methods are obtained. Numerical experiments demonstrate the effectiveness and superiority of the BTS method and the PBTS iterative methods.

Key words: complex symmetric linear system, convergence analysis, block triangular splitting, preconditioning

中图分类号: 

  • O242.2
[1] SOMMERFELD A. Partial differential equations[M]. New York: Academic Press, 1949:1-10.
[2] ARRIDGE S R. Optical tomography in medical imaging[J]. Inverse Problems, 1999, 15(2):41-93.
[3] HIPTMAIR R. Finite elements in computational electromagnetism[J]. Acta Numerica, 2002, 11:237-339.
[4] ARANSON I S, KRAMER L. The world of the complex Ginzburg-Landau equation[J]. Reviews of Modern Physics, 2002, 74(1):99-143.
[5] KURAMOTO Y. Chemical oscillations, waves, and turbulence[M]. New York: Dover Publications, Inc, 2003:89-140.
[6] SAAD Y, SCHULTZ M H. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems[J]. SIAM Journal on Scientific and Statistical Computing, 1986, 7(3):856-869.
[7] FREUND R W, NACHTIGAL N M. QMR: a quasi-minimal residual method for non-Hermitian linear systems[J]. Numerische Mathematik, 1991, 60(1):315-339.
[8] AXELSSON O, KHCHEROV A. Real valued iterative methods for solving complex symmetric linear systems[J]. Numerical Linear Algebra with Applications, 2000, 7(4):197-218.
[9] AXELSSON O, NEYTCHEVA M, AHMAD B. A comparison of iterative methods to solve complex valued linear algebraic systems[J]. Numerical Algorithms, 2014, 66(4):811-841.
[10] AXELSSON O, BAI Z Z, QIU S X. A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part[J]. Numerical Algorithms, 2004, 35:351-372.
[11] BAI Z Z, GOLUB G H, NG M K. Hermitian and skew-hermitian splitting methods for non-hermitian positive definite linear systems[J]. SIAM Journal on Matrix Analysis and Applications, 2003, 24(3):603-626.
[12] 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.
[13] BAI Z Z, BENZI M, CHEN F. On preconditioned MHSS iteration methods for complex symmetric linear systems[J]. Numerical Algorithms, 2011, 56:297-317.
[14] BAI Z Z, BENZI M, CHEN F, et al. Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems[J]. IMA Journal of Numerical Analysis, 2013, 33(1):343-369.
[15] SALKUYEH D K, HEZARI D, EDALATPOUR V. Generalized successive over-relaxation iterative method for a class of complex symmetric linear system of equations[J]. International Journal of Computer Mathematics, 2015, 92(4):802-815.
[16] HEZARI D, EDALATPOUR V, SALKUYEH D K. Preconditioned GSOR iterative method for a class of complex symmetric linear system of linear equations[J]. Numerical Linear Algebra with Applications, 2015, 22(4):761-776.
[17] LI Xian, WEI Hongzhang, WU Yujiang. On symmetric block triangular splitting iteration method for a class of complex symmetric system of linear equations[J]. Applied Mathematics Letters, 2018, 79:131-137.
[18] ZHANG Jianhua, WANG Zewen, ZHAO Jing. Preconditioned symmetric block triangular splitting iteration method for a class of complex symmetric system of linear equations[J]. Applied Mathematics Letters, 2018, 86:95-102.
[19] ZHENG Qingqing, MA Changfeng. A class of triangular splitting methods for saddle point problems[J]. Journal of Computational and Applied Mathematics, 2016, 298:13-23.
[20] LI Jingtao, MA Changfeng. The parameterized upper and lower triangular splitting methods for saddle point problems[J]. Numerical Algorithms, 2017, 76:413-425.
[21] BAI Z Z. On preconditioned iteration methods for complex linear systems[J]. Journal of Engineering Mathematics, 2015, 93:41-60.
[22] LIANG Zhaozheng, ZHANG Guofeng. On SSOR iteration method for a class of block two-by-two linear systems[J]. Numerical Algorithms, 2016, 71:655-671.
[23] WU Yujiang, LI Xu, YUAN Jinyun. A non-alternating preconditioned HSS iteration method for non-Hermitian positive definite linear systems[J]. Computational and Applied Mathematics, 2017, 36(1):367-381.
[24] BENZI M, BERTACCINI D. Block preconditioning of real-valued iterative algorithms for complex linear systems[J]. IMA Journal of Numerical Analysis, 2008, 28(3):598-618.
[25] LI Xu, YANG Aili, WU Yujiang. Lopsided PMHSS iteration method for a class of complex symmetric linear systems[J]. Numerical Algorithms, 2014, 66(3):555-568.
[26] ZHENG Qingqing, LU Linzhang. A shift-splitting preconditioned for a class of block two-by-two linear systems[J]. Applied Math Letters, 2016, 66(3):54-60.
[27] 王洋, 赵彦军, 冯毅夫. 基于超松弛迭代的 MHSS 加速方法[J]. 山东大学学报(理学版), 2016, 51(8):61-65. WANG Yang, ZHAO Yanjun, FENG Yifu. On successive-overrelaxation acceleration of MHSS iteration[J]. Journal of Shandong University(Natural Science), 2016, 51(8):61-65.
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