非Hermite线性方程组的若干预处理迭代算法

张迎春; 李英; 肖曼玉; 谢公南

doi:10.21656/1000-0887.390222

非Hermite线性方程组的若干预处理迭代算法

doi: 10.21656/1000-0887.390222

¹西北工业大学航海学院机械与动力工程系, 西安 710072;²商丘师范学院信息技术学院, 河南商丘 476000;³西北工业大学应用数学系, 西安 710129

基金项目: 国家自然科学基金(51676163)

详细信息

作者简介:
张迎春（1992—），女，博士生(E-mail: zhangyingchun@mail.nwpu.edu.cn)；谢公南（1980—），男，教授，博士，博士生导师(通讯作者. E-mail: xgn@nwpu.edu.cn).

中图分类号: O246
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出版历程
- 收稿日期: 2018-08-23
- 修回日期: 2018-09-02
- 刊出日期: 2019-03-01

Some Preconditioning Iterative Algorithms for Non-Hermitian Linear Equations

¹Department of Mechanical and Power Engineering, School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, P.R.China;²School of Information Technology, Shangqiu Normal University, Shangqiu, Henan 476000, P.R.China;³Department of Applied Mathematics, Northwestern Polytechnical University,Xi’an 710129, P.R.China

Funds: The National Natural Science Foundation of China(51676163)

摘要

摘要: 非Hermite线性方程组在科学和工程计算中有着重要的理论研究意义和使用价值，因此如何高效求解该类线性方程组，一直是研究者所探索的方向.通过提出一种预处理方法，对非Hermite线性方程组和具有多个右端项的复线性方程组求解的若干迭代算法进行预处理，旨在提高原算法的收敛速度.最后通过数值试验表明，所提出的若干预处理迭代算法与原算法相比较，预处理算法迭代次数大大降低，且收敛速度明显优于原算法.除此之外，广义共轭A-正交残量平方法(GCORS2)的预处理算法与其他算法相比，具有良好的收敛性行为和较好的稳定性.
- 非Hermite线性方程组 /
- 广义共轭A-正交残量平方法 /
- 预处理方法
Abstract: Non-Hermitian linear equations have extensive application in scientific and engineering calculations and are expected to be solved with high efficiency. To accelerate the convergence rate of original algorithms, a preconditioning technique was developed and applied to some iterative methods chosen to solve the nonHermitian linear equations and complex linear systems with multiple righthand sides. Several numerical experiments show that the preconditioned iterative methods are superior to the original methods in terms of both the convergence rate and the number of iterations. In addition, the preconditioned generalized conjugate A-orthogonal residual squared method (GCORS2) has better convergent behavior and stability than other preconditioned methods.
- non-Hermitian linear equations /
- generalized conjugate A-orthogonal residual squared method /
- preconditioning method

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参考文献(18)

[1]	BAZN F S V, KLEEFELD A, LEEM K H, et al. Sampling method based projection approach for the reconstruction of 3D acoustically penetrable scatterers[J]. Linear Algebra and Its Applications,2016,495(15): 289-323.
[2]	SADD Y. A flexible inner-outer preconditioned GMRES algorithm[J]. SIAM Journal on Scientific Computing,1993,14(2): 461-469.
[3]	BAI A, DAY D, DONGARRA J, et al. A test matrix collection for non-Hermitian eigenvalue problems: CS-97-355[R]. Knoxville, TN: Department of Computer Science, University of Tennessee, 1997.
[4]	BAYLISS A, GLODSTEIN C I, TURKEL E. An iteration method for the Helmholtz equation[J]. Journal of Computational Physics,1983,49(3): 443-457.
[5]	HU Q Y, YUAN L. A plane-wave least-squares method for time-harmonic Maxwell’s equations in absorbing media[J]. SIAM Journal on Scientific Computing,2014,36(4): 1937-1959.
[6]	HUTTUNEN T, MALINEN M, MONK P. Solving Maxwell’s equations using the ultra weak variational formulation[J]. Journal of Computational Physics,2007,223(2): 731-758.
[7]	SAAD Y, SCHULTZ M H. A generalized minimum residual algorithm for solving nonsymmetirc linear systems[J]. SIAM Journal on Scientific and Statistical Computing,1986,7(3): 856-869.
[8]	DU K. GMRES with adaptively deflated restarting and its performance on an electromagnetic cavity problem[J]. Applied Numerical Mathematics,2011,61(9): 977-988.
[9]	VAN DERVORST H A. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems[J].SIAM Journal on Scientific and Statistical Computing,1992,13(2): 631-644.
[10]	DEHGHAN M, MOHAMMADI-ARANI R. Generalized product-type methods based on bi-conjugate gradient (GPBiCG) for solving shifted linear systems[J]. Computational and Applied Mathematics,2017,36(4): 1591-1606.
[11]	ZHAO L, HUANG T Z, JING Y F, et al. A generalized product-type BiCOR method and its application in signal deconvolution[J]. Computers and Mathematics With Applications,2013,66(8): 1372-1388.
[12]	GU X M, HUANG T Z, CARPENTIERI B, et al. A hybridized iterative algorithm of the BiCORSTAB and GPBiCOR methods for solving non-Hermitian linear systems[J].Computers and Mathematics With Applications,2015,70(12): 3019-3031.
[13]	ZHANG J H, DAI H. Generalized conjugate A-orthogonal residual squared method for complex non-Hermitian linear systems[J]. Journal of Computational Mathematics,2014,32(3): 248-265.
[14]	张建华, 戴华. 求解具有多个右端项线性方程组的总体CGS算法[J]. 高等学校计算数学学报, 2008,30(4): 390-399.(ZHANG Jianhua, DAI Hua. Global CGS algorithm for linear systems with multiple right-hand sides[J]. Numerical Mathematics a Journal of Chinese Universities,2008,30(4): 390-399.(in Chinese))
[15]	ZHANG J H, DAI H. Global GPBiCG method for complex non-Hermitian linear systems with multiple right-hand sides[J]. Computational and Applied Mathematics,2016,35(1): 171-185.
[16]	张建华. 非Hermitian线性方程组的若干迭代方法及其预处理[D]. 博士学位论文. 南京: 南京航空航天大学, 2016.(ZHANG Jianhua. Some iterative methods and their preconditioned variants for non-Hermitian linear systems[D]. PhD Thesis. Nanjing: Nanjing University of Aeronautics and Astronautics, 2016.(in Chinese))
[17]	富明慧, 李勇息. 求解病态线性方程组的预处理精细积分法[J]. 应用数学和力学, 2018,39(4): 462-469.(FU Minghui, LI Yongxi. A preconditioned precise integration method for solving ill-conditioned linear equations[J]. Applied Mathematics and Mechanics,2018,39(4): 462-469.(in Chinese))
[18]	DAVIS T A, HU Y F. The university of Florida sparse matrix collection[J].ACM Transactions on Mathematical Software,2011,38(1): 1-25.