求解复对称线性系统的极小残差非平衡修正HSS迭代法
Minimum Residual Lopsided Modified HSS Iteration Method for a Class of Complex Symmetric Linear Systems
摘要: 结合非平衡MHSS (LMHSS)迭代法和极小残差策略，本文提出了极小残差LMHSS (MRLMHSS)迭代法，并推导了两个广义参数的计算公式以及分析了所提出方法的收敛性。数值实验验证了所提出方法的有效性和优越性。
Abstract: Combining the minimum residual technique and the lopsided MHSS (LMHSS) method, the minimum residual LMHSS (MRLMHSS) iteration method is established. The formulas for computing these two parameters are derived, and a detailed convergence analysis is presented. Numerical results demonstrate the effectiveness and superiority of the proposed methods.
文章引用：刘小杰. 求解复对称线性系统的极小残差非平衡修正HSS迭代法[J]. 应用数学进展, 2025, 14(6): 1-12. https://doi.org/10.12677/aam.2025.146295

参考文献

[1] Benzi, M. and Bertaccini, D. (2008) Block Preconditioning of Real-Valued Iterative Algorithms for Complex Linear Systems. IMA Journal of Numerical Analysis, 28, 598-618. [Google Scholar] [CrossRef] 
[2] Arridge, S.R. (1999) Optical Tomography in Medical Imaging. Inverse Problems, 15, R41-R93. [Google Scholar] [CrossRef] 
[3] Bertaccini, D. (2004) Efficient Solvers for Sequences of Complex Symmetric Linear Systems. Electronic Transactions on Numerical Analysis, 18, 49-64.
[4] You, Y.-L. and Kaveh, M. (2000) Fourth-Order Partial Differential Equations for Noise Removal. IEEE Transactions on Image Processing, 9, 1723-1730. [Google Scholar] [CrossRef] [PubMed]
[5] Bai, Z. (2011) Block Alternating Splitting Implicit Iteration Methods for Saddle‐Point Problems from Time‐Harmonic Eddy Current Models. Numerical Linear Algebra with Applications, 19, 914-936. [Google Scholar] [CrossRef] 
[6] Feriani, A., Perotti, F. and Simoncini, V. (2000) Iterative System Solvers for the Frequency Analysis of Linear Mechanical Systems. Computer Methods in Applied Mechanics and Engineering, 190, 1719-1739. [Google Scholar] [CrossRef] 
[7] Poirier, B. (2000) Efficient Preconditioning Scheme for Block Partitioned Matrices with Structured Sparsity. Numerical Linear Algebra with Applications, 7, 715-726. [Google Scholar] [CrossRef] 
[8] Day, D. and Heroux, M.A. (2001) Solving Complex-Valued Linear Systems via Equivalent Real Formulations. SIAM Journal on Scientific Computing, 23, 480-498. [Google Scholar] [CrossRef] 
[9] Bai, Z., Golub, G.H. and Ng, M.K. (2003) Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems. SIAM Journal on Matrix Analysis and Applications, 24, 603-626. [Google Scholar] [CrossRef] 
[10] Li, L., Huang, T. and Liu, X. (2007) Modified Hermitian and Skew‐Hermitian Splitting Methods for Non‐Hermitian Positive‐Definite Linear Systems. Numerical Linear Algebra with Applications, 14, 217-235. [Google Scholar] [CrossRef] 
[11] Bai, Z., Benzi, M. and Chen, F. (2010) Modified HSS Iteration Methods for a Class of Complex Symmetric Linear Systems. Computing, 87, 93-111. [Google Scholar] [CrossRef] 
[12] Li, X., Yang, A. and Wu, Y. (2014) Lopsided PMHSS Iteration Method for a Class of Complex Symmetric Linear Systems. Numerical Algorithms, 66, 555-568. [Google Scholar] [CrossRef] 
[13] Yang, A., Cao, Y. and Wu, Y. (2019) Minimum Residual Hermitian and Skew-Hermitian Splitting Iteration Method for Non-Hermitian Positive Definite Linear Systems. BIT Numerical Mathematics, 59, 299-319. [Google Scholar] [CrossRef] 
[14] Hu, Q. and Zou, J. (2001) An Iterative Method with Variable Relaxation Parameters for Saddle-Point Problems. SIAM Journal on Matrix Analysis and Applications, 23, 317-338. [Google Scholar] [CrossRef] 
[15] Gutknecht, M.H. and Rozložník, M. (2001) By How Much Can Residual Minimization Accelerate the Convergence of Orthogonal Residual Methods? Numerical Algorithms, 27, 189-213. [Google Scholar] [CrossRef] 
[16] Gutknecht, M.H. and Rosložnik, M. (2001) Residual Smoothing Techniques: Do They Improve the Limiting Accuracy of Iterative Solvers? Bit Numerical Mathematics, 41, 86-114. [Google Scholar] [CrossRef] 
[17] Bai, Z., Benzi, M. and Chen, F. (2011) On Preconditioned MHSS Iteration Methods for Complex Symmetric Linear Systems. Numerical Algorithms, 56, 297-317. [Google Scholar] [CrossRef] 
[18] Zhang, W., Yang, A. and Wu, Y. (2024) Novel Minimum Residual MHSS Iteration Method for Solving Complex Symmetric Linear Systems. Applied Mathematics Letters, 148, Article 108869. [Google Scholar] [CrossRef] 
[19] Huang, Z. (2021) Modified Two-Step Scale-Splitting Iteration Method for Solving Complex Symmetric Linear Systems. Computational and Applied Mathematics, 40, Article No. 122. [Google Scholar] [CrossRef] 
[20] Huang, N. (2022) Variable-Parameter HSS Methods for Non-Hermitian Positive Definite Linear Systems. Linear and Multilinear Algebra, 70, 6664-6681. [Google Scholar] [CrossRef] 
[21] Bai, Z. and RozložnÍk, M. (2015) On the Numerical Behavior of Matrix Splitting Iteration Methods for Solving Linear Systems. SIAM Journal on Numerical Analysis, 53, 1716-1737. [Google Scholar] [CrossRef] 
[22] Zhang, W., Yang, A. and Wu, Y. (2021) Minimum Residual Modified HSS Iteration Method for a Class of Complex Symmetric Linear Systems. Numerical Algorithms, 86, 1543-1559. [Google Scholar] [CrossRef] 

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