位移全局GPBiCG方法求解Stein矩阵方程X + AXB = C
Shifted Global GPBiCG Method for Solving the Stein Matrix Equation X + AXB = C
摘要: 广义点积型双共轭梯度法(Generalized Product-type Biconjugate Gradient, GPBiCG)作为共轭梯度平方法(Conjugate Gradient Squared, CGS)和稳定双共轭梯度法(Biconjugate Gradient Stabilized, BiCGStab)的推广，收敛速度快于BiCGStab，收敛曲线比CGS更光滑。因此全局型GPBiCG方法被用于求解具有多个右端项的线性系统。基于全局GPBiCG方法，本文提出一种位移全局GPBiCG方法(Shifted Global GPBiCG, SGl-GPBiCG)求解Stein矩阵方程。该方法充分利用了Stein矩阵方程的位移结构。最后，我们给出数值算例验证了新方法的有效性。
Abstract: Generalized Product-type Biconjugate Gradient (GPBiCG) method can be regarded as generalizations of Conjugate Gradient Squared (CGS) and Biconjugate Gradient Stabilized (BiCGStab) methods which is faster than BiCGStab and its convergence is smoother than CGS. Then the global variant of GPBiCG method is proposed for linear systems with multiple right-hand sides. In this paper, we present a shifted global GPBiCG (SGl-GPBiCG) method for solving the Stein matrix equation based on the global GPBiCG method. The new method makes full use of the shifted structure of the Stein matrix equation. Finally, numerical examples are given to illustrate the effectiveness of the method.
文章引用：王平东, 李胜坤. 位移全局GPBiCG方法求解Stein矩阵方程X + AXB = C[J]. 应用数学进展, 2025, 14(12): 311-323. https://doi.org/10.12677/aam.2025.1412509

参考文献

[1] Bouhamidi, A. and Jbilou, K. (2007) Sylvester Tikhonov-Regularization Methods in Image Restoration. Journal of Computational and Applied Mathematics, 206, 86-98. [Google Scholar] [CrossRef] 
[2] Zhao, X., Wang, F., Huang, T., Ng, M.K. and Plemmons, R.J. (2013) Deblurring and Sparse Unmixing for Hyperspectral Images. IEEE Transactions on Geoscience and Remote Sensing, 51, 4045-4058. [Google Scholar] [CrossRef] 
[3] Datta, B. (2004) Numerical Methods for Linear Control Systems. Academic Press.
[4] Kong, H., Zhou, B. and Zhang, M.R. (2010) A Stein Equation Approach for Solutions to the Diophantine Equations. 2010 Chinese Control and Decision Conference, Xuzhou, 26-28 May 2010, 3024-3028. [Google Scholar] [CrossRef] 
[5] Zhang, Y.N., Jiang, D.C. and Wang, J. (2002) A Recurrent Neural Network for Solving Sylvester Equation with Time-Varying Coefficients. IEEE Transactions on Neural Networks, 13, 1053-1063. [Google Scholar] [CrossRef] [PubMed]
[6] Van Dooren, P.M. (2000) Gramian Based Model Reduction of Large-Scale Dynamical Systems. In: Barbosa, V.C., et al., Eds., Chapman and Hall CRC Research Notes in Mathematics, Routledge, 231-248.
[7] Saad, Y. (2003) Iterative Methods for Sparse Linear Systems. 2nd Edition, SIAM.
[8] O'Leary, D.P. (1980) The Block Conjugate Gradient Algorithm and Related Methods. Linear Algebra and Its Applications, 29, 293-322. [Google Scholar] [CrossRef] 
[9] Simoncini, V. and Gallopoulos, E. (1996) Convergence Properties of Block GMRES and Matrix Polynomials. Linear Algebra and Its Applications, 247, 97-119. [Google Scholar] [CrossRef] 
[10] Freund, R.W. and Malhotra, M. (1997) A Block QMR Algorithm for Non-Hermitian Linear Systems with Multiple Right-Hand Sides. Linear Algebra and Its Applications, 254, 119-157. [Google Scholar] [CrossRef] 
[11] Guennouni, A.E., Jbilou, K. and Sadok, H. (2003) A Block Version of BICGSTAB for Linear Systems with Multiple Right-Hand Sides. Electronic Transactions on Numerical Analysis, 16, 129-142.
[12] Jbilou, K., Messaoudi, A. and Sadok, H. (1999) Global FOM and GMRES Algorithms for Matrix Equations. Applied Numerical Mathematics, 31, 49-63. [Google Scholar] [CrossRef] 
[13] Heyouni, M. (2001) The Global Hessenberg and CMRH Methods for Linear Systems with Multiple Right-Hand Sides. Numerical Algorithms, 26, 317-332. [Google Scholar] [CrossRef] 
[14] Jbilou, K., Sadok, H. and Tinzefte, A. (2005) Oblique Projection Methods for Linear Systems with Multiple Right-Hand Sides. Electronic Transactions on Numerical Analysis, 20, 119-138.
[15] Golub, G., Nash, S. and Van Loan, C. (1979) A Hessenberg-Schur Method for the Problem AX + XB= C. IEEE Transactions on Automatic Control, 24, 909-913. [Google Scholar] [CrossRef] 
[16] Bartels, R.H. and Stewart, G.W. (1972) Solution of the Matrix Equation AX + XB = C. Communications of the ACM, 15, 820-826.
[17] 黄飞虎, 汪晓虹. 求解大型Stein方程的块Krylov子空间方法[J]. 数值计算与计算机应用, 2013, 34(1): 47-58.
[18] Jbilou, K. (2006) Low Rank Approximate Solutions to Large Sylvester Matrix Equations. Applied Mathematics and Computation, 177, 365-376. [Google Scholar] [CrossRef] 
[19] Bao, L., Lin, Y. and Wei, Y. (2007) A New Projection Method for Solving Large Sylvester Equations. Applied Numerical Mathematics, 57, 521-532. [Google Scholar] [CrossRef] 
[20] Bouhamidi, A., Hached, M., Heyouni, M. and Jbilou, K. (2011) A Preconditioned Block Arnoldi Method for Large Sylvester Matrix Equations. Numerical Linear Algebra with Applications, 20, 208-219. [Google Scholar] [CrossRef] 
[21] Zhou, B., Lam, J. and Duan, G. (2009) On Smith-Type Iterative Algorithms for the Stein Matrix Equation. Applied Mathematics Letters, 22, 1038-1044. [Google Scholar] [CrossRef] 
[22] Zhang, S. (1997) GPBi-CG: Generalized Product-Type Methods Based on Bi-CG for Solving Nonsymmetric Linear Systems. SIAM Journal on Scientific Computing, 18, 537-551. [Google Scholar] [CrossRef] 
[23] Dehghan, M. and Mohammadi-Arani, R. (2016) Generalized Product-Type Methods Based on Bi-Conjugate Gradient (GPBiCG) for Solving Shifted Linear Systems. Computational and Applied Mathematics, 36, 1591-1606. [Google Scholar] [CrossRef] 
[24] Frommer, A. (2003) BiCGStab(ℓ) for Families of Shifted Linear Systems. Computing, 70, 87-109. [Google Scholar] [CrossRef] 
[25] Zhang, J. and Dai, H. (2014) Global GPBiCG Method for Complex Non-Hermitian Linear Systems with Multiple Right-Hand Sides. Computational and Applied Mathematics, 35, 171-185. [Google Scholar] [CrossRef] 

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