BiCR算法求解Sylvester矩阵方程组的Perhermitian解
The BiCR Algorithm for Solving the Perhermitian Solutions of Sylvester Matrix Equations
摘要: 对于给定的矩阵X∈Cn×n，如果SXS=SH，其中S是给定的反射矩阵，即SH=S，S2=I，则称矩阵X为perhermitian矩阵。本文提出一种用于求解Sylvester矩阵方程组的perhermitian解的双共轭残差(BiCR)算法，并且证明了该算法的收敛性。通过选择任意初始perhermitian矩阵，可以在有限步求解出Sylvester矩阵方程组的唯一最小范数perhermitian解。最后，我们给出了一些数值算例来验证该算法的有效性和可行性。
Abstract: For a given matrix X∈Cn×n , matrix X is said to be perhermitian if SXS=SH , where S is a given re-flection matrix, i.e., SH=S , S2=I . In this paper, we propose the Bi-Conjugate Residual (BiCR) algorithm for solving the perhermitian solutions of Sylvester matrix equations and prove the con-vergence of the algorithm. By choosing any initial perhermitian matrices, the unique mini-mum-norm perhermitian solutions of the Sylvester matrix equations can be solved in finite steps. Finally, we give some numerical examples to verify the validity and feasibility of the algorithm.
文章引用：唐孝伟, 李胜坤. BiCR算法求解Sylvester矩阵方程组的Perhermitian解[J]. 应用数学进展, 2023, 12(12): 4967-4986. https://doi.org/10.12677/AAM.2023.1212489

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