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

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。

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.