Hermitian Toeplitz线性方程组的新预处理方法

摘要

本文研究解Hermitian Toeplitz线性方程组Ax =b的预处理共轭梯度法.基于Hermitian Toeplitz 矩阵可通过酉相似转化为一个实Toeplitz矩阵与一个Hankel矩阵的和(UAU*=T+H)的结论,我们首先将Ax=b转化为实线性方程组(T+H)[x1,x2]=[b1,b2].然后,我们提出一个新预处理子来求解这两个方程组.特剐地,我们采用DCT和DST求解,只涉及到实运算.我们分析预处理矩阵的谱性质,并讨论每步迭代的计算复杂度.数值实验表明该预处理子是有效的.%In this paper we give a preconditioned conjugate gradient method (PCG) to solve the Hermitian Toeplitz system Ax =b.Based on the fact that an Hermitian Toeplitz matrix A can be reduced into a real Toeplitz matrix plus a Hankel matrix (i.e.,UAU* =T+ H) by a unitary similarity transformation (this unitary matrix is U=(I-iJ)/√2),we first reduce the system Ax=b to a real linear systems (T+H)[x1,x2]=[b1,b2].Then we propose a new preconditioner for solving those two systems.In particular,our solver only involves real arithmetics when the discrete sine transform (DST) and discrete cosine transform (DCT) are used.The spectral properties of the preconditioned matrix are analyzed,and the computational complexity is discussed.Numerical experiments show that our preconditioner performs well for solving the Hermitian Toeplitz systems.