期刊论文

A New Preconditioning Technique for Hermitian Toeplitz Systems

中文

Shuxue Lilun yu Yingyong

1006-8074

2018

38

03

50-58

国家自然科学基金资助项目（11371075)；

本校为第一机构

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 si...