Hermitian Toeplitz矩阵向量乘积的快速算法

数学理论与应用 ›› 2017, Vol. 37 ›› Issue (3-4): 38-42.

Hermitian Toeplitz矩阵向量乘积的快速算法

刘仲云¹，陈思恒¹ ，徐伟进¹，张育林²

1.长沙理工大学数学与统计学院
2.Minho大学数学中心,葡萄牙

出版日期:2017-12-30 发布日期:2020-09-21
基金资助:
国家自然科学基金资助项目(11371075)

A Fast Algorithm for Computing Products of Hermitian Toeplitz Matrices and Vectors

Liu Zhongyun， Chen Siheng， Xu Weijin，Zhang Yulin

1.School of Mathematics and Statistics,Changsha University of Science and Technology
2.Centro de Matemática,Universidade do Minho

Online:2017-12-30 Published:2020-09-21

摘要/Abstract

摘要： 众所周知,大规模 HermitianToeplitz矩阵向量乘积Ax 可由快速 Fourier变换 (FFT)进行计算.事实上,HermitianToeplitz矩阵在酉相似变换下可约化为一个实的 Toeplitz矩阵与 Hankel矩阵之和.基于此, 本文利用 DCT和 DST,构造了一个更有效的方法,只需O(n)的复运算.

关键词: Hermitian Toeplitz矩阵, 矩阵向量乘法, DCT, DST, 实运算

Abstract: It is known that the product Axof a large scale Hermitian Toeplitz matrix Aand a vector xcan be computed effectively by using the Fast Fourier Transform(FFT).In this paper,based on the fact that an Her-mitian Toeplitz matrix Acan be reduced into a real Toeplitz-plus-Hankel matrix(A=T+H)by a unitarysimilarity transformation(the unitary matrix is U=1/√2(I-iJ), we develop a more efficient algorithm thatonly O(n)complex arithmetics are included for computing the product Ax by employing the DCT and DST.

Key words: Hermitian Toeplitz matri, x Matrix-vector multiplication, , DCT, DST, Real operation

刘仲云, 陈思恒, 徐伟进, 张育林. Hermitian Toeplitz矩阵向量乘积的快速算法[J]. 数学理论与应用, 2017, 37(3-4): 38-42.

Liu Zhongyun, Chen Siheng, Xu Weijin, Zhang Yulin. A Fast Algorithm for Computing Products of Hermitian Toeplitz Matrices and Vectors[J]. Mathematical Theory and Applications, 2017, 37(3-4): 38-42.