Improved bisection eigenvalue method for band symmetric Toeplitz matrices

Electronic Transactions on Numerical Analysis

article

Yuli Eidelman¹  Iulian Haimovici¹ 

[1] School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University

关键词: Toeplitz;    quasiseparable;    banded matrices;    eigenstructure;    inequalities;    Sturm with bisection;   

DOI  :  10.1553/etna_vol58s316

学科分类：数学（综合）

来源: Kent State University * Institute of Computational Mathematics

【 摘 要 】

We apply a general bisection eigenvalue algorithm, developed for Hermitian matrices with quasiseparable representations, to the particular case of real band symmetric Toeplitz matrices. We show that every band symmetric Toeplitz matrix $T_q$ with bandwidth $q$ admits the representation $T_q=A_q+H_q$, where the eigendata of $A_q$ are obtained explicitly and the matrix $H_q$ has nonzero entries only in two diagonal blocks of size $(q-1)\times (q-1)$. Based on this representation, one obtains an interlacing property of the eigenvalues of the matrix $T_q$ and the known eigenvalues of the matrix $A_q$. This allows us to essentially improve the performance of the bisection eigenvalue algorithm. We also present an algorithm to compute the corresponding eigenvectors.

【 授权许可】

Unknown