TY - JOUR

T1 - Perturbation Analysis for Antitriangular Schur Decomposition

AU - Chen, Xiao Shan

AU - Li, Wen

AU - Ng, Michael K.

N1 - Funding information:
t School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China ([email protected], [email protected]). The work of these authors was supported by the Natural Science Foundation of Guangdong Province (91510631000021), the National Natural Science Foundation of China (10971075), Research Fund for the Doctoral Program of Higher Education of China Grant (20104407110001), and the Opening Project of Guangdong Province Key Laboratory of Computational Science of Sun Yat-sen University (201106005).
*Centre for Mathematical Imaging and Vision, and Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong ([email protected]). This author’s research was supported in part by an HKBU FRG grant.
Publisher copyright:
Copyright © 2012 Society for Industrial and Applied Mathematics

PY - 2012/4/17

Y1 - 2012/4/17

N2 - Let Z be an n × n complex matrix. A decomposition Z = ŪMU H is called an antitriangular Schur decomposition of Z if U is an n × n unitary matrix and M is an n × n antitriangular matrix. The antitriangular Schur decomposition is a useful tool for solving palindromic eigenvalue problems. However, there is no perturbation result for an antitriangular Schur decomposition in the literature. The main contribution of this paper is to give a perturbation bound of such decomposition and show that the bound depends inversely on f(M) := min ∥XN∥ F=1 ∥(Aup(MX L - X̄ UM), Aup(M TX L - X̄ UM T))∥ F, where X L and X U are the strictly lower triangular and upper triangular parts of X, X N = X L + X U, and Aup(Y ) denotes the strictly upper antitriangular part of Y. The quantity √2/f(M) can be used to characterize the condition number of the decomposition, i.e., when √2/f(M) is large (or small), the decomposition problem is ill-conditioned (or well-conditioned). Numerical examples are presented to illustrate the theoretical result.

AB - Let Z be an n × n complex matrix. A decomposition Z = ŪMU H is called an antitriangular Schur decomposition of Z if U is an n × n unitary matrix and M is an n × n antitriangular matrix. The antitriangular Schur decomposition is a useful tool for solving palindromic eigenvalue problems. However, there is no perturbation result for an antitriangular Schur decomposition in the literature. The main contribution of this paper is to give a perturbation bound of such decomposition and show that the bound depends inversely on f(M) := min ∥XN∥ F=1 ∥(Aup(MX L - X̄ UM), Aup(M TX L - X̄ UM T))∥ F, where X L and X U are the strictly lower triangular and upper triangular parts of X, X N = X L + X U, and Aup(Y ) denotes the strictly upper antitriangular part of Y. The quantity √2/f(M) can be used to characterize the condition number of the decomposition, i.e., when √2/f(M) is large (or small), the decomposition problem is ill-conditioned (or well-conditioned). Numerical examples are presented to illustrate the theoretical result.

KW - Antitriangular Schur form

KW - Condition number

KW - Perturbation analysis

UR - http://www.scopus.com/inward/record.url?scp=84865466823&partnerID=8YFLogxK

U2 - 10.1137/110841370

DO - 10.1137/110841370

M3 - Journal article

AN - SCOPUS:84865466823

SN - 0895-4798

VL - 33

SP - 325

EP - 335

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 2

ER -