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 -