dc.contributor.author 
Guo, ChunHua 

dc.contributor.author 
Lin, WenWei 

dc.date.accessioned 
20140427T23:16:01Z 

dc.date.available 
20140427T23:16:01Z 

dc.date.issued 
2010 

dc.identifier.citation 
SIAM J. Matrix Anal. Appl. 
en_US 
dc.identifier.uri 
http://hdl.handle.net/10294/5264 

dc.description.abstract 
In studying the vibration of fast trains, we encounter a palindromic quadratic eigenvalue problem (QEP) $(\lambda^2 A^T + \lambda Q + A)z = 0$, where $A, Q \in \mathbb{C}^{n \times n}$ and $Q^T = Q$. Moreover, the matrix $Q$ is block tridiagonal and block Toeplitz,
and the matrix $A$ has only one nonzero block in the upperright corner. So most of the
eigenvalues of the QEP are zero or infinity. In a linearization approach, one typically starts with
deflating these known eigenvalues, for the sake of efficiency. However, this initial deflation process involves the inverses
of two potentially illconditioned matrices. As a result, large error might be introduced into the data for the
reduced problem. In this paper we propose using the solvent approach
directly on the original QEP, without any deflation process. We apply a structurepreserving doubling
algorithm to compute the stabilizing solution of the matrix equation
$X+A^TX^{1}A=Q$, whose existence is guaranteed by a result on the WienerHopf factorization of
rational matrix functions associated with semiinfinite block Toeplitz matrices and a generalization of Bendixson's
theorem to bounded linear operators on Hilbert spaces. The doubling algorithm is shown to
be well defined and quadratically convergent. The complexity of the doubling algorithm is
drastically reduced by using the ShermanMorrisonWoodbury formula and the special structures
of the problem. Once the stabilizing solution is obtained, all nonzero finite eigenvalues
of the QEP can be found efficiently and with the automatic reciprocal relationship, while
the known eigenvalues at zero or infinity remain intact. 
en_US 
dc.description.sponsorship 
NSERC, NSC (Taiwan), NCTS (Taiwan) 
en_US 
dc.language.iso 
en 
en_US 
dc.publisher 
SIAM 
en_US 
dc.title 
Solving a structured quadratic eigenvalue problem by a structurepreserving doubling algorithm 
en_US 
dc.type 
Article 
en_US 
dc.description.authorstatus 
Faculty 
en_US 
dc.description.peerreview 
yes 
en_US 