# Solving a Structured Quadratic Eigenvalue Problem by a Structure-Preserving Doubling Algorithm

@article{Guo2010SolvingAS,
title={Solving a Structured Quadratic Eigenvalue Problem by a Structure-Preserving Doubling Algorithm},
author={Chun-Hua Guo and Wen-Wei Lin},
journal={SIAM J. Matrix Anal. Appl.},
year={2010},
volume={31},
pages={2784-2801}
}
• Published 1 July 2010
• Mathematics
• SIAM J. Matrix Anal. Appl.
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 upper-right 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…
26 Citations

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## References

SHOWING 1-10 OF 39 REFERENCES

### Numerical Solution of a Quadratic Matrix Equation

This paper is concerned with the efficient numerical solution of the matrix equation $AX^2 + BX + C = 0$, where A, B, C and X are all square matrices. Such a matrix X is called a solvent. This

### Algorithms for hyperbolic quadratic eigenvalue problems

• Mathematics
Math. Comput.
• 2005
It is shown that a relatively efficient test forhyperbolicity can be obtained by computing the eigenvalues of the QEP, and that a hyperbolic QEP is overdamped if and only if its largest eigenvalue is nonpositive.

### Structure-Preserving Algorithms for Palindromic Quadratic Eigenvalue Problems Arising from Vibration of Fast Trains

• Computer Science
SIAM J. Matrix Anal. Appl.
• 2008
Numerical experiments show that the proposed structure-preserving algorithms perform well on the palindromic QEP arising from a finite element model of high-speed trains and rails.

### Structured backward error for palindromic polynomial eigenvalue problems

• Computer Science
Numerische Mathematik
• 2010
The analysis reveals distinctive features of PPEP from general polynomial eigenvalue problems (PEP) investigated by Tisseur and Liu and Wang and obtained Computable formulas and bounds for the structured backward errors are obtained.

### Implicit QR algorithms for palindromic and even eigenvalue problems

• Computer Science
Numerical Algorithms
• 2008
By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version and the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence.

### Numerical analysis of a quadratic matrix equation

• Mathematics
• 2000
The quadratic matrix equation AX2+ BX + C = 0in n x nmatrices arises in applications and is of intrinsic interest as one of the simplest nonlinear matrix equations. We give a complete

### Numerical methods for palindromic eigenvalue problems: Computing the anti‐triangular Schur form

• Mathematics
Numer. Linear Algebra Appl.
• 2009
It is shown how a combination of unstructured methods followed by a structured refinement can be used to solve ill-conditioned problems with eigenvalues near the unit circle, in particular near $\pm 1$.