# 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} }

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

### An Improved Structure-Preserving Doubling Algorithm For a Structured Palindromic Quadratic Eigenvalue Problem

- Computer Science, Mathematics
- 2014

An improved version of Guo and Lin’s efficient solvent approach which solves the palindromic quadratic eigenvalue problem (PQEP) by computing the so-called stabilizing solution to the mk×mk nonlinear matrix equation X + ATX−1A = Q via the doubling algorithm.

### A Fast Algorithm For Fast Train Palindromic Quadratic Eigenvalue Problems

- MathematicsSIAM J. Sci. Comput.
- 2016

This paper presents a solvent approach to solve a palindromic quadratic eigenvalue problem in vibration analysis of high speed trains that has eigenvalues 0 and $\infty$ each of multiplicity $(m-1)k$ just by examining $A$, but it is its remaining $2k$ eigen values that are of interest.

### A new look at the doubling algorithm for a structured palindromic quadratic eigenvalue problem

- MathematicsNumer. Linear Algebra Appl.
- 2015

Replacing Guo and Lin's key intermediate step by a modified one leads to an alternative method for the PQEP that is faster, but the improvement in speed is not as dramatic as just for solving the respective nonlinear matrix equations and levels off as m increases.

### A decoupled form of the structure-preserving doubling algorithm with low-rank structures

- Computer ScienceArXiv
- 2020

A new decoupled form of the SDA is proposed, building on the associated Krylov subspaces thus leading to the inherent low-rank structures, and the resulting dSDA is much more efficient since only one quantity is computed iteratively.

### Structured backward error for palindromic polynomial eigenvalue problems, II: Approximate eigentriplets

- Computer ScienceFrontiers of Mathematics in China
- 2018

Several informative and very sharp upper bounds are revealed that reveal the critical cases in which there is no structured backward perturbation such that the given approximate eigentriplet becomes an exact one of any perturbed PPEP, unless further additional conditions are imposed.

### Updating $$\star $$⋆-palindromic quadratic systems with no spill-over

- MathematicsComputational and Applied Mathematics
- 2018

This paper concerns the model updating problems of the $$\star $$⋆-palindromic quadratic system $$P(\lambda )=\lambda ^2 A^{\star }+\lambda Q+A$$P(λ)=λ2A⋆+λQ+A, where $$A, Q\in {\mathbb {C}}^{n\times…

### Structured backward error for palindromic polynomial eigenvalue problems

- Computer ScienceNumerische 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.

### On the tripling algorithm for large-scale nonlinear matrix equations with low rank structure

- Computer ScienceJ. Comput. Appl. Math.
- 2015

### On Inverse Eigenvalue Problems of Quadratic Palindromic Systems with Partially Prescribed Eigenstructure

- MathematicsTaiwanese Journal of Mathematics
- 2019

. The palindromic inverse eigenvalue problem (PIEP) of constructing matrices A and Q of size n × n for the quadratic palindromic polynomial P ( λ ) = λ 2 A (cid:63) + λQ + A so that P ( λ ) has p…

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