A Fast Algorithm For Fast Train Palindromic Quadratic Eigenvalue Problems

In the vibration analysis of high speed trains arises such a palindromic quadratic eigenvalue problem (PQEP) $(\lambda^2 A^{\rm T}+ \lambda Q + A)z=0$, where $A,\, Q\in {\mathbb C}^{n\times n}$ have special structures: both $Q$ and $A$ are $m\times m$ block matrices with each block being $k \times k$ (thus $n=m \times k$), and $Q$ is complex symmetric and tridiagonal block-Toeplitz, and $A$ has only one nonzero block in the $(1,m)$th block position which is the same as the subdiagonal block of $Q$. This PQEP has eigenvalues 0 and $\infty$ each of multiplicity $(m-1)k$ just by examining $A$, but it is its remaining $2k$ eigenvalues, usually nonzero and finite but with an extreme wide range in magnitude, that are of interest. The problem is notoriously difficult numerically. Earlier methods that seek to deflate eigenvalues $0$ and $\infty$ first often produce eigenvalues that are too inaccurate to be useful due to the large errors introduced in the deflation process. The solvent approach proposed by Guo and...