In this paper, based on Patel's algorithm (1993), we propose a structure-preserving algorithm or solving palindromic quadratic eigenvalue problems (QEPs). We also show the relationship between the structure-preserving algorithm and the URV-based structure-preserving algorithm by Schröder (2007). For large sparse palindromic QEPs, we develop a generalized Τ-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi algorithm for solving the resulting Τ-skew-Hamiltonian pencils. Numerical experiments show that our proposed structure-preserving algorithms perform well on the palindromic QEP arising from a finite element model of high-speed trains and rails.
- Τ-skew-Hamiltonian Pencil
- Τ-symplectic pencil
- Palindromic quadratic eigenvalue problem