The authors deal with a particular structured quadratic eigenvalue problem whose application arises in the context of vibration analysis of high speed trains. This well studied problem has not been solved satisfactorily yet. Here the solvent approach is applied and it is shown that the suggested implementation has a complexity which can be compared with other current algorithms. This algorithm exploits the sparsity of the problem and is structure preserving. Theoretical results are proved by a generalization of the classical Bendixson’s theorem [J. Bendixson, Acta Math. 25, 359–365 (1902; JFM 33.0106.01)]. In particular, since the algorithm is shown to be quadratically convergent, it enjoys a better accuracy than existing methods. Effective numerical examples highlight the efficiency and the accuracy properties of the presented method.