During this thesis, we concerned about the backward error of the polynomial eigenvalue problem with certain structures. First, we depict the importance of the backward error for the polynomial eigenvalue problem. Second, we introduce structured polynomial eigenvalue problem, such as Hermitian and skew-Hermitian which were arousing from the computation of threedimensional vertex singularities of anisotropic elastic elds; the P-Conjugate-P structure came from the time delay biology systems. Third, we propose some processes to deduce the size of the backward error with the structured perturbations. In the end, some numerical examples are given to show the performance of the results.

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