Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems

In this paper, we introduce a generalized Krylov subspace ${\mathcal{G}_{m}(\mathbf{A};\mathbf{u})}$ based on a square matrix sequence {Aj} and a vector sequence {uj}. Next we present a generalized Arnoldi procedure for generating an orthonormal basis of ${\mathcal{G}_{m}(\mathbf{A};\mathbf{u})}$. By applying the projection and the refined technique, we derive a restarted generalized Arnoldi method and a restarted refined generalized Arnoldi method for solving a large-scale polynomial eigenvalue problem (PEP). These two methods are applied to solve the PEP directly. Hence they preserve essential structures and properties of the PEP. Furthermore, restarting reduces the storage requirements. Some theoretical results are presented. Numerical tests report the effectiveness of these methods.

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