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On Implicit Restarting Of Second Order Arnoldi Procedure For Quadratic Eigenvalue Problem

Quadratic eigenvalue problem (QEP) is often solved by linearizing and then deploying well known techniques to solve the resulting linear (generalized) eigenproblem. However, a generic linear eigensolver is unaware of the underlying structure of the quadratic problem, which may cause loss of important structural spectral properties of the original problem. Bai and Su (2005.) first realized that in the case of iterative Arnoldi-type methods, it is advantageous to apply the Rayleigh-Ritz projection directly to the initial QEP. To that end, they introduced second order Krylov subspaces, and the corresponding second order Arnoldi procedure for generating orthonormal bases. The resulting method, Second Order Arnoldi (SOAR), is further modified yielding TOAR (Lu, Su and Bai, 2016). SOAR procedure is also used for dimension reduction of large scale second order dynamical systems. The key feature of this approach is perseverance of the structure of the dynamical system. In this talk, we will present implicit restarting in SOAR(TOAR) in context of computing the prescribed number of eigenvalues of QEP. The wanted number of eigenvalues is much smaller than the dimension of the original problem. The emphasize of the talk will be the issues of better choices of starting vectors, choosing shifts to construct polynomial filters during the restart process, and extracting the wanted eigenvalues and eigenvectors, with particular attention to the peculiarities of the quadratic problem.

eigenvalues ; quadratic eigenvalue problem ; Arnoldi algorithm ; Krylov subspace ; SOAR

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