In this work we consider a class of delay eigenvalue problems that admit a spectrum similar to that of a Hamiltonian matrix, in the sense that the spectrum is symmetric with respect to both the real and imaginary axis. More precisely, we present a method to iteratively approximate the eigenvalues of such delay eigenvalue problems closest to a given purely real or imaginary shift, while preserving the symmetries of the spectrum. To this end the presented method exploits the equivalence between the considered delay eigenvalue problem and the eigenvalue problem associated with a linear but infinite-dimensional operator. To compute the eigenvalues closest to the given shift, we apply a specifically chosen shift-invert transformation to this linear operator and compute the eigenvalues with the largest modulus of the new shifted and inverted operator using an (infinite) Arnoldi procedure. The advantage of the chosen shift-invert transformation is that the spectrum of the transformed operator has a "real skew-Hamiltonian"-like structure. Furthermore, it is proven that the Krylov space constructed by applying this operator, satisfies an orthogonality property in terms of a specifically chosen bilinear form. By taking this property into account during the orthogonalization process, it is ensured that even in the presence of rounding errors, the obtained approximation for, e.g., a simple, purely imaginary eigenvalue is simple and purely imaginary. The presented work can thus be seen as an extension of [V. Mehrmann and D. Watkins, "Structure-Preserving Methods for Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils", SIAM J. Sci. Comput. (22.6), 2001], to the considered class of delay eigenvalue problems. Although the presented method is initially defined on function spaces, it can be implemented using finite dimensional linear algebra operations.
翻译:在这项工作中,我们考虑的是一种延迟值问题,它包含与汉密尔顿矩阵相似的频谱,即频谱是真实和想象轴的对称性。更准确地说,我们提出了一个方法,可以迭接这种延迟值的对数值,最接近于给定的纯真实或想象变化,同时保留频谱的对称。为此,我们提出的方法利用了所考虑的延迟值问题和与直线但无限的运算符有关的直径差的偏差值问题之间的等值。为了对最接近给定的变换值进行对称性对称性。我们为这个直线轴操作者应用了专门选择的变换值变换法,用新变换和旋转操作器的最大变换的调数来计算。所选择的变换法的优点是,变换的操作器的频谱可以“真实的Skew-Hatonyal ” 结构。此外,我们通过应用这个运算法来具体地平流法的变现, 算法可以显示的是“Oralalalalalalalalalal ” 的变变现过程,可以显示它是一个变变现的变现过程,可以显示一个变现的变现,可以显示一个变现。