The eigenvalue shift technique is the most well-known and fundamental tool for matrix computations. Applications include the search of eigeninformation, the acceleration of numerical algorithms, the study of Google's PageRank. The shift strategy arises from the concept investigated by Brauer [1] for changing the value of an eigenvalue of a matrix to the desired one, while keeping the remaining eigenvalues and the original eigenvectors unchanged. The idea of shifting distinct eigenvalues can easily be generalized by Brauer's idea. However, shifting an eigenvalue with multiple multiplicities is a challenge issue and worthy of our investigation. In this work, we propose a new way for updating an eigenvalue with multiple multiplicities and thoroughly analyze its corresponding Jordan canonical form after the update procedure.
A structure preserving shift-invert infinite Arnoldi algorithm for a class of delay eigenvalue problems with Hamiltonian symmetry
在这项工作中，我们考虑一类延迟本征值问题，承认频谱类似的哈密顿矩阵，在这个意义上说，频谱是对称的实轴和虚轴。更确切地说，我们提出了一种方法来迭代近似的特征值，这样的延迟本征值问题最接近一个给定的纯实或虚的移位，同时保持对称性的频谱。为此，所提出的方法利用考虑延迟本征值问题和本征值问题与线性，但无限维运营商之间的等价性。为了计算最接近给定移位的本征值，我们对该线性算子应用专门选择的移位-反转变换，并使用（无限）Arnoldi过程计算具有新移位和反转算子的最大模的本征值。所选择的移位-反转变换的优点在于变换后的算子的谱具有“实斜哈密顿”类结构。进一步证明了利用该算子构造的Krylov空间满足关于特定双线性型的正交性。通过在正交化过程中考虑该性质，确保了即使在存在舍入误差的情况下，简单的、纯虚的本征值是简单的和纯虚的。因此，所提出的工作可以被看作是[V. Mehrmann和D. Watkins，“Structure-Preserving Methods for Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils”，SIAM J. Sci. Comput.（22.6），2001]，应用于所考虑的时滞本征值问题。虽然所提出的方法最初是在函数空间上定义的，但它可以使用有限维线性代数运算来实现。
A Multigrid Method Based On Shifted-Inverse Power Technique for Eigenvalue Problems
Xiamen University（厦门大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Tianjin University of Finance and Economics（天津财经大学）
Accurate eigenvalue decomposition of arrowhead matrices and applications
Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split（斯普利特大学电气、机械和造船工程学院）；Department of Computer Science and Engineering, The Pennsylvania State University（宾夕法尼亚州立大学计算机科学与工程系）
Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices
Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split（斯普利特大学电气、机械与航海工程学院）；Department of Computer Science and Engineering, The Pennsylvania State University（宾夕法尼亚州立大学计算机科学与工程系）
How long does it take to compute the eigenvalues of a random symmetric matrix?
J.P. Morgan Securities LLC（摩根大通证券）；Courant Institute for Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）
