论文标题
单数二次特征值问题：线性化和弱条件数
Singular quadratic eigenvalue problems: Linearization and weak condition numbers
论文作者
论文摘要
奇异特征值问题的数值解决方案使该系数的小扰动可能对特征值准确性有任意不良影响。但是，很长一段时间以来，尽管不可避免地存在圆形误差，但这种扰动是出色的和标准的特征值求解器（例如QZ算法）倾向于获得良好准确性。最近，Lotz和Noferini通过引入$δ$ - 效率特征值的概念来量化了这种现象。在这项工作中，我们考虑了奇异的二次特征值问题和两个流行的线性化。我们的结果表明，正确选择的线性化增加了$δ$ - 效果特征值条件数量仅略微地增加，这在数值求解器中也证明了这些线性化的使用也是在奇异的情况下。我们提出了一种非常简单但通常有效的算法，用于计算奇异二次特征值问题的良好条件特征值，通过在系数中添加小的随机扰动。我们证明，特征值条件号具有很高的概率，是可靠的标准，用于检测和排除从单数部分创建的虚假特征值。
The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $δ$-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $δ$-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.
