Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems
Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems
复制标题
二次特征值问题数值求解的标度、灵敏度和稳定性
DOI：
10.1002/nme.2076
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发表时间：
2007
影响因子：
2.9
通讯作者：
Higham N
中科院分区：
文献类型：
--
作者：
Higham N
求解二次特征值问题（QEP）（λ2M+ λD+K）x= 0 最常见的方法是将其转换为两倍维度的线性问题（λX+Y）z= 0，并通过 QZ 算法或 Krylov 方法求解线性问题。在此过程中，了解线性化过程对计算解的准确性和稳定性的影响非常重要。我们针对三个特定的线性化讨论这些问题：标准伴随线性化和保持问题对称性的两个线性化。为了便于说明，我们采用 QEP 模型来描述两端简单支撑、中点阻尼的梁的运动。我们表明，上述线性化导致梁问题的数值结果较差，但 Fan、Lin 和 Van Dooren 提出的双参数缩放解决了不稳定性。我们还表明，梁 QEP 的特征值有一半是纯虚数，并且是无阻尼问题的特征值。我们的分析利用了最近发展的理论来解释线性化的灵敏度和稳定性，并总结了其主要结论。除了主张应该常规使用缩放之外，我们还提供了有关如何选择线性化的指导，并说明了条件数和后向误差的实际价值。版权所有 © 2007 约翰·威利父子有限公司
The most common way of solving the quadratic eigenvalue problem (QEP) (λ2M+ λD+K)x= 0 is to convert it into a linear problem (λX+Y)z= 0 of twice the dimension and solve the linear problem by the QZ algorithm or a Krylov method. In doing so, it is important to understand the influence of the linearization process on the accuracy and stability of the computed solution. We discuss these issues for three particular linearizations: the standard companion linearization and two linearizations that preserve symmetry in the problem. For illustration we employ a model QEP describing the motion of a beam simply supported at both ends and damped at the midpoint. We show that the above linearizations lead to poor numerical results for the beam problem, but that a two‐parameter scaling proposed by Fan, Lin and Van Dooren cures the instabilities. We also show that half of the eigenvalues of the beam QEP are pure imaginary and are eigenvalues of the undamped problem. Our analysis makes use of recently developed theory explaining the sensitivity and stability of linearizations, the main conclusions of which are summarized. As well as arguing that scaling should routinely be used, we give guidance on how to choose a linearization and illustrate the practical value of condition numbers and backward errors. Copyright © 2007 John Wiley & Sons, Ltd.