A defective eigenvalue is well documented to be hypersensitive to data perturbations and round-off? errors, making it a formidable challenge in numerical computation particularly when the matrix is known through approximate data. This paper establishes a finitely bounded sensitivity of a defective eigenvalue with respect to perturbations that preserve the geometric multiplicity and the smallest Jordan block size. Based on this perturbation theory, numerical computation of a defective eigenvalue is regularized as a well-posed least squares problem so that it can be accurately carried out using floating point arithmetic even if the matrix is perturbed.

中文翻译：

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缺陷特征值的灵敏度和计算

有缺陷的特征值已被证明对数据扰动和四舍五入非常敏感？误差，这在数值计算中是一个巨大的挑战，特别是当通过近似数据知道矩阵时。本文建立了一个缺陷特征值相对于扰动的有限界灵敏度，该扰动保持了几何多重性和最小的Jordan块大小。基于这种扰动理论，缺陷特征值的数值计算被规范化为一个摆正的最小二乘问题，因此，即使对矩阵进行了扰动，也可以使用浮点算术来精确地执行。