We critically discuss the problem of finding the λ-index of a real symmetric matrix M , defined as the number of eigenvalues smaller than λ, using the entries of M as only input. We show that a widely used formula based on the branch-cut structure of the complex logarithm should be handled with care, as it generically fails to produce the correct result if the same branch is chosen for the two logarithms. We improve the formula using multidimensional Fresnel integrals, showing that even the new version provides at most a self-consistency equation for , whose solution is not guaranteed to be unique. Our results are corroborated by explicit examples and numerical evaluations.

我们危重讨论找到的问题λ -index一个实对称矩阵的中号，定义为特征值小于数量λ，使用条目的中号作为唯一的输入。我们表明，应谨慎处理基于复杂对数的分支割结构的广泛使用的公式，因为如果为两个对数选择相同的分支，则通常无法产生正确的结果。我们使用多维菲涅尔积分改进了公式，表明即使新版本也最多提供了一个自洽方程 ，其解决方案不能保证是唯一的。明确的例子和数值评估证实了我们的结果。