Floating point algorithms are studied for computational problems arising in Density Functional Theory (DFT), a powerful technique to determine the electronic structure of solids, e.g., metals, oxides, or semiconductors. Specifically, we seek algorithms with provable properties for the density matrix and the corresponding electron density in atomic systems described by the Kohn-Sham equations expressed in a localized basis set. The underlying problem is a Hermitian generalized eigenvalue problem of the form $HC=SCE$, where $H$ is Hermitian and $S$ is Hermitian positive-definite (HPD). Different methods are developed and combined to solve this problem. We first describe a Hermitian pseudospectral shattering method in finite precision, and use it to obtain a new gap-independent floating point algorithm to compute all eigenvalues of a Hermitian matrix within an additive error $\delta$ in $O(T_{MM}(n)\log^2(\tfrac{n}{\delta}))$. Here $T_{MM}(n) = O(n^{\omega+\eta})$, for any $\eta>0$, and $\omega\leq 2.371552$ is the matrix multiplication exponent.To the best of our knowledge, this is the first algorithm to achieve nearly $O(n^\omega)$ bit complexity for all Hermitian eigenvalues. As by-products, we also demonstrate additive error approximations for all singular values of rectangular matrices, and, for full-rank matrices, relative error approximations for all eigenvalues, all singular values, the spectral norm, and the condition number. We finally provide a novel analysis of a logarithmically-stable Cholesky factorization algorithm, and show that it can be used to accurately transform the HPD generalized eigenproblem to a Hermitian eigenproblem in $O(T_{MM}(n))$. All these tools are combined to obtain the first provably accurate floating point algorithms with nearly $O(T_{MM}(n))$ bit complexity for the density matrix and the electron density of atomic systems.

翻译：针对密度泛函理论（DFT）中出现的计算问题，本文研究了浮点算法。密度泛函理论是一种用于确定固体（如金属、氧化物或半导体）电子结构的强大技术。具体而言，我们寻求针对原子系统中密度矩阵及相应电子密度的可证明性质算法，这些系统由采用局域基组表达的Kohn-Sham方程描述。其根本问题是形式为$HC=SCE$的厄米广义特征值问题，其中$H$为厄米矩阵，$S$为厄米正定矩阵。我们开发并组合了多种方法来求解该问题。首先，我们描述了有限精度下的厄米伪谱破碎方法，并利用该方法获得了一种新的与间隙无关的浮点算法，可在$O(T_{MM}(n)\log^2(\tfrac{n}{\delta}))$时间内计算厄米矩阵的所有特征值，附加误差为$\delta$。此处$T_{MM}(n) = O(n^{\omega+\eta})$（对任意$\eta>0$），且$\omega\leq 2.371552$为矩阵乘法指数。据我们所知，这是首个对所有厄米特征值实现近$O(n^\omega)$比特复杂度的算法。作为副产品，我们还展示了矩形矩阵所有奇异值的附加误差近似，以及满秩矩阵所有特征值、所有奇异值、谱范数和条件数的相对误差近似。最后，我们提供了一种对数稳定Cholesky分解算法的新分析，并证明该算法可用于在$O(T_{MM}(n))$时间内将厄米正定广义特征值问题精确转化为厄米特征值问题。所有这些工具被结合，首次为原子系统的密度矩阵和电子密度提供了具有近$O(T_{MM}(n))$比特复杂度的可证明精确浮点算法。