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：本文研究密度泛函理论（DFT）中计算问题的浮点算法，DFT是确定固体（如金属、氧化物或半导体）电子结构的强效技术。具体而言，针对局域基组中描述的Kohn-Sham方程对应的原子体系，我们寻求密度矩阵及相应电子密度的可证明性质算法。其核心问题是形如$HC=SCE$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。我们开发并组合多种方法以求解该问题。首先描述有限精度下的厄米伪谱分裂方法，并利用它获得一种新的无关谱隙的浮点算法，能在$O(T_{MM}(n)\log^2(\tfrac{n}{\delta}))$内计算厄米矩阵所有特征值的附加误差$\delta$，其中对任意$\eta>0$有$T_{MM}(n) = O(n^{\omega+\eta})$，$\omega\leq 2.371552$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$O(n^\omega)$位复杂度的算法。作为副产品，我们还给出了矩形矩阵所有奇异值的附加误差近似，以及对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，我们对对数稳定的Cholesky分解算法进行了新颖分析，证明其能在$O(T_{MM}(n))$内将厄米正定广义特征问题精确转化为厄米特征问题。所有工具组合后，首次实现了对原子体系密度矩阵和电子密度的近$O(T_{MM}(n))$位复杂度的可证明精确浮点算法。