SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page C265-C293, January 2020.
In this paper we propose an accurate, highly parallel algorithm for the generalized eigendecomposition of a matrix pair $(H, S)$, given in a factored form $(F^{\ast} J F, G^{\ast} G)$. Matrices $H$ and $S$ are generally complex and Hermitian, and $S$ is positive definite. This type of matrix emerges from the representation of the Hamiltonian of a quantum mechanical system in terms of an overcomplete set of basis functions. This expansion is part of a class of models within the broad field of density functional theory, which is considered the gold standard in condensed matter physics. The overall algorithm consists of four phases, the second and fourth being optional, where the two last phases are a computation of the generalized hyperbolic singular value decomposition (SVD) of a complex matrix pair $(F,G)$, according to a given matrix $J$ defining the hyperbolic scalar product. If $J = I$, then these two phases compute the generalized SVD (GSVD) in parallel very accurately and efficiently.

中文翻译：

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基于Hari-Zimmermann广义双曲SVD的特征分解LAPW方法

SIAM科学计算杂志，第42卷，第5期，第C265-C293页，2020年1月。
在本文中，我们为矩阵对$（H，S）$的广义特征分解提出了一种精确，高度并行的算法，以分解形式$（F ^ {\ ast} JF，G ^ {\ ast} G）给出$。矩阵$ H $和$ S $通常是复数和Hermitian矩阵，而$ S $是正定矩阵。这种类型的矩阵是从量子力学系统的哈密顿量的表示形式中得出的，它涉及一组超完备的基函数。这种扩展是密度泛函理论广泛领域中的一类模型的一部分，密度泛函理论被认为是凝聚态物理的金标准。总体算法由四个阶段组成，第二个和第四个阶段是可选的，最后两个阶段是对复杂矩阵对$（F，G）$的广义双曲奇异值分解（SVD）的计算，根据给定的矩阵$ J $定义双曲标量积。如果$ J = I $，则这两个阶段可以非常准确，高效地并行计算广义SVD（GSVD）。