The LAPW method with eigendecomposition based on the HariZimmermann generalized hyperbolic SVD
Abstract
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 matrices 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 golden standard in condensed matter physics. The overall algorithm consists of four phases, the second and the fourth being optional, where the two last phases are computation of the generalized hyperbolic 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 GSVD in parallel very accurately and efficiently.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1907.08560
 Bibcode:
 2019arXiv190708560S
 Keywords:

 Mathematics  Numerical Analysis;
 Computer Science  Mathematical Software;
 65F15;
 65F25;
 65Y05;
 65Z05
 EPrint:
 The supplementary material is available at https://web.math.pmf.unizg.hr/mfbda/papers/smSISC.pdf due to its size. This revised manuscript is currently being considered for publication