# The LAPW method with eigendecomposition based on the Hari--Zimmermann generalized hyperbolic SVD

## TL;DR

This paper introduces a highly parallel and accurate algorithm for the generalized eigendecomposition of complex Hermitian matrix pairs, leveraging a hyperbolic SVD approach within the context of Density Functional Theory.

## Contribution

It presents a novel eigendecomposition algorithm based on the Hari--Zimmermann hyperbolic SVD, optimized for parallel computation and applicable to quantum mechanical Hamiltonians.

## Key findings

- Algorithm achieves high accuracy in eigendecomposition
- Parallel implementation improves computational efficiency
- Applicable to complex Hermitian matrices in quantum physics

## 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.