In this paper we propose an accurate, highly parallel algorithm for the generalized eigendecomposition of a matrix pair , given in a factored form . Matrices and are generally complex and Hermitian, and 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 , according to a given matrix defining the hyperbolic scalar product. If , then these two phases compute the generalized SVD (GSVD) in parallel very accurately and efficiently.