3 research outputs found
The LAPW method with eigendecomposition based on the Hari--Zimmermann generalized hyperbolic SVD
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 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 , according to a given matrix defining the hyperbolic scalar
product. If , then these two phases compute the GSVD in parallel very
accurately and efficiently.Comment: The supplementary material is available at
https://web.math.pmf.unizg.hr/mfbda/papers/sm-SISC.pdf due to its size. This
revised manuscript is currently being considered for publicatio
A Kogbetliantz-type algorithm for the hyperbolic SVD
In this paper a two-sided, parallel Kogbetliantz-type algorithm for the
hyperbolic singular value decomposition (HSVD) of real and complex square
matrices is developed, with a single assumption that the input matrix, of order
, admits such a decomposition into the product of a unitary, a non-negative
diagonal, and a -unitary matrix, where is a given diagonal matrix of
positive and negative signs. When , the proposed algorithm computes
the ordinary SVD. The paper's most important contribution -- a derivation of
formulas for the HSVD of matrices -- is presented first, followed
by the details of their implementation in floating-point arithmetic. Next, the
effects of the hyperbolic transformations on the columns of the iteration
matrix are discussed. These effects then guide a redesign of the dynamic pivot
ordering, being already a well-established pivot strategy for the ordinary
Kogbetliantz algorithm, for the general, HSVD. A heuristic but
sound convergence criterion is then proposed, which contributes to high
accuracy demonstrated in the numerical testing results. Such a -Kogbetliantz
algorithm as presented here is intrinsically slow, but is nevertheless usable
for matrices of small orders.Comment: a heavily revised version with 32 pages and 4 figure