engleski

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 n, admits such a decomposition into the product of a unitary, a non-negative diagonal, and a J-unitary matrix, where J is a given diagonal matrix of positive and negative signs. When J=±I, the proposed algorithm computes the ordinary SVD. The paper's most important contribution---a derivation of formulas for the HSVD of 2 x 2 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, n x n HSVD. A heuristic but sound convergence criterion is then proposed, which contributes to high accuracy demonstrated in the numerical testing results. Such a J- Kogbetliantz algorithm as presented here is intrinsically slow, but is nevertheless usable for matrices of small orders.

hyperbolic singular value decomposition ; Kogbetliantz algorithm ; Hermitian eigenproblem ; OpenMP multicore parallelization

Rad je prihvaćen 9. rujna 2021. i objavljen online 15. listopada 2021. Preprint je dostupan na: https://arxiv.org/abs/2003.06701