Naslov
A Kogbetliantz-type algorithm for the hyperbolic SVD

Autori
Novaković, Vedran ; Singer, Sanja

Izvornik
Numerical algorithms (1017-1398) 90 (2022), 2; 523-561

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

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

Sažetak
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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika, Računarstvo

Napomena
Rad je prihvaćen 9. rujna 2021. i objavljen online 15. listopada 2021.

Preprint je dostupan na: https://arxiv.org/abs/2003.06701

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