A Kogbetliantz-type algorithm for the hyperbolic SVD
| Autor: | Novaković, Vedran, Singer, Sanja |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Numer. Algoritms 90 (2022), 2; 523-561 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s11075-021-01197-4 |
| Popis: | 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 $J=\pm I$, the proposed algorithm computes the ordinary SVD. The paper's most important contribution -- a derivation of formulas for the HSVD of $2\times 2$ 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, $n\times n$ 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. Comment: Accepted for publication in Numerical Algorithms. This version slightly differs from the accepted one, with several small corrections and an alternative (hopefully more stable) data server listed |
| Databáze: | arXiv |
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