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.

翻译：在本文中,为真实和复杂的平方矩阵的超曲单值分解(HSVD)开发了双向、平行的Kogbetliantz型算法,其中单一假设输入矩阵以美元为序,承认这种分解为单一、非负对数和美元统一矩阵的产物,其中J美元是正负迹象的对角矩阵。当拟议算法计算了普通的SVD。文件的最重要贡献 -- -- 以2美元为序的HSVD公式的衍生,然后在浮动点算术中详细介绍了其实施细节。接下来,讨论了超曲变对循环矩阵列的影响。这些影响随后指导了动态平流顺序的重新设计,已经成为普通Kogbetliantz算法的固定策略。对于一般来说, $n\betimetrial 值为2美元, 美元基调算法的公式是高正数的硬值。