论文标题
双曲线SVD的Kogbetliantz型算法
A Kogbetliantz-type algorithm for the hyperbolic SVD
论文作者
论文摘要
在本文中，开发了一种两面，平行的kogbetliantz型算法，用于开发真实和复杂的平方矩阵的双波利奇异价值分解（HSVD），一个假设是一个假设，即$ n $的输入矩阵，即$ n $的输入矩阵，将这种分解为单位，$ jo $ juy $ j的$ j $ jo $ j $ j-正和负符号的对角矩阵。当$ j = \ pm i $时，提出的算法计算普通SVD。该论文最重要的贡献 - 首先提出了$ 2 \ times 2 $矩阵的HSVD公式的推导，其次是在浮点算术中实现的细节。接下来，讨论双曲线转换对迭代矩阵列的影响。然后，这些效果指导了动态枢轴订购的重新设计，这已经是普通Kogbetliantz算法的公认的枢轴策略，用于一般，$ n \ times n $ hsvd。然后提出了启发式但声音收敛标准，这在数值测试结果中有助于高精度。此处介绍的这样的$ J $ -KOGBETLIANTZ算法在本质上很慢，但是对于小订单的矩阵仍然可用。
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.
