论文标题
采用更强大的算法来计算受限的单数值分解
Towards a more robust algorithm for computing the restricted singular value decomposition
论文作者
论文摘要
提出了一种新的算法来计算密集矩阵的受限奇异值分解。就像Zha的方法\ Cite {Zha92}一样，新算法使用隐式Kogbetliantz迭代，但具有四个主要创新。第一个创新是一种有用的准三角概括形式，仅需要正常的转换才能计算。根据应用程序，可以使用此Schur形式而不是完整的分解。第二个创新是一个新的预处理阶段，比以前的方法需要更少的等级确定。第三个创新是一种数字稳定的RSVD算法，价格为$ 2 \ times 2 $上三角矩阵，它构成了隐式Kogbetliantz迭代的关键组成部分。第四次创新是限制单数三重序的替代缩放，该缩放为其计算提供了优雅的公式。除了这四项创新之外，还广泛讨论了算法的定性（数值）特征。 （可选）后处理阶段中的一些数值挑战也被考虑。但是，他们的解决方案需要进一步研究。数值测试和示例证实了该方法的有效性。
A new algorithm to compute the restricted singular value decomposition of dense matrices is presented. Like Zha's method \cite{Zha92}, the new algorithm uses an implicit Kogbetliantz iteration, but with four major innovations. The first innovation is a useful quasi-upper triangular generalized Schur form that just requires orthonormal transformations to compute. Depending on the application, this Schur form can be used instead of the full decomposition. The second innovation is a new preprocessing phase that requires fewer rank determinations than previous methods. The third innovation is a numerically stable RSVD algorithm for $2\times 2$ upper-triangular matrices, which forms a key component of the implicit Kogbetliantz iteration. The fourth innovation is an alternative scaling for the restricted singular triplets that results in elegant formulas for their computation. Beyond these four innovations, the qualitative (numerical) characteristics of the algorithm are discussed extensively. Some numerical challenges in the (optional) postprocessing phase are considered too; though, their solutions require further research. Numerical tests and examples confirm the effectiveness of the method.
