2020/02/12 by Ian N. Zwaan, Zwaan, Ian N.
Computer Science · Mathematics · Physics and Astronomy · #65F15 #65F22 #65F30 #65F50 #65R30 #65R32 #Advanced Optimization Algorithms Research #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2002.04828
openalex publication_date 2020/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A new algorithm to compute the restricted singular value decomposition of\ndense matrices is presented. Like Zha's method citeZha92, the new algorithm\nuses an implicit Kogbetliantz iteration, but with four major innovations. The\nfirst innovation is a useful quasi-upper triangular generalized Schur form that\njust requires orthonormal transformations to compute. Depending on the\napplication, this Schur form can be used instead of the full decomposition. The\nsecond innovation is a new preprocessing phase that requires fewer rank\ndeterminations than previous methods. The third innovation is a numerically\nstable RSVD algorithm for 2\× 2 upper-triangular matrices, which forms a\nkey component of the implicit Kogbetliantz iteration. The fourth innovation is\nan alternative scaling for the restricted singular triplets that results in\nelegant formulas for their computation. Beyond these four innovations, the\nqualitative (numerical) characteristics of the algorithm are discussed\nextensively. Some numerical challenges in the (optional) postprocessing phase\nare considered too; though, their solutions require further research. Numerical\ntests and examples confirm the effectiveness of the method.\n