• DOI:10.1007/s11075-021-01197-4
• Corpus ID: 212725931

A Kogbetliantz-type algorithm for the hyperbolic SVD

@article{Novakovic2020AKA,
  title={A Kogbetliantz-type algorithm for the hyperbolic SVD},
  author={Vedran Novakovi'c and Sanja Singer},
  journal={Numerical Algorithms},
  year={2020},
  volume={90},
  pages={523 - 561}
}

• Vedran Novakovi'c, Sanja Singer
• Published 14 March 2020
• Computer Science
• Numerical Algorithms

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 = ±I, the proposed algorithm computes the ordinary SVD. The paper’s most important contribution—a derivation of formulas for the HSVD of 2 × 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 × 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.