Abstract

We discuss some progress in the design and implementation of block-Jacobi SVD algorithms in the scope of the last 25 years. Two ideas were crucial for enhancing the efficiency of two-sided and one-sided serial or parallel block-Jacobi methods: the so-called dynamic ordering of subproblems solved in each iteration step, and a suitable preconditioning of an original matrix. These two ideas led to a substantial decrease of (serial or parallel) iteration steps needed for the convergence. Consequently, especially the one-sided block-Jacobi SVD algorithm became competitive in speed with some SVD algorithms based on the matrix bi-diagonalization. We also discuss new theoretical results w.r.t. the asymptotic quadratic convergence of block-Jacobi SVD algorithms regardless of the distribution of singular values of the original matrix.

Keywords

MSC

65F30

Cite this article

@article{cai-8403,
  author  = {Gabriel Okša and Martin Bečka},
  title   = {Block-Jacobi SVD Algorithms: A Review},
  journal = {Computing and Informatics},
  volume  = {44},
  number  = {5},
  pages   = {1254--1272},
  year    = {2025},
  issn    = {1335-9150},
  doi     = {10.31577/cai_2025_5_1254}
}
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