Block-Jacobi SVD Algorithms: A Review

Abstract

We discuss some progress in the design and implementation of block-Jacobi SVD algorithms in last 25 years. Two ideas were crucial for enhancing the efficiency of both two- 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 to the distribution of singular values of an original matrix.