Convergence to Singular Triplets in the Two-Sided Block-Jacobi SVD Algorithm with Dynamic Ordering

Abstract

We study the convergence of computed quantities to singular triplets in the serial and parallel block-Jacobi singular value decomposition (SVD) algorithm with dynamic ordering. After eliminating possible zero singular values by two finite decompositions of a matrix $A\in\mathbb{C}^{m\times n},\, m\geq n$, which reduce the matrix dimensions to $n\times n$, it is shown that an iterated nonsingular matrix $A^{(k)}$ converges to a fixed diagonal matrix and its diagonal elements are the singular values of an initial matrix $A$. For the case of simple singular values, it is proved that the corresponding columns of the matrices of accumulated unitary transformations converge to corresponding left and right singular vectors. When a multiple singular value (or a cluster of singular values) is well separated from the other singular values, the convergence of two sequences of appropriate orthogonal projectors towards the orthogonal projectors onto the corresponding left and right subspaces is proved. Additionally, the convergence of orthogonal projectors leads to the convergence of certain computed subspaces towards the singular left and right subspaces spanned by left and right singular vectors corresponding to a multiple singular value or a cluster. An example computed in MATLAB illustrates the developed theory.