Exact deflation for accurate SVD computation of nonnegative bidiagonal products of arbitrary rank

Abstract

Dealing with zero singular values can be quite challenging, as they have the potential to cause numerous numerical difficulties. This paper presents a method for computing the singular value decomposition (SVD) of a nonnegative bidiagonal product of arbitrary rank, regardless of whether the factors are of full rank or rank-deficient, square or rectangular. A key feature of our method is its ability to exactly deflate all zero singular values with a favorable complexity, irrespective of rank deficiency and ill conditioning. Furthermore, it ensures the computation of nonzero singular values, no matter how small they may be, with high relative accuracy. Additionally, our method is well-suited for accurately computing the SVDs of arbitrary submatrices, leveraging an approach to extract their representations from the original product. We have conducted error analysis and numerical experiments to validate the claimed high relative accuracy.