An Accelerated Divide-and-Conquer Algorithm for the Bidiagonal SVD Problem

Abstract

In this paper, aiming at solving the bidiagonal SVD problem, a classical divide-and-conquer (DC) algorithm is modified, which needs to compute the SVD of broken arrow matrices by solving secular equations. The main cost of DC lies in the updating of singular vectors, which involves two matrix-matrix multiplications. We find that the singular vector matrices of a broken arrow matrix are Cauchy-like matrices and have an off-diagonal low-rank property, so they can be approximated efficiently by hierarchically semiseparable (HSS) matrices. Hereby, by using the HSS techniques, the complexity of computing singular vectors can be reduced significantly. An accelerated DC algorithm is proposed, denoted by ADC. Furthermore, we use a structured low-rank approximation method to construct these HSS approximations. Numerous experiments show ADC is both fast and numerically stable. When dealing with large matrices with few deflations, ADC can be 3x faster than DC in the optimized LAPACK libraries such as Intel MKL without any degradation in accuracy. These techniques can be used to similarly solve the symmetric tridiagonal eigenvalue problem. (A corrected PDF is attached.)