Fundamental Theory of a Singular Value Algorithm : Convergence Analysis of the dqds Algorithm

Abstract

Matrix singular values play an important role in many applications. Accordingly, numerical methods for computing singular values are of great importance in practice. In 1994, Fernando and Parlett discovered an efficient algorithm, which is now called the differential quotient difference with shifts (dqds) algorithm. The dqds algorithm has received majority support due to its accuracy, speed, and numerical stability, and is implemented as DLASQ in LAPACK. This paper is concerned with convergence theorems of the dqds algorithm for singular values. Specifically, we discuss the global convergence theorem given by the present author and survey a variety of theorems on convergence rate of the dqds algorithm.