Chapter 23 - Computing the Singular Value Decomposition

Abstract

This chapter develops two algorithms for the computation of the SVD. The first of these is the one-sided Jacobi method. By properly choosing c and s, right Jacobi rotations can be constructed that reduce A to a diagonal matrix of singular values. The result is the same if AT*A is reduced to the matrix of singular values using orthogonality transformations. This algorithm is particularly effective when computing small singular values. The computation of singular values is well conditioned, but the computation of left and right singular vectors can be ill-conditioned if some singular values are close together. There is a variant of the one-sided Jacobi algorithm that provides higher accuracy and speed than the algorithm we have described. The algorithm uses rank-revealing QR with column pivoting. The standard algorithms for computing the SVD first transform the matrix to upper-bidiagonal form using pre- and postmultiplication by Householder reflections. The chapter presents the Demmel and Kahan zero-shift QR downward sweep algorithm that transforms the upper-bidiagonal matrix to a diagonal matrix of singular values using bulge chasing.