Improving the Accuracy of Computed Singular Values *

This paper describes a computational method for improving the accuracy of a given singular value and its associated left and right singular vectors. The method is analogous to iterative improvement for the solution of linear systems. That is, by means of a low-precision computation, an iterative algorithm is applied to increase the accuracy of the singular value and vectors; extended precision computations are used in the residual calculation. The method is related to Newton's method applied to the singular value problem and inverse iteration for the eigenvalue problem. 1. The basic algorithm. In a recent paper, Dongarra, Moler and Wilkinson [1] described an algorithm for improving an approximation to a simple eigenvalue and the corresponding eigenvector. In this paper we extend and modify the algorithm to cover the singular value problem. We know that the singular values of a matrix are well conditioned in the sense that small changes in the matrix result in small changes in the singular values. The singular vectors may not be well determined and may vary drastically with small changes in the matrix. In [3], Stewart describes a somewhat analogous procedure for determining error bounds and obtaining corrections to the singular values and vectors associated with invariant subspaces. Here we describe a procedure for improving a single or arbitrary singular value and singular vectors using the previously computed factorization. We begin with a brief description of the basic algorithm. Given an m x n rectangular matrix A, we are interested in the decomposition (1.1) A=UXVT, where U and V are unitary matrices and E is a rectangular diagonal matrix of the same dimension as A with real nonnegative diagonal entries. The equations can also be written as (1.2) Avi-O'iUi and (1.3) A ui trivi for each singular value ri. If o', u, and v have been derived from some computation on a computer with finite precision or by some insight into the problem, they are generally not the true singular value and vectors, but approximations. We know, however, that there exist/z 1,/:, y, and z such that (1.4) A(v + y) (tr + tz 1)(u + z) and (1.5) A T (u + z) (o" + tzz)(V + y),