Ming Gu

Computing the Singular Value Decomposition with High Relative Accuracy

We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, which in general only guarantee correct digits in the singular values with large enough magniturds. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as finite element problems and quantum mechanics, it is the smallest singular values that have physical meaning, and should be determined accurately by the data. Many recent papers have identified special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite different, motivating us to seek a common perturbation theory and common algorithm. We provide these in this paper, and show that high relative accuracy is possible in many new cases as well. The briefest way to describe our results is that we can compute the SVD to high relative accuracy provided we can compute a high accuracy pivoted LDU decomposition. We provide many examples of matrix classes permitting such an LDU decomposition.

Fast and stable eigendecomposition of symmetric banded plus semi-separable matrices

Abstract A new fast and stable algorithm to reduce a symmetric banded plus semi-separable matrix to tridiagonal form via orthogonal similarity transformations is presented.

Templates for Linear Algebra Problems

The increasing availability of advanced-architecture computers is having a very significant effect on all spheres of scientific computation, including algorithm research and software development in numerical linear algebra. Linear algebra -in particular, the solution of linear systems of equations and eigenvalue problems — lies at the heart of most calculations in scientific computing. This paper discusses some of the recent developments in linear algebra designed to help the user on advanced-architecture computers.