Harmonic and refined extraction methods for the singular value problem, with applications in least squares problems

For the accurate approximation of the minimal singular triple (singular value and left and right singular vector), we may use two separate search spaces, one for the left, and one for the right singular vector. In Lanczos bidiagonalization, for example, such search spaces are constructed. In [3], the author proposes a Jacobi–Davidson type method for the singular value problem, where solutions to certain correction equations are used to expand the search spaces. As noted in [3], the standard Galerkin subspace extraction works well for the computation of large singular triples, but may lead to unsatisfactory approximations to small and interior triples. To overcome this problem for the smallest triples, we propose three harmonic and a refined approach. All methods are derived in a number of different ways. Two of these methods can also be applied when we are interested in interior singular triples. Theoretical results as well as numerical experiments indicate that the results of the alternative extraction processes are often better than the standard approach. We show that when Lanczos bidiagonalization is used to approximate the smallest singular triples, the standard, harmonic, and refined extraction methods are all essentially equivalent. This gives more insight in the success of the use of Lanczos bidiagonalization to find the smallest singular triples. Finally, we present a novel method for the least squares problem, the success of which is based on a good extraction process for the smallest singular triples. The truncated SVD is also discussed in this context.