Generalized Davidson and multidirectional-type methods for the generalized singular value decomposition

TL;DR

This paper introduces two new iterative algorithms for efficiently computing extremal generalized singular values and vectors, improving convergence and search space management, with theoretical analysis and numerical validation.

Contribution

The paper presents a generalized Davidson-type algorithm and a multidirectional subspace expansion method for generalized SVD, with convergence analysis and practical truncation strategies.

Findings

Both methods demonstrate monotonic and linear convergence.

Numerical experiments show the methods are competitive.

The algorithms effectively handle large-scale problems with controlled search space growth.

Abstract

We propose new iterative methods for computing nontrivial extremal generalized singular values and vectors. The first method is a generalized Davidson-type algorithm and the second method employs a multidirectional subspace expansion technique. Essential to the latter is a fast truncation step designed to remove a low quality search direction and to ensure moderate growth of the search space. Both methods rely on thick restarts and may be combined with two different deflation approaches. We argue that the methods have monotonic and (asymptotic) linear convergence, derive and discuss locally optimal expansion vectors, and explain why the fast truncation step ideally removes search directions orthogonal to the desired generalized singular vector. Furthermore, we identify the relation between our generalized Davidson-type algorithm and the Jacobi--Davidson algorithm for the generalized singular value decomposition. Finally, we generalize several known convergence results for the Hermitian eigenvalue problem to the Hermitian positive definite generalized eigenvalue problem. Numerical experiments indicate that both methods are competitive.