An Implicitly Restarted Lanczos Bidiagonalization Method for Computing Smallest Singular Triplets

We describe the development of a method for the efficient computation of the smallest singular values and corresponding vectors for large sparse matrices [4]. The method combines state-of-the-art techniques that make it a useful computational tool appropriate for large scale computations. The method relies upon Lanczos bidiagonalization (LBD) with partial reorthogonalization [6], enhanced with implicit restarts and harmonic Ritz values. We note that although LBD has been successfully used for the approximation of largest singular values [5], our target in this paper is the computation of the smallest singular values. Thus, in order to design a matrix free method by avoiding shift-and-invert techniques we rely on harmonic Ritz values. Using LBD for the approximation of the smallest singular values often causes the lengths of the Lanczos bases to become quite large in order to obtain accurate approximations. For that reason, we embed an implicit restarting mechanism in LBD [12], which maintains memory requirements constant at each restart. In order to avoid the explicit inversion of A, we employ a harmonic Ritz value shift strategy [10, 9]. Harmonic Ritz values and vectors have been reported (see e.g. [8]) to be