An implicitly restarted Lanczos bidiagonalization method with refined harmonic shifts for computing smallest singular triplets

E. Kokiopoulou et al. developed an implicitly restarted Lanczos bidiagonalization method (named IRLANB) for computing the smallest singular triplets. They use the wanted harmonic Ritz values and the corresponding refined harmonic Ritz vectors to approximate the desired smallest singular triplets. However, they use the unwanted harmonic Ritz values as shifts, named the harmonic shifts, to implicitly restart their algorithm. Therefore, although they replace the harmonic Ritz vectors by the refined harmonic Ritz vectors, which are always better than the harmonic Ritz vectors, to improve the convergence of IRLANB, the subspace after implicitly restarting is the same as that without this replacement. In this paper, using the information of the refined harmonic Ritz vectors, we present a new shift strategy, named the refined harmonic shifts, to replace the harmonic shifts used in IRLANB and obtain a new method, called IRRLANB. The refined harmonic shifts are better than the harmonic shifts and can be computed cheaply and reliably. The numerical experiments are reported to indicate that IRRLANB is superior than IRLANB.