  The harmonic Lanczos bidiagonalization method can be used to compute the
smallest singular triplets of a large matrix $A$. We prove that for good enough
projection subspaces harmonic Ritz values converge if the columns of $A$ are
strongly linearly independent. On the other hand, harmonic Ritz values may miss
some desired singular values when the columns of $A$ almost linearly dependent.
Furthermore, harmonic Ritz vectors may converge irregularly and even may fail
to converge. Based on the refined projection principle for large matrix
eigenproblems due to the first author, we propose a refined harmonic Lanczos
bidiagonalization method that takes the Rayleigh quotients of the harmonic Ritz
vectors as approximate singular values and extracts the best approximate
singular vectors, called the refined harmonic Ritz approximations, from the
given subspaces in the sense of residual minimizations. The refined
approximations are shown to converge to the desired singular vectors once the
subspaces are sufficiently good and the Rayleigh quotients converge. An
implicitly restarted refined harmonic Lanczos bidiagonalization algorithm
(IRRHLB) is developed. We study how to select the best possible shifts, and
suggest refined harmonic shifts that are theoretically better than the harmonic
shifts used within the implicitly restarted Lanczos bidiagonalization algorithm
(IRHLB). We propose a novel procedure that can numerically compute the refined
harmonic shifts efficiently and accurately. Numerical experiments are reported
that compare IRRHLB with five other algorithms based on the Lanczos
bidiagonalization process. It appears that IRRHLB is at least competitive with
them and can be considerably more efficient when computing the smallest
singular triplets.
