A harmonic Lanczos bidiagonalization method for computing interior singular triplets of large matrices

摘要

We propose a harmonic Lanczos bidiagonalization method for computing some interior singular triplets of large matrices. We prove that the approximate singular triplet is convergent if the norm of a certain Rayleigh quotient matrix is uniformly bounded and the approximate singular values are well separated. Combining with the implicit restarting technique, we develop an implicitly restarted harmonic Lanczos bidiagonalization algorithm and suggest a strategy to select shifts. Numerical experiments show that one can use this algorithm to compute the interior singular triplets efficiently.