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

TL;DR

This paper introduces a harmonic Lanczos bidiagonalization method combined with implicit restarting to efficiently compute interior singular triplets of large matrices, with proven convergence under certain conditions.

Contribution

It presents a novel harmonic Lanczos bidiagonalization approach with an implicit restart technique for interior singular triplet computation, including a shift selection strategy.

Findings

Converges when the Rayleigh quotient matrix is bounded and singular values are well separated.

Numerical experiments demonstrate efficiency in computing interior singular triplets.

The method outperforms existing algorithms in certain large-scale scenarios.

Abstract

This paper proposes a harmonic Lanczos bidiagonalization method for computing some interior singular triplets of large matrices. It is shown that the approximate singular triplets are convergent if 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 selection strategy of shifts. Numerical experiments show that one can use this algorithm to compute interior singular triplets efficiently.