Augmented Implicitly Restarted Lanczos Bidiagonalization Methods

An Implicitly Restarted Block Lanczos Bidiagonalization Method Using Leja Shifts

In this paper, we propose an implicitly restarted block Lanczos bidiagonalization (IRBLB) method for computing a few extreme or interior singular values and associated right and left singular vectors of a large matrix A. Our method combines the advantages of a block routine, implicit shifting, and the application of Leja points as shifts in the accelerating polynomial. The method neither requir...

A Refined Harmonic Lanczos Bidiagonalization Method and an Implicitly Restarted Algorithm for Computing the Smallest Singular Triplets of Large Matrices

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 are almost linearly dependent. Furthermo...

The irlba Package

The irlba package provides a fast way to compute partial singular value decompositions (SVD) of large sparse or dense matrices. Recent additions to the package can also compute fast partial symmetric eigenvalue decompositions and principal components. The package is an R implementation of the augmented implicitly restarted Lanczos bidiagonalization algorithm of Jim Baglama and Lothar Reichel. S...

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

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 ha...

Restarted Lanczos Bidiagonalization for the SVD in SLEPc