Implicit application of polynomial filters in a k-step Arnoldi method
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
The symmetric eigenvalue problem
The symmetric eigenvalue problem
A refined subspace iteration algorithm for large sparse eigenproblems
Applied Numerical Mathematics
Implicitly Restarted GMRES and Arnoldi Methods for Nonsymmetric Systems of Equations
SIAM Journal on Matrix Analysis and Applications
ACM Transactions on Mathematical Software (TOMS)
Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems
ACM Transactions on Mathematical Software (TOMS)
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
Low-Rank Matrix Approximation Using the Lanczos Bidiagonalization Process with Applications
SIAM Journal on Scientific Computing
An analysis of the Rayleigh—Ritz method for approximating eigenspaces
Mathematics of Computation
Matrix algorithms
A Jacobi--Davidson Type SVD Method
SIAM Journal on Scientific Computing
A Krylov--Schur Algorithm for Large Eigenproblems
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
A Test Matrix Collection for Non-Hermitian Eigenvalue Problems
A Test Matrix Collection for Non-Hermitian Eigenvalue Problems
Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization
Applied Numerical Mathematics - Numerical algorithms, parallelism and applications
Augmented Implicitly Restarted Lanczos Bidiagonalization Methods
SIAM Journal on Scientific Computing
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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. Furthermore, harmonic Ritz vectors may converge irregularly and may even 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 harmonic 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.