An Implicitly Restarted Refined Bidiagonalization Lanczos Method for Computing a Partial Singular Value Decomposition

Augmented Implicitly Restarted Lanczos Bidiagonalization Methods

“…The shifts are applied by "chasing the bulge" in a curtailed QR-algorithm. Other implementations of restarted Lanczos bidiagonalization are discussed in [13,14,15,16,28]. The present paper describes mathematically equivalent, but numerically more robust, implementations of the methods discussed by Kokiopoulou, Bekas, and Gallopoulos [17].…”

“…However, for a small value of m, the desired singular triplets of A may be approximated poorly by computed approximate singular triplets {σ j }. In order to circumvent this difficulty, several methods have been proposed that are based on the computation of partial Lanczos bidiagonalizations (1.3)-(1.4) with m small for a sequence of initial vectors p 1 ; see, e.g., [5,13,14,15,16,17,28]. These methods are commonly referred to as restarted partial Lanczos bidiagonalization methods.…”

The design of a distributed MATLAB-based environment for computing pseudospectra

“…In fact, the computation of s(z), even by means of advanced methods (e.g. those presented in [2,23,25,26]), is very expensive, especially when the smallest singular values are clustered. Moreover, we need to compute s(z) for many values of z.…”

“…The environment also incorporates and lets one use, via appropriate selections in PPsGUI, many state-of-the-art algorithms for computing singular values. So, in addition to the transfer function framework, one can use MATLAB's own direct and iterative methods svd, svds as well as methods described in [2,23,25,26]. PPsGUI also allows the superposition of plots as well as three-dimensional plotting.…”

The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors

“…The principle of refined projection methods has, very recently, been extended to compute a partial singular value decomposition of a large matrix. We proposed a refined bidiagonalization Lanczos method and developed an implicitly restarted bidiagonalization Lanczos algorithm, which was more efficient than the implicitly restarted bidiagonalization Lanczos algorithm [13]. In a general setting, a unified convergence theory of general refined projection methods was established when A is diagonalizable [8] and when A is a general matrix [12].…”