We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenval...

Recently, matrix-based methods have gained wide attentions in pattern recognition and machine learning communities. The generalized low rank approximations of matrices (GLRAM) and the bilinear Lanczos components (BLC) algorithm are two popular algorithms that treat data as the native twodimensional matrix patterns. However, these two algorithms often require heavy computation time and memory sp...

In this overview we discuss iterative methods for solving large linear systems with sparse (or, possibly, structured) nonsymmetric (or, non-Hermitian) matrix that are based on the Lanczos process. They feature short recurrences for the generation of the Krylov space and for the sequence of approximations to the solution. This means low cost and low memory requirement. For very large sparse non-...

The Lanczos method is one of the standard approaches for computing a few eigenpairs of a large, sparse, symmetric matrix. It is typically used with restarting to avoid unbounded growth of memory and computational requirements. Thick-restart Lanczos is a popular restarted variant because of its simplicity and numerically robustness. However, convergence can be slow for highly clustered eigenvalu...

In this paper we show that the two-sided Lanczos procedure combined with implicit restarts, offers significant advantages over Pad6 approximations used typically for model reduction in circuit simulation. Keywords-Dynamical systems, Model reduction, Numerical methods, Pad6 approximation, Lanczos algorithm.

We show the importance that the Lanczos invariant has in the study of 4 R embedded into 5 E , in the analysis of non-null constant vectors, and in the existence of the Lanczos potential for the Weyl tensor.

The nonsymmetric Lanczos algorithm, which belongs to the class of Krylov subspace methods, is increasingly being used for model reduction of large scale systems of the form f(s) = c (sI−A)−1b, to exploit the sparse structure and reduce the computational burden. However, a good approximation is, usually, achieved only with relatively high order reduced models. Moreover, the computational cost of...

We give a brief description of a non-symmetric Lanczos algorithm that does not require strict bi-orthogonality among the generated vectors. We show how the vectors generated are algebraically related to “Controllable Space” and “Observable Space” for a related linear dynamical system. The algorithm described is particularly appropriate for large sparse systems. 1. Intr~uction The Lanczos Algori...