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...

In Part I [6] of this paper, we have presented an implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices. Here, we show how the look-ahead Lanczos process | combined with a quasi-minimal residual (QMR) approach | can be used to develop a robust black box solver for large sparse non-Hermitian linear systems. Details of an implementation of the resulting QMR algorithm are p...

First several algorithms based on the unsymmetric Lanczos process are surveyed the biorthog onalization BO algorithm for constructing a tridiagonal matrix T similar to a given matrix A whose ex treme spectrum is sought typically the BOBC algorithm which generates directly the LU factors of T the Biores Lanczos Orthores Biomin Lanczos Orthomin or biconjugate gradient BCG and the Biodir Lanczos O...

In 1950 Lanczos [22] proposed a method for computing the eigenvalues of symmetric and nonsymmetric matrices. The idea was to reduce the given matrix to a tridiagonal form, from which the eigenvalues could be determined. A characterization of the breakdowns in the Lanczos algorithm in terms of algebraic conditions of controllability and observability was addressed in [6] and [26]. Hankel matrice...

A new version of the unsymmetric Lanczos algorithm without look-ahead is described combining elements of numerical stability and parallel algorithm design. Firstly, stability is obtained by a coupled two-term procedure that generates Lanczos vectors scaled to unit length. Secondly, the algorithm is derived by making all inner products of a single iteration step independent such that global sync...

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...

The classic Lanczos method is an effective method for tridiagonalizing real symmetric matrices. Its block algorithm can significantly improve performance by exploiting memory hierarchies. In this paper, we present a block Lanczos method for tridiagonalizing complex symmetric matrices. Also, we propose a novel componentwise technique for detecting the loss of orthogonality to stablize the block ...

The Lanczos algorithm is becoming accepted as a powerful tool for finding the eigenvalues and for solving linear systems of equations. Any practical implementation of the algorithm suffers however from roundoff errors, which usually cause the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, full reorthogonalization (FRO) and selective orthogo...