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...

In 1929 Lanczos showed how to derive Dirac’s equation from a more fundamental system that predicted that spin 1 2 particles should come in pairs. Today, these pairs can unambiguously be interpreted as isospin doublets. From the same fundamental equation Lanczos derived also the correct form of the wave equation of massive spin 1 particles that would be rediscovered by Proca in 1936. Lanczos’s f...

This work presents an adaptive block Lanczos method for large-scale non-Hermitian Eigenvalue problems (henceforth the ABLE method). The ABLE method is a block version of the non-Hermitian Lanczos algorithm. There are three innovations. First, an adaptive blocksize scheme cures (near) breakdown and adapts the blocksize to the order of multiple or clustered eigenvalues. Second, stopping criteria ...

The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential insta-bilities. We present an implementation of a look-ahead version of the Lanczos algorithm which overcomes these problems by skipping over th...

It is well known that the Lanczos process suffers from loss of orthogonality in the case of finite-precision arithmetic. Several approaches have been proposed in order to address this issue, thus enabling the successful computation of approximate eigensolutions. However, these techniques have been studied mainly in the context of long Lanczos runs, but not for restarted Lanczos eigensolvers. Se...

We study the Lanczos method for computing extreme eigenvalues of a symmetric or Hermitian matrix. It is not guaranteed that the extreme Ritz values are close to the extreme eigenvalues—even when the norms of the corresponding residual vectors are small. Assuming that the starting vector has been chosen randomly, we compute probabilistic bounds for the extreme eigenvalues from data available dur...

The three-term Lanczos process for a symmetric matrix leads to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving a reduced system in one way or another. This leads to well-known methods: MINRES (minimal residual), GMRES (generalized minimal residual), a...

The Lanczos algorithm for matrix tridiagonalisation suffers from strong numerical instability in finite precision arithmetic when applied to evaluate matrix eigenvalues. The mechanism by which this instability arises is well documented in the literature. A recent application of the Lanczos algorithm proposed by Bai, Fahey and Golub allows quadrature evaluation of inner products of the form ψ · ...

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...

In this paper, we investigate the block Lanczos algorithm for solving large sparse symmetric linear systems with multiple right-hand sides, and show how to incorporate deeation to drop converged linear systems using a natural convergence criterion, and present an adaptive block Lanczos algorithm. We propose also a block version of Paige and Saun-ders' MINRES method for iterative solution of sym...