This paper shows that there is a close relationship between the Euclidean algorithm for polynomials and the Lanczos method for solving sparse linear systems, especially when working over finite fields. It uses this relationship to account rigorously for the appearance of self-orthogonal vectors arising in the course of the Lanczos algorithm. It presents an improved Lanczos method which overcome...

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 ...

In recent years, there has been a true revival of the nonsymmetric Lanczos method. On the one hand, the possible breakdowns in the classical algorithm are now better understood, and so-called look-ahead variants of the Lanczos process have been developed, which remedy this problem. On the other hand, various new Lanczos-based iterative schemes for solving nonsymmetric linear systems have been p...

For the solution of large, sparse, non-Hermitian linear systems, Lanczos-type product methods that are based on the Lanczos three-term recurrence are derived in a systematic way. These methods either square the Lanczos process or combine it with a local minimization of the residual. For them a quasi-minimal residual (QMR) smoothing is proposed that can also be implemented by short-term recurren...

When the isotypic subspaces of a representation are viewed as the eigenspaces of a symmetric linear transformation, isotypic projections may be achieved as eigenspace projections and computed using the Lanczos iteration. In this paper, we show how this approach gives rise to an efficient isotypic projection method for permutation representations of distance transitive graphs and the symmetric g...

The spectral transformation Lanczos method is very popular for solving large scale Her-mitian generalized eigenvalue problems. The method uses a special inner product so that the symmetric Lanczos method can be used. Sometimes, a semi-inner product must be used. This may lead to instabilities and breakdown. In this paper, we suggest a cure for breakdown by use of an implicit restart in the Lanc...

In this paper we review some of the research that has emerged to form Lanczos potential theory. From Lanczos’ pioneering work on quadratic Lagrangians, which ultimately led to the discovery of his famed tensor, through to the current developments in the area of exact solutions of the Weyl–Lanczos equations, we aim to exhibit what are generally considered to be the pivotal advances in the theory...

We discuss a Krylov-Schur like restarting technique applied within the symplectic Lanczos algorithm for the Hamiltonian eigenvalue problem. This allows to easily implement a purging and locking strategy in order to improve the convergence properties of the symplectic Lanczos algorithm. The Krylov-Schur-like restarting is based on the SR algorithm. Some ingredients of the latter need to be adapt...