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...

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...

The symmetric band Lanczos process is an extension of the classical Lanczos algorithm for symmetric matrices and single starting vectors to multiple starting vectors. After n iterations, the symmetric band Lanczos process has generated an n× n projection, Tn, of the given symmetric matrix onto the n-dimensional subspace spanned by the first n Lanczos vectors. This subspace is closely related to...

The Riemann-Lanczos problem for 4-dimensional manifolds was discussed by Bampi and Caviglia. Using exterior differential systems they showed that it was not an involutory differential system until a suitable prolongation was made. Here, we introduce the alternative Janet-Riquier theory and use it to consider the Riemann-Lanczos problem in 2 and 3 dimensions. We find that in 2 dimensions, the Ri...

In a semiorthogonal Lanczos algorithm, the orthogonality of the Lanczos vectors is allowed to deteriorate to roughly the square root of the rounding unit, after which the current vectors are reorthogonalized. A theorem of Simon 4] shows that the Rayleigh quotient | i.e., the tridiagonal matrix produced by the Lanczos recursion | contains fully accurate approximations to the Ritz values in spite...

This paper describes a parallel implementation of a generalized Lanczos procedure for structural dynamic analysis on a distributed memory parallel computer. One major cost of the generalized Lanczos procedure is the factorization of the (shifted) stiffness matrix and the forward and backward solution of triangular systems. In this paper, we discuss load assignment of a sparse matrix and propose...