Variations of Zhang's Lanczos-type product method

The Lanczos Method with Semi-inner Product

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...

Look-Ahead Procedures for Lanczos-Type Product Methods Based on Three-Term Lanczos Recurrences

Lanczos-type product methods for solving large sparse non-Hermitian linear systems have as residual polynomials either the squares of the Lanczos polynomials or the products of the latter with another sequence of polynomials, which is normally chosen to enforce some local minimization of the residual norm. In either case, these methods inherit from the underlying Lanczos process the danger of b...

The Lanczos Method 10 . 3 1 Lanczos Method

A Lanczos-type method for multiple starting vectors

Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of biorthogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to multiple starting vectors. G...

8 The Lanczos Method