New look-ahead implementations of Lanczos-type methods for Unsymmetric Systems

A breakdown, that is a division by zero, can arise in the algorithms for implementing Lanczos' method because of the non--existence of some formal orthogonal polynomials. Such a breakdown can be avoided by jumping over the non--existent polynomials. This strategy, known as look-ahead, was already used in some algorithms such as the MRZ and the MRZ--stab. In these methods, the number of vectors to be stored depends on the length of the jump. Here, we propose new algorithms, called HMRZ and HMRZ-stab, which only need the storage of a fixed number of vectors, independent of this length. These new algorithms are based on Horner's rule, and they seem to be more stable than the old ones. Our algorithms, in the case without breakdown, compare well with other known algorithms with respect to the computational kernel and to the storage requirements. A complete exposition of this subject could be found in [5].