A look-ahead Lánczos algorithm for unsymmetric matrices

Abstract : The two sided Lanczos algorithm is known to suffer from serious breakdowns. These occur when the associated moment matrix does not permit triangular factorization. We modify the algorithm slightly so that it corresponds to using a 2 X 2 'pivot' in triangular factorization whenever a 1 X 1 pivot would be dangerous. The incidence of breakdown is greatly reduced. The price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 X 2 pivot is used. (Author)