A unified approach to some strategies for the treatment of breakdown in Lanczos-type algorithms
The Lanczos method for solving systems of linear equations is implemented by using some recurrence relationships between polynomials of a family of formal orthogonal polynomials or between those of two adjacent families of formal orthogonal polynomials. A division by zero can occur in these relations, thus producing a breakdown in the algorithm which has to be stopped. In this paper, three strategies to avoid this drawback are discussed: the MRZ and its variants, the normalized and unnormalized BIORES algorithm and the composite step biconjugate algorithm. We prove that all these algorithms can be derived from a unified framework; in fact, we give a formalism for finding all the recurrence relationships used in these algorithms, which shows that the three strategies use the same techniques.