A unified approach to some strategies for the treatment of breakdown in Lanczos-type algorithms

Volume 26 / 1999

A. El Guennouni Applicationes Mathematicae 26 (1999), 477-488 DOI: 10.4064/am-26-4-477-488


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.


  • A. El Guennouni

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