Restarting from Specific Points to Cure Breakdown in Lanczos-type Algorithms


  • Maharani Maharani Department of Mathematical Sciences, University of Essex, Colchester, CO4 3SQ
  • Abdellah Salhi Department of Mathematical Sciences, University of Essex, Colchester, CO4 3SQ, U.K. phone: +44-1206-873032; fax: +44-1206-3040



breakdown, formal orthogonal polynomial, Lanczos-type algorithms, systems of linear equations, restarting.


Breakdown in Lanczos-type algorithms is a common phenomenon which is due to the non-existence of some orthogonal polynomials. It causes the
solution process to halt. It is, therefore, important to deal with it to improve the resilience of the algorithms and increase their usability. In this paper, we consider restarting from a number of different approximate solutions that seem to be attractive starting points. They are: (a) the last iterate preceding breakdown, (b) the iterate with minimum residual norm found so far, and (c) the approximate solution whose entries are the median values of entries of all iterates generated by the Lanczos-type algorithm considered. Although it has been shown theoretically in the context of Arnoldi-type algorithms as well as Lanczos-type algorithms that restarting mitigates breakdown and allows the iterative process to continue and converge to good solutions, here we give an alternative theorem to that effect and a proof of it. However, emphasis is on the quality of the restarting points. Numerical results are included.

Author Biography

Maharani Maharani, Department of Mathematical Sciences, University of Essex, Colchester, CO4 3SQ

Department of Mathematical Sciences, the 10th rank of research in the UK universities.


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