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A preemptive restarting approach to beating the inherent instability of Lanczos-type algorithms | ||
| Iranian Journal of Science | ||
| مقاله 5، دوره 37، شماره 3.1، آذر 2013، صفحه 349-358 اصل مقاله (206.45 K) | ||
| نوع مقاله: Regular Paper | ||
| شناسه دیجیتال (DOI): 10.22099/ijsts.2013.1634 | ||
| نویسندگان | ||
| M. Farooq* 1؛ A. Salhi2 | ||
| 1Department of Mathematics, University of Peshawar, 25120, Khyber Pakhtunkhwa | ||
| 2Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, UK | ||
| چکیده | ||
| Lanczos-type algorithms are well known for their inherent instability. They typically breakdown occurs when relevant orthogonal polynomials do not exist. Current approaches to curing breakdown rely on jumping over the non-existent polynomials to resume computation. This may have to be used many times during the solution process. We suggest an alternative to jumping, which consists of restarting the algorithms that fail. Three different strategies can be taken: (ST1) Restarting following breakdown of the algorithm in use; (ST2) pre-emptive restarting after a fixed number of iterations; (ST3) restarting when near breakdown is detected through monitoring. We describe a restarting framework with a generic algorithm that invokes one or the other of the three strategies suggested. Four of the most prominent recently developed Lanczos-type algorithms namely, and will be presented and then deployed in the restarting framework. However, we will only report on results obtained with strategy ST2 as it is the only viable one at the moment. | ||
| کلیدواژهها | ||
| Lanczos algorithm؛ Systems of Linear Equations؛ Formal Orthogonal Polynomials, Restarting, Switching, Breakdown | ||
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