A preemptive restarting approach to beating the inherent instability of Lanczos-type algorithms

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, A4 , A12 , A5 / B10 and A8 / B10 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.