LAUSR

The Lanczos-type algorithms for Systems of Linear Equations (SLEs) are efficient but fragile. A number of ways to resolve this issue have been suggested. But, the problem is still not fully sorted, in our view. Here, we suggest a way that takes advantage of the sequence of approximate solutions that have been computed prior to breakdown by embedding interpolation/extrapolation to avoid it.  The approach, referred to as Embedded Interpolation-Extrapolation Model in Lanczos-type Algorithm (EIEMLA), generates new iterates which are at least as good as the best in the current sequence. This process is repeated after appending the new iterates to the sequence of approximate solutions until some convergence tolerance is achieved. To improve EIEMLA's convergence and stability, a restart version of REIEMLA is also considered. These algorithms are more robust than other Lanczos-type algorithms, including those with restarting and switching strategies. Both algorithms have been implemented to run in parallel on a Cloud computing platform. Our tests involve SLEs with up to $10^6$ variables and equations. The results show that breakdown is mitigated and efficiency gains can be achieved through parallelization.