High Performances of Stabilized Lanczos-Types for Solving High Dimension Problems: A Survey

Lanczos-type algorithms are an iterative methods for solving sym-metric and unsymmetric systems of linear equations (SLEs) which par-ticularly involving large numbers of variables and equations. One bigissue with these algorithms is known as breakdown; it causes the algo-rithms to fail before obtaining (converging to) a good solution. This,obviously reduces the performances of the algorithms. Several strate-gies have been suggested and investigated. Some recent ones includerestarting the algorithms using quality points, switching between vari-ants of these algorithms and embedding interpolation and extrapola-tion processes within these Lanzcos-type algorithms. The Embeddingof Interpolation and Extrapolation Methods within Lanczos-type Al-gorithms or EIEMLA approach and its modification (or MEIEMLA),as well as the hybrid restarting-MEIEMLA are some of the lateststrategies introduced to combat breakdown and improve the conver-gence rates of Lanczos-type algorithms. All of those methods werereviewed in this paper, put them into context and highlights their ad-vantages and disadvantages to direct the readers to potential furtherinvestigations. Beside, we introduce the use of support vector ma-chine (SVR) in similarly purposes as EIEMLA and MEIEMLA.Thecomparison of the methods is on efficiency and robustness. Numerical results are included