Iterative Schemes for Large Linear Systems

Abstract

Sparse linear systems occur in a multitude of applications in computational science and engineering. While the need for solving such linear systems is most prevalent in numerical simulations using mathematical models based on differential equations, it also arises in other areas. For example, the PageRank vector that was used by Google to order the nodes of a network based on its link structure can be interpreted as the solution of a linear system of order equal to the number of nodes (Brin and Page, The anatomy of a large-scale hypertextual web search engine. In: Proceedings of 7th Intnational Conference World Wide Web, (1998), Langville and Meyer, Google’s PageRank and Beyond: The Science of Search Engine Rankings, (2006) [1, 2]). The system can be very large and sparse so that parallel iterative methods become necessary; see e.g. (Gleich et al. Fast parallel pagerank: a linear system approach, (2004), Gleich et al. SIAM J. Sci. Comput. 32(1), (2010) [3, 4]). We note that we will not describe any parallel asynchronous iterative methods that are sometimes proposed for solving such systems when they are very large, possibly distributed across several computers that are connected by means of a relatively slow network; cf. (Bahi et al. Parallel Iterative Algorithms, (2008), Bertsekas and Tsitsiklis, Parallel and Distributed Computation, (1989), Kollias et al. PARCO, (2005), Ishii and Tempo, IEEE Trans. Autom. Control, 55(9), (2010)[5, 6, 7, 8]).