On the Convergence of Iterative Simulation-Based Methods for Singular Linear Systems

We consider simulation-based algorithms for linear systems of equations, Ax = b, where A is singular. The convergence properties of iterative solution methods can be impaired when the methods are implemented with simulation, as is often done in important classes of large-scale problems. We focus on special cases of singular systems, including some arising in approximate dynamic programming, where convergence of the residual sequence may be obtained without a stabilization mechanism, while the sequence of iterates may diverge. For some of these special cases, under additional assumptions, we show that the sequence is guaranteed to converge. For situations where the sequence of iterates diverges, we propose schemes for extracting from the divergent sequence another sequence that converges to a solution of Ax = b. LIDS Report 2879, December 2011 (Revised March 2012) Massachusetts Institute of Technology, Cambridge, MA Laboratory for Information and Decision Systems