Solution of Infinite Linear Systems by Automatic Adaptive Iterations

The problem of approximating the solution of infinite linear systems finitely expressed by a sparse coefficient matrix in block Hessenberg form is considered. The convergence of the solutions of a sequence of truncated problems to the infinite problem solution is investigated. A family of algorithms, some of which are adaptive, is introduced, based on the application of the Gauss-Seidel method to a sequence of truncated problems of increasing size $n_i$ with non-increasing tolerance $10^{-t_i}$. These algorithms do not require special structural properties of the coefficient matrix and they differ in the way the sequences $n_i$ and $t_i$ are generated. The testing has been performed on both infinite problems arising from the discretization of elliptical equations on unbounded domains and stochastic problems arising from queueing theory. Extensive numerical experiments permit the evaluation of the various strategies and suggest that the best trade-off between accuracy and computational cost is reached by some of the adaptive algorithms.