A Probabilistic Analysis of the Neumann Series Iteration

Given a random matrix A with eigenvalues between -1 and 1, we analyze the number of iterations needed to solve the linear equation (I-A)x=b with the Neumann series iteration. We give sufficient conditions for convergence of an upper bound of the iteration count in distribution. Specifically, our results show that when the scaled extreme eigenvalues of A converge in distribution, this scaled upper bound on the number of iterations will converge to the reciprocal of the limiting distribution of the largest eigenvalue.