Inner and Outer Iterations for the Chebyshev Algorithm

We analyze the Chebyshev iteration in which the linear system involving the splitting matrix is solved inexactly by an inner iteration. We assume that the tolerance for the inner iteration may change from one outer iteration to the other. When the tolerance converges to zero, the asymptotic convergence rate is unaaected. Motivated by this result, we seek the sequence of tolerance values that yields the lowest cost. We nd that among all sequences of slowly varying tolerances, a constant one is optimal. Numerical calculations that verify our results are shown. Our analysis is based on asymptotic methods, such as the W.K.B method, for linear recurrence equations and an estimate of the accuracy of the resulting asymptotic result.