INNER AND OUTER ITERATIONS FOR THE CHEBYSHEV ALGORITHM

We analyze the preconditioned Chebyshev iteration in which at each ste p the linear system involving the preconditioner is solved inexactly b y an inner iteration. We allow the tolerance used in the inner iterati on to decrease from one outer iteration to the next. When the toleranc e converges to zero, the asymptotic convergence rate is the same as fo r the exact method. Motivated by this result, we seek the sequence of tolerance values that yields the lowest cost to achieve a specified ac curacy. We find that among all sequences of slowly varying tolerances, a constant one is optimal. Numerical calculations that verify our res ults are presented. Asymptotic methods, such as the W.K.B. method for linear recurrence equations, are used with an estimate of the accuracy of the asymptotic result.