A note on conjugate gradient convergence

The one-dimensional discrete Poisson equation on a uniform grid with n points produces a linear system of equations with a symmetric, positive-definite coefficient matrix. Hence, the conjugate gradient method can be used, and standard analysis gives an upper bound of O(n) on the number of iterations required for convergence. This paper introduces a systematically defined set of solutions dependent on a parameter beta, and for several values of beta, presents exact analytic expressions for the number of steps k(beta, tau, n) needed to achieve accuracy tau. The asymptotic behavior of these expressions has the form O(n(alpha)) as n --> infinity and O(tau gamma) as tau --> 0. In particular, two choices of beta corresponding to nonsmooth solutions give alpha = 0, i.e., iteration counts independent of n; this is in contrast to the standard bounds. The standard asymptotic convergence behavior, alpha = 1, is seen for a relatively smooth solution. Numerical examples illustrate and supplement the analysis.