On Meinardus' examples for the conjugate gradient method

Abstract

. The conjugate gradient (CG) method is widely used to solve a positive definite linear system Ax = b of order N. It is well known that the relative residual of the kth approximate solution by CG (with the initial approximation x 0 = 0) is bounded above by 2[∇ k k +∇ -k k ] -1 with ∇k=√k+1 √k-1 where K = K ( A) = ∥A∥ 2 ∥A -1 ∥ 2 is A's spectral condition number. In 1963, Meinardus (Numer. Math., 5 (1963), pp. 14-23) gave an example to achieve this bound for k = N - 1 but without saying anything about all other 1 < k < N - 1. This very example can be used to show that the bound is sharp for any given k by constructing examples to attain the bound, but such examples depend on k and for them the (k + 1)th residual is exactly zero. Therefore it would be interesting to know if there is any example on which the CG relative residuals are comparable to the bound for all 1 ≤ k < N - 1. There are two contributions in this paper: (1) A closed formula for the CG residuals for all 1 ≤ k < N- 1 on Meinardus' example is obtained, and in particular it implies that the bound is always within a factor of -√2 of the actual residuals; (2) A complete characterization of extreme positive linear systems for which the kth CG residual achieves the bound is also presented.