The Convergence Rate of Block Preconditioned Systems Arising from Lmf-based Ode Codes

The solution of ordinary and partial differential equations using implicit linear multistep formulas (LMF) is considered. More precisely, boundary value methods (BVMs), a class of methods based on implicit formulas will be taken into account in this paper. These methods require the solution of large and sparse linear systems M̂x = b. Block-circulant preconditioners have been proposed to solve these linear systems. By investigating the spectral condition number of M̂ , we show that the conjugate gradient method, when applied to solving the normalized preconditioned system, converges in at most O(log s) steps, where the integration step size is O(1/s). Numerical results are given to illustrate the effectiveness of the analysis. AMS subject classification: 65F10, 65N22.