Abstract: The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric, large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value methods. In this paper, we propose a new circulant preconditioner to speed up the convergence rate of the GMRES method, which is a convex linear combination of P-circulant and Strang-type circulant preconditioners. Theoretical and practical arguments are given to show that this preconditioner is feasible and effective in some cases. 

Key words: circulant preconditioner, boundary value method, ordinary differential equation