A Circulant Preconditioner for the Systems of LMF-Based ODE Codes

In this paper, a recently introduced block circulant preconditioner for the linear systems of the codes for ordinary differential equations (ODEs) is investigated. Most ODE codes based on implicit formulas, at each integration step, need the solution of one or more unsymmetric linear systems that are often large and sparse. Here, the boundary value methods, a class of implicit methods for the numerical integration of ODEs based on linear multistep formulas, are considered more in detail for initial value problems. Theoretical and practical arguments are given to show that the block circulant preconditioner can give fast preconditioned iterations for various classes of differential problems. Moreover, the P-circulants, a recently introduced circulant approximation for unsymmetric Toeplitz matrices, are shown to be more suitable sometimes than other circulant matrices for the underlying block preconditioner.