The discrete problem associated with boundary value methods for the solution of initial value problems,which is equivalent to a linear system Gy = f, is a quasi-Toeplitz system when a constant stepsize is used. In this paper we split the matrix G in the form of G = P+E where P is a circulant matrix. In [1],P is used to be the preconditioner of G.I analyze the proconditioners and investigate the spectra of the preconditioned coefficient matrices. It is proved that the matrix P^-1E has the eigenvalue zero with at least (n-2)m and the formula of the nonzeor eigenvalues is given. When the limit n approaches infinite, the nonzeor eigenvalues approaches two curve segments.

References.

[1] Daniele Bertaccini and Michael K. Ng, The Convergence of Block Preconditioned Systems Arising from LMF-based ODE Codes.