Since Moulinec & Suquet (1994, 1998) introduced an iterative method based on Fourier transforms to compute the mechanical properties of heterogeneous materials, improved algorithms have been proposed to increase the convergence rate of the scheme. This paper is devoted to the comparison of the accelerated schemes proposed by Eyre & Milton (1999), by Michel et al (2000) and by Monchiet & Bonnet (2012). It shows that the algorithms by Eyre-Milton and by Michel et al are particular cases of Monchiet-Bonnet algorithm, corresponding to particular choices of parameters of the method. An upper bound of the spectral radius of the schemes is determined, which enables to propose sufficient conditions of convergence of the schemes. Conditions are found for minimizing this upper bound. This study shows that the scheme which minimizes this upper bound is the scheme of Eyre & Milton. The paper discusses the choice of the convergence test used in the schemes.