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A new approximation of the cluster variational method is introduced for thethree-dimensional Ising model on the simple cubic lattice. The maximal clusteris, as far as we know, the largest ever used in this method. A message-passingalgorithm, generalized belief propagation, is used to minimize the variationalfree energy. Convergence properties and performance of the algorithm areinvestigated. The approximation is used to compute the spontaneousmagnetization, which is then compared to previous results. Using the presentresults as the last step in a sequence of three cluster variationalapproximations, an extrapolation is obtained which captures the leadingcritical behaviour with a good accuracy.

Generalized belief propagation for the magnetization of the simple cubic Ising model