A new algorithm for the maximum a posteriori (MAP) decoding of linear block codes is presented. The proposed algorithm can be regarded as a conventional BCJR algorithm for a section trellis diagram, where branch metrics of the trellis are computed by the recursive MAP algorithm proposed by the authors. The decoding complexity of the proposed algorithm depends on the sectionalization of the trellis. A systematic way to find the optimum sectionalization which minimizes the complexity is also presented. Since the algorithm can be regarded as a generalization of both of the BCJR and the recursive MAP algorithms, the complexity of the proposed algorithm cannot be larger than those algorithms, as far as the sectionalization is chosen appropriately.