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A new family of fast algorithms for batch and adaptive computation of multichannel linear- and affine-phase filters is developed. Both lattice and transversal realizations in ordinary and suitable normalized forms are studied. An important feature of the proposed recursive theme (besides its notable computational advantages) is that it offers a remarkable symmetry and unification of expressions and thus greatly facilitates the implementation process. Moreover, it demonstrates the extensive commonalities between adaptive and batch methods that result from the coincidences of their internal structure.