We propose a block ILU factorization technique for block tridiagonal matrices that need not necessarily be M-matrices. The technique explores reduction by a coarse-vector restriction of the block size of the approximate Schur complements computed throughout the factorization process. Then on the basis of the Sherman-Morrison-Woodbury formula these are easily inverted. We prove the existence of the proposed factorization techniques in the case of (nonsymmetric, in general) M-matrices. For block

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... ridiagonal matrices with positive definite symmetric part we show the existence of a limit version of the factorization (exact inverses of the reduced matrices are needed). The theory is illustrated with numerical tests.