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### The idea behind block factorizations

The starting point for an incomplete block factorization is a partitioning of the matrix, as mentioned in §. Then an incomplete factorization is performed using the matrix blocks as basic entities (see Axelsson [12] and Concus, Golub and Meurant [55] as basic references).

The most important difference with point methods arises in the inversion of the pivot blocks. Whereas inverting a scalar is easily done, in the block case two problems arise. First, inverting the pivot block is likely to be a costly operation. Second, initially all diagonal blocks of the matrix may be sparse and we would like to maintain this type of structure. Hence the need for approximations of inverses arises.

As in the case of incomplete point factorizations, the existence of incomplete block methods is guaranteed if the coefficient matrix is an -matrix. For a general proof, see Axelsson [13].