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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  and Concus, Golub and Meurant  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 the diagonal blocks of the matrix are likely to be be sparse and we would like to maintain this type of structure throughout the factorization. Hence the need for approximations of inverses arises. Figure: Block version of a - factorization

In addition to this, often fill-in in off-diagonal blocks is discarded altogether. Figure describes an incomplete block factorization that is analogous to the - factorization (section ) in that it only updates the diagonal blocks.

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 .

Jack Dongarra
Mon Nov 20 08:52:54 EST 1995