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One reason that block methods are of interest is that they are potentially more suitable for vector computers and parallel architectures. Consider the block factorization
where is the block diagonal matrix of pivot blocks.
Making the transition to an incomplete factorization we can replace the diagonal of pivots by either the diagonal of incomplete factorization pivots , or the inverse of , the diagonal of approximations to the inverses of the pivots. In the first case we find for the incomplete factorization
and in the second case
We see that for factorizations of the first type (which covers all methods in Concus, Golub and Meurant ) solving a systems means solving smaller systems with the matrices. For the second type (which was discussed by Meurant , Axelsson and Polman  and Axelsson and Eijkhout ) solving a system with entails multiplying by the blocks. Therefore, the second type has a much higher potential for vectorizability.