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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 [55]) solving a
systems means solving
smaller systems with the matrices. For the second type (which
was discussed by Meurant [152], Axelsson and
Polman [20] and Axelsson and
Eijkhout [15]) solving
a system with
entails multiplying by the
blocks.
Therefore, the second type has a much higher potential for
vectorizability.