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,
and
, 
are the block lower and upper triangle of the factorization;
they coincide with 
, 
in the case of a block tridiagonal
matrix.
We can turn this into an incomplete factorization by replacing the
block diagonal matrix of pivots by the block diagonal matrix of
incomplete factorization pivots 
, giving
For factorizations of this type (which covers all
methods in Concus, Golub and Meurant [57] and
Kolotilina and Yeremin [141]) solving
a linear system means solving
smaller systems with the 
matrices.
Alternatively, we can replace by a
the inverse of the block diagonal matrix of the
approximations to the inverses of the pivots, 
, giving
For this second type (which
was discussed by Meurant [155], Axelsson and
Polman [21] 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. Unfortunately, such a factorization is theoretically
more troublesome; see the above references or Eijkhout and
Vassilevski [90].