Block versions of the Jacobi preconditioner can be derived by a
partitioning of the variables. If the index set
is partitioned as with the sets mutually
disjoint, then
The preconditioner is now a block-diagonal matrix.
Often, natural choices for the partitioning suggest themselves:
In problems with multiple physical variables per node,
blocks can be formed by grouping the equations per node.
In structured matrices, such as those from partial differential
equations on regular grids, a partitioning can be based on the
physical domain. Examples are a partitioning along lines in the
2D case, or planes in the 3D case. This will be discussed
further in §.
On parallel computers it is natural to let the
partitioning coincide with the division of variables
over the processors.
is partitioned as 
with the sets 
mutually
disjoint, then
.