The basic operation to be performed is given by A x = y .
, x and y distributed like vectors:
For this case,
assume that x and y are identically distributed according to
the inducing vector distribution that induced the distribution of matrix A .
Notice that by spreading vector x within columns, we duplicate
all necessary elements of x so that local matrix vector multiplication
can commence on each node. After this, a reduction (summation) within rows of nodes
of the local partial results yields the desired vector y .
However, since only a portion of vector y needs to be known to
each node, a
distributed reduction (MPI_Reduce_scatter)
within rows of nodes suffices. This process is illustrated
in Figure 1.5.
In this figure, the matrix 
denotes the sub-matrix
of A assigned to node (i,j) .
In general,
After spreading the sub-vectors of x within columns of nodes, node (i,j) holds the following sub-vectors:
Thus, all sub-vectors of x required for the
local matrix-vector multiply are in place.
After executing the local matrix-vector multiply,
each node owns a local contribution to part of y ,
so that
a summation of the results within rows of nodes
completes the matrix-vector multiply,
leaving the appropriate piece of
the result vector on each node,
We will see that this summation within one dimension of the mesh
becomes a basic operation in PLAPACK, in Chapter 
.
, matrix row x and matrix column y :
Again, we wish to perform A x = y , but this time we
assume that x and y are a row and column of a matrix, respectively,
where the distribution of that matrix is induced by the same inducing vector distribution as that of matrix A .
Notice that by spreading (broadcasting) matrix row x within columns, we duplicate
all necessary elements of x so that local matrix vector multiplication
can commence on each node. After this, a summation within rows of nodes
of the local partial results yields the desired vector y .
Since y is a column, existing on only one column of nodes, a
summation to one node (MPI_Reduce) within each row of nodes can be utilized.
, matrix column x and matrix row y :
Now we
assume that x and y are a column and row of a matrix, respectively,
where the distribution of that matrix is induced by the same inducing vector distribution as that of matrix A .
Notice that by spreading matrix column x within rows of nodes, we duplicate
all necessary elements of x so that local matrix vector multiplication
can commence on each node. After this, a summation within rows of nodes
(MPI_Reduce_scatter) must occur,
leaving the result distributed like the inducing vector.
The final operation
is to redistribute (gather) the result to the row of the target matrix.