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## The Block Mapping

The one-dimensional distribution scheme is a mapping of a set of blocks onto the processes. The previous section informally described this mapping as well as some of its properties. To be complete, we shall describe the precise mapping that associates to a matrix entry identified by its global indexes the coordinates of the process that owns it and its local position within that process's memory.

Suppose we have a two dimensional array A of size to be distributed on a process grid in a block-column fashion. By convention, the array columns are numbered 1 through N and the processes are numbered 0 through P-1. First, the array is divided into contiguous blocks of NB columns with . When NB does not divide N evenly, the last block of columns will only contain columns instead of NB. By convention, these blocks are numbered starting from zero and dealt out to the processes. In other words, if we assume that the process 0 receives the first block, the block is assigned to the process of coordinate (0,p). The mapping of a column of the array globally indexed by J is defined by the following analytical equation:
where J is a global column index in the array, p is the column coordinate of the process owning that column, and finally x is the column coordinate within that block of columns where the global array column of index J is to be found. It is then fairly easy to establish the analytical relationship between these variables. One obtains:
These equations allow to determine the local information, i.e. the local index x as well as the process column coordinate p corresponding to a global column identified by its global index J and conversely. Table 4.9 illustrates this mapping layout when P=2 and N=16 and NB=8. At most one block is assigned to each process.

Table 4.9: One-dimensional block-column mapping example for P=2 and N=16

This example of the one-dimensional block-column distribution mapping can be expressed in HPF by using the following statements:

          REAL :: A( M, N )
    !HPF$ PROCESSORS PROC( 1, P )
    !HPF$ DISTRIBUTE A( *, BLOCK( NB ) ) ONTO PROC

A similar example of block-row distribution can easily be constructed. For an array B, such an example can be expressed in HPF by using the following statements:

          REAL :: B( N, NRHS )
    !HPF$ PROCESSORS PROC( P, 1 )
    !HPF$ DISTRIBUTE B( BLOCK( NB ), * ) ONTO PROC

There is in fact no real reason to always deal out the blocks starting with the process 0. In fact, it is sometimes useful to start the data distribution with the process of arbitrary coordinate SRC, in which case Equation 4.3 becomes:

Table 4.10 illustrates Equation 4.4 for the block-cyclic layout when , , and .

Table 4.10: One-dimensional block-column mapping for P=2, SRC=1, N=16 and NB=8

This example of the one-dimensional block-column distribution mapping can be expressed in HPF by using the following statements:

          REAL :: A( M, N )
    !HPF$ PROCESSORS PROC( 1, P )
    !HPF$ TEMPLATE T( M, N + P*NB )
    !HPF$ DISTRIBUTE T( *, BLOCK( NB ) ) ONTO PROC
    !HPF$ ALIGN A( I, J ) WITH T( I, SRC*NB + J )

A similar example of block-row distribution can easily be constructed. For an array B, such an example can be expressed in HPF by using the following statements:

          REAL :: B( N, NRHS )
    !HPF$ PROCESSORS PROC( P, 1 )
    !HPF$ TEMPLATE T( N + P*NB, NRHS )
    !HPF$ DISTRIBUTE T( BLOCK( NB ), * ) ONTO PROC
    !HPF$ ALIGN A( I, J ) WITH T( SRC*NB + I, J )

In ScaLAPACK, the local storage convention of the one-dimensional block distributed matrix in every process's memory is assumed to be Fortran-like, that is, ``column major'' .

Determining the number of rows or columns of a global band matrix that a specific process receives is an essential task for the user. The notation LOC() is used for block-row distributions and LOC() is used for block-column distributions. These local quantities occur throughout the leading comments of the source code, and are reflected in the sample argument description in section 4.4.7.

For block distribution, a matrix can be distributed unevenly. More specifically, one process in the process grid can receive an array that is smaller than other processes. It is also possible that some processes receive no data. For further information on one-dimensional block-column or block-row data distribution, please refer to section 4.4.1.

Block-Column Distribution: LOC(N_A) denotes the number of columns that a process would receive if N_A columns of a matrix is distributed over columns of its process row.

For example, let us assume that the coefficient matrix A is band symmetric of order N and has been block-column distributed on a process grid.

In the ideal case where the matrix is evenly distributed to all processes in the process grid, and . Thus, each process receives a block of size of the matrix A. Therefore,

LOC(N_A) = NB_A.

However, if , at least one of the processes in the process grid will receive a block of size smaller than . Thus,

     if (  and  ) then

       		 processes (0,0), ... , (0,K-1) receive

       		           		 LOC(N_A) = NB_A

       		 and process (0,K) receives

       		           		 LOC(N_A) = N_A - K  NB_A.

       		 if  then processes  do
          not receive any data.

    end if

Block-Row Distribution: LOC(M_B) denotes the number of rows that a process would receive if M_B rows of a matrix is distributed over rows of its process column.

Let us assume that the N-by-NRHS right-hand-side matrix B has been block-row distributed on a process grid.

In the ideal case where the matrix is evenly distributed to all processes in the process grid, and . Thus, each process receives a block of size of the matrix B. Therefore,

LOC(M_B) = MB_B.

However, if , then at least one of the processes in the process grid will receive a block of size smaller than . Thus,

     if (  and  ) then

       		 processes t<ex2html_image_mark>#tex2html_wrap_inline15295# receive

       		           		 LOC(M_B) = MB_B

       		 and process (K,0) receives

       		           		 LOC(M_B) = M_B - K  MB_B.

       		 if  then processes  do
          not receive any data.

    end if

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