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## The Block Column and Row Distributions

ScaLAPACK assumes a one-dimensional block distribution for the band and tridiagonal routines. The block distribution is used when the computational load is distributed homogeneously over the global data. This distribution leads to a highly efficient implementation of the divide-and-conquer algorithms used in ScaLAPACK.

For convenience we will number the processes from 0 to P-1, and the matrix rows from 1 to M and the matrix columns from 1 to N. Figure 4.8 shows the two data layouts used in ScaLAPACK for solving narrow band linear systems. In all cases, each submatrix is labeled with the number of the process that contains it. Process 0 owns the shaded submatrices.

Consider the layout illustrated on the left of figure 4.8, the one-dimensional block column distribution. This distribution

Figure 4.8: The one-dimensional block-column and block-row distributions

assigns a block of NB contiguous columns of a matrix to successive processes arranged in a one-dimensional process grid. Each process receives at most one block of columns of the matrix, i.e., . Column k is stored on process . The maximum number of columns stored per process is given by . In the figure M=N=16 and P=4. This distribution assigns blocks of columns of size NB to successive processes. If the value of P evenly divides the value of N and NB = N / P, then each process owns a block of equal size. However, if this is not the case, then either the last process to receive a portion of the matrix will receive a smaller block than other processes, or some processes may receive an empty portion of the matrix. The transpose of this layout, the one-dimensional block-row distribution, is shown on the right of figure 4.8.

The block-column distribution scheme is the data layout that is used in the ScaLAPACK library for the coefficient matrix of the narrow band and tridiagonal solvers.

The block-row distribution scheme is the data layout that is used in the ScaLAPACK library for the right-hand-side matrix of the narrow band and tridiagonal solvers.

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