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If the sparse matrix is comprised of square dense blocks of nonzeros in some regular pattern, we can modify the CRS (or CCS) format to exploit such block patterns. Block matrices typically arise from the discretization of partial differential equations in which there are several degrees of freedom associated with a grid point. We then partition the matrix in small blocks with a size equal to the number of degrees of freedom, and treat each block as a dense matrix, even though it may have some zeros.
If is the dimension of each block and is the number of nonzero blocks in the matrix , then the total storage needed is . The block dimension of is then defined by .
Similar to the CRS format, we require 3 arrays for the BCRS format: a rectangular array for floating-point numbers ( val(,,)) which stores the nonzero blocks in (block) row-wise fashion, an integer array ( col_ind()) which stores the actual column indices in the original matrix of the () elements of the nonzero blocks, and a pointer array ( row_blk()) whose entries point to the beginning of each block row in val(:,:,:) and col_ind(:). The savings in storage locations and reduction in indirect addressing for BCRS over CRS can be significant for matrices with a large .