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Matrix-vector products

The matrix--vector products are often easily parallelized on shared-memory machines by splitting the matrix in strips corresponding to the vector segments. Each processor then computes the matrix--vector product of one strip. For distributed-memory machines, there may be a problem if each processor has only a segment of the vector in its memory. Depending on the bandwidth of the matrix, we may need communication for other elements of the vector, which may lead to communication bottlenecks. However, many sparse matrix problems arise from a network in which only nearby nodes are connected. For example, matrices stemming from finite difference or finite element problems typically involve only local connections: matrix element is nonzero only if variables and are physically close. In such a case, it seems natural to subdivide the network, or grid, into suitable blocks and to distribute them over the processors. When computing , each processor requires the values of at some nodes in neighboring blocks. If the number of connections to these neighboring blocks is small compared to the number of internal nodes, then the communication time can be overlapped with computational work. For more detailed discussions on implementation aspects for distributed memory systems, see de Sturler [60] and Pommerell [170].