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Matrix-vector Multiplication参考
Parallel Programmingin C with MPI and OpenMP Michael J. Quinn Chapter 8 Matrix-vector Multiplication Chapter Objectives Review matrix-vector multiplicaiton Propose replication of vectors Develop three parallel programs, each based on a different data decomposition Outline Sequential algorithm and its complexity Design, analysis, and implementation of three parallel programs Rowwise block striped Columnwise block striped Checkerboard block Sequential Algorithm Storing Vectors Divide vector elements among processes Replicate vector elements Vector replication acceptable because vectors have only n elements, versus n2 elements in matrices Rowwise Block Striped Matrix Partitioning through domain decomposition Primitive task associated with Row of matrix Entire vector Phases of Parallel Algorithm Agglomeration and Mapping Static number of tasks Regular communication pattern (all-gather) Computation time per task is constant Strategy: Agglomerate groups of rows Create one task per MPI process Complexity Analysis Sequential algorithm complexity: ?(n2) Parallel algorithm computational complexity: ?(n2/p) Communication complexity of all-gather: ?(log p + n) Overall complexity: ?(n2/p + log p) Isoefficiency Analysis Sequential time complexity: ?(n2) Only parallel overhead is all-gather When n is large, message transmission time dominates message latency Parallel communication time: ?(n) n2 ? Cpn ? n ? Cp and M(n) = n2 System is not highly scalable Block-to-replicated Transformation MPI_Allgatherv MPI_Allgatherv MPI_Allgatherv in Action Function create_mixed_xfer_arrays First array How many elements contributed by each process Uses utility macro BLOCK_SIZE Second array Starting position of each process’ block Assume blocks in process rank order Function replicate_block_vector Create space for entire vector Create “mixed transfer” arrays Call MPI_Allgatherv Function read_replicated_vector Process p-1 Opens file Reads vector length Broadcast vector length (root process = p-
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