Combining building blocks for parallel multi-level matrix multiplication

EXTENDED ABSTRACT Matrix-matrix multiplication is one of the core computations in many algorithms from scientific computing or numerical analysis and many efficient realizations have been invented over the years, including many parallel ones. The current trend to use clusters of PCs or SMPs for scientific computing suggests to revisit matrix-matrix multiplication and investigate efficiency and scalability of different versions on clusters. In this talk we present parallel algorithms for matrix-matrix multiplication which are built up from several algorithms in a multilevel structure. On a single processor ATLAS or PHiPAC create very efficient implementations by adjusting the computation order to the specific memory hierarchy and by exploiting functional parallelism of the processor (SSE2). Parallel approaches include many methods based on decomposition like Cannon's algorithm, or the algorithm of Fox. Efficient implementation variants of the latter are SUMMA or PUMMA. Matrix-matrix multiplication by Strassen or Strassen-Winograd benefits from a reduced number of operations but require a special schedule for a parallel implementation. In the context of clusters of SMPs, mixed programming models like mixed task and data parallelism are important since efficiency and scalability can be improved by using multiprocessor tasks (M-tasks). Task parallel implementations or mixed matrix-matrix multiplications have already been proposed in literature. One possibility of parallelizing Strassen's algorithm is to distribute the seven intermediate results onto a group of processors of size 7 i , preferably in a ring or torus configuration. Other approaches include to mix the common Fox BMR method (broadcast multiply roll) with Strassen's algorithm. Another mixed parallel