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In this paper, we present several new and generalized parallel dense matrix multiplication algorithms of the form C = AB + C on two-dimensional process grid topologies. These algorithms can deal with rectangular matrices distributed on rectangular grids. We classify these algorithms coherently into three categories according to the communication primitives used and thus we o er a taxonomy for this family of related algorithms. All these algorithms are represented in the data distribution independent approach and thus do not require a speci c data distribution for correctness. The algorithmic compatibility condition result shown here ensures the correctness of the matrix multiplication. We de ne and extend the data distribution functions and introduce permutation compatibility and algorithmic compatibility. We also discuss a permutation compatible data distribution (modi ed virtual 2D data distribution). We conclude that no single algorithm always achieves the best performance on di erent matrix and grid shapes. A practical approach to resolve this dilemma is to use poly-algorithms. We analyze the characteristics of each of these matrix multiplication algorithms and provide initial heuristics for using the poly-algorithm. All these matrix multiplication algorithms have been tested on the IBM SP2 system. The experimental results are presented in order to demonstrate their relative performance characteristics, motivating the combined value of the taxonomy and new algorithms introduced here. Revision and extension of our original paper entitled \The Multicomputer Toolbox Approach to Concurrent BLAS" submitted September 1, 1993. Address correspondence to: Mississippi State University, Engineering Research Center, PO Box 6176, Mississippi State, MS 39762. Tel: 601-325-8435. Email: tony@cs.msstate.edu.

A poly-algorithm for parallel dense matrix multiplication on two-dimensional process grid topologies