A case study in mechanically deriving dense linear algebra code

Design by Transformation (DxT) is a top-down approach to mechanically derive high-performance algorithms for dense linear algebra. We use DxT to derive the implementation of a representative matrix operation, twosided Trmm. We start with a knowledge base of transformations that were encoded for a simpler set of operations, the level-3 BLAS, and add only a few transformations to accommodate the more complex twosided Trmm. These additions explode the search space of our prototype system, DxTer, requiring the novel techniques defined in this paper to eliminate large segments of the search space that contain suboptimal algorithms. Performance results for the mechanically optimized implementations on 8,192 cores of a BlueGene/P architecture are given.