Algorithm, Architecture, and Floating-Point Unit Codesign of a Matrix Factorization Accelerator

This paper examines the mapping of algorithms encountered when solving dense linear systems and linear least-squares problems to a custom Linear Algebra Processor. Specifically, the focus is on Cholesky, LU (with partial pivoting), and QR factorizations and their blocked algorithms. As part of the study, we expose the benefits of redesigning floating point units and their surrounding data-paths to support these complicated operations. We show how adding moderate complexity to the architecture greatly alleviates complexities in the algorithm. We study design tradeoffs and the effectiveness of architectural modifications to demonstrate that we can improve power and performance efficiency to a level that can otherwise only be expected of full-custom ASIC designs. A feasibility study of inner kernels is extended to blocked level and shows that, at block level, the Linear Algebra Core (LAC) can achieve high efficiencies with up to 45 GFLOPS/W for both Cholesky and LU factorization, and over 35 GFLOPS/W for QR factorization. While maintaining such efficiencies, our extensions to the MAC units can achieve up to 10, 12, and 20 percent speedup for the blocked algorithms of Cholesky, LU, and QR factorization, respectively.