An adaptive blocking strategy for matrix factorizations

Abstract

On most high-performance architectures, data movement is slow compared to floating-point (in particular, vector) performance. On these architectures block algorithms have been successful for matrix computations. By considering a matrix as a collection of submatrices (the so-called blocks) one naturally arrives at algorithms that require little data movement. The optimal blocking strategy, however, depends on the computing environment and on the problem parameters. Current approaches use fixed-width blocking strategies that are not optimal. This paper presents an “adaptive blocking” methodology for determining in a systematic manner an optimal blocking strategy for a uniprocessor machine. We demonstrate this technique on a block QR factorization routine on a uniprocessor. After generating timing models for the high-level kernels of the algorithm we can formulate the optimal blocking strategy in a recurrence relation that we can solve inexpensively with a dynamic programming technique. Experiments on one processor of a CRAY-2 show that in fact the resulting blocking strategy is as good as any fixed-width blocking strategy. So while we do not know the optimum fixed-width blocking strategy unless we re-run the same problem several times, adaptive blocking provides optimum performance in the very first run.

This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38