Making Strassen Matrix Multiplication Safe

Strassen's recursive algorithm for matrix-matrix multiplication has seen slow adoption in practical applications despite being asymptotically faster than the traditional algorithm. A primary cause for this is the comparatively weaker numerical stability of its results. Techniques that aim to improve the errors of Strassen stand the risk of losing any potential performance gain. Moreover, current methods of evaluating such techniques for safety are overly pessimistic or error prone and generally do not allow for quick and accurate comparisons. In this paper we present an efficient technique to obtain rigorous error bounds for floating point computations based on an implementation of unum arithmetic. Using it, we evaluate three techniques - exact dot product, fused multiply-add, and matrix quadrant rotation - that can potentially improve the numerical stability of Strassen's algorithm for practical use. We also propose a novel error-based heuristic rotation scheme for matrix quadrant rotation. Finally we apply techniques that improve numerical safety with low overhead to a LINPACK linear solver to demonstrate the usefulness of the Strassen algorithm in practice.