Work-efficient matrix inversion in polylogarithmic time

Abstract

We present an algorithm for matrix inversion that combines the practical requirement of an optimal number of arithmetic operations and the theoretical goal of a polylogarithmic critical path length. The algorithm reduces inversion to matrix multiplication. It uses Strassen's recursion scheme but on the critical path, it breaks the recursion early switching to an asymptotically inefficient yet fast use of Newton's method. We also show that the algorithm is numerically stable. Overall, we get a candidate for a massively parallel algorithm that scales to exascale systems even on relatively small inputs. Preliminary experiments on multicore machines give the surprising result that even on such moderately parallel machines the algorithm outperforms Intel's Math Kernel Library and that Strassen's algorithm seems to be numerically more stable than one might expect.