Work-Efficient Matrix Inversion in Polylogarithmic Time

Abstract

We present an algorithm for inversion of symmetric positive definite matrices 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.