A mixed precision algorithm for the matrix square root

Abstract

Mixed precision algorithms can significantly enhance the performance of linear algebra solvers by leveraging increasingly powerful low precision hardware while recovering working precision accuracy through, for example, iterative refinement. In this paper, we propose a novel mixed precision algorithm for computing matrix square roots. Our algorithm combines a Schur decomposition approach in low precision with iterative refinement performed through an approximate Newton method. We perform a detailed convergence analysis of the approximate Newton method. For the special case of symmetric positive definite matrices, this analysis implies that one can recover full working precision accuracy under mild conditions. Numerical experiments on x86-64 architectures indicate that our algorithm frequently reduces execution time compared with a fixed working-precision Schur algorithm.