A Stable Scaling of Newton-schulz for Improving the Sign Function Computation of a Hermitian Matrix

The Newton-Schulz iteration is a quadratically convergent, inversion-free method for computing the sign function of a matrix. It is advantageous over other methods for high-performance computing because it is rich in matrix-matrix multiplications. In this paper we propose a variant for Hermitian matrices that improves the initially slow convergence of the iteration. The main idea is to scale the iteration to have steeper derivatives of the mapping function at the origin such that the convergence of the eigenvalues with small magnitudes is accelerated. The scaling is stable based on a backward stability result of Y. Nakatsukasa and N. J. Higham. Generally, the number of iterations is reduced by half compared with standard Newton-Schulz. With proper shifts of the matrix, this number may be further reduced. We demonstrate numerical calculations with matrices of size up to approximately 105 on medium-sized computing clusters and also apply the algorithm to electronic structure calculations.