Parallel approximation of the exponential of Hermitian matrices

TL;DR

This paper introduces a parallel algorithm for computing the exponential of Hermitian matrices using rational approximation and partial fraction decomposition, emphasizing efficiency and error analysis.

Contribution

It presents a novel parallelizable method for Hermitian matrix exponentials based on rational approximation and linear system resolutions, with comprehensive error and performance analysis.

Findings

The method is efficient and scalable for large matrices.

Error analysis shows robustness against rounding errors.

Numerical tests outperform Krylov algorithms in certain cases.

Abstract

In this work, we consider a rational approximation of the exponential function to design an algorithm for computing matrix exponential in the Hermitian case. Using partial fraction decomposition, we obtain a parallelizable method, where the computation reduces to independent resolutions of linear systems. We analyze the effects of rounding errors on the accuracy of our algorithm. We complete this work with numerical tests showing the efficiency of our method and a comparison of its performances with Krylov algorithms.