Computing the matrix exponential with the double exponential formula

TL;DR

This paper introduces a novel method for computing the matrix exponential of non-Hermitian matrices using the double exponential formula for Fourier integrals, including error analysis and adaptive algorithms.

Contribution

It develops two algorithms utilizing the DE formula for matrix exponential computation, addressing non-Hermitian cases and automatic parameter selection.

Findings

Analyzes truncation error for the DE formula in matrix exponential computation.

Proposes fixed mesh size and adaptive algorithms for improved accuracy.

Provides theoretical foundation for applying DE formula to oscillatory integrals in matrices.

Abstract

This paper considers the computation of the matrix exponential $\mathrm{e}^A$ with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore it is necessary to truncate the infinite interval. In this paper, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates $\mathrm{e}^A$ with the fixed mesh size which is a parameter in the DE formula affecting the accuracy. Second one computes $\mathrm{e}^A$ based on the first one with automatic selection of the mesh size depending on the given error tolerance.