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.

翻译：本文考虑利用数值求积方法计算矩阵指数 $\mathrm{e}^A$。尽管已有多种基于求积的算法被提出，但它们主要针对（近）埃尔米特矩阵。为处理非埃尔米特矩阵，我们采用包含振荡项的另一种积分表示，并应用专门针对傅里叶积分的双指数（DE）公式。DE公式将给定积分转化为区间无穷的另一个积分，因此必须截断无穷区间。为利用DE公式，本文分析了截断误差并提出两种算法：第一种使用固定网格尺寸（DE公式中影响精度的参数）近似 $\mathrm{e}^A$；第二种在第一种算法基础上，根据给定的误差容限自动选择网格尺寸计算 $\mathrm{e}^A$。