We propose a quadrature-based formula for computing the exponential function of matrices with a non-oscillatory integral on an infinite interval and an oscillatory integral on a finite interval. In the literature, existing quadrature-based formulas are based on the inverse Laplace transform or the Fourier transform. We show these expressions are essentially equivalent in terms of complex integrals and choose the former as a starting point to reduce computational cost. By choosing a simple integral path, we derive an integral expression mentioned above. Then, we can easily apply the double-exponential formula and the Gauss-Legendre formula, which have rigorous error bounds. As numerical experiments show, the proposed formula outperforms the existing formulas when the imaginary parts of the eigenvalues of matrices have large absolute values.

翻译：本文提出了一种基于数值积分的公式，用于计算矩阵指数函数。该公式包含一个无限区间上的非振荡积分和一个有限区间上的振荡积分。现有文献中基于数值积分的公式主要建立在逆拉普拉斯变换或傅里叶变换的基础上。我们证明这些表达式在复积分意义下本质等价，并选择前者作为计算起点以降低计算成本。通过选取简单的积分路径，我们推导出上述积分表达式。随后，可以方便地应用具有严格误差界的双指数公式和高斯-勒让德公式进行数值计算。数值实验表明，当矩阵特征值的虚部绝对值较大时，所提公式的计算性能优于现有方法。