Abstract This paper investigates the performance of the numerical inverse Laplace transformation (ILT) method based on concentrated matrix exponential (CME) distributions, referred to as the CME method. The CME method does not generate overshoot and undershoot (i.e., avoids Gibbs oscillation), preserves monotonicity of functions, its accuracy is gradually improving with the order, and it is numerically stable even for order 1000 when using machine precision arithmetic, while other methods get unstable already for order 100 using the same arithmetic. For ILT based tail approximation, the paper recommends an abscissa shifting approach which improves the accuracy of most ILT methods and proposes a heuristic procedure to approximate the numerical accuracy of some ILT methods.

中文翻译：

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使用集中矩阵指数分布的数值逆拉普拉斯变换

摘要 本文研究了基于集中矩阵指数 (CME) 分布的数值逆拉普拉斯变换 (ILT) 方法的性能，称为 CME 方法。CME 方法不产生过冲和下冲（即避免 Gibbs 振荡），保持函数的单调性，其精度随着阶数逐渐提高，在使用机器精度算法时即使对于 1000 阶数也数值稳定，而其他方法得到使用相同的算法对于 100 阶已经不稳定。对于基于 ILT 的尾部逼近，本文推荐了一种横坐标移动方法，该方法提高了大多数 ILT 方法的精度，并提出了一种启发式程序来逼近某些 ILT 方法的数值精度。