In this paper, we study the relative error in the numerical solution of a linear ordinary differential equation y '( t ) = Ay ( t ), t ≥ 0, where A is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g., a polynomial or a rational approximation. The error of the numerical solution with respect to the exact solution is due to this approximation as well as to a possible perturbation in the initial value. For an unperturbed initial value, we have found: (1) unlike the absolute error, the relative error always grows linearly in time; (2) in the long-time, the contributions to the relative error relevant to non-rightmost eigenvalues of A disappear.

中文翻译：

---

数值积分矩阵指数近似的相对误差分析

本文研究线性常微分方程y'(t)=Ay(t)，t≥0的数值解的相对误差，其中A为正规矩阵。通过在任何步骤使用矩阵指数的近似，例如多项式或有理近似，来获得数值解。数值解相对于精确解的误差是由于这种近似以及初始值的可能扰动。对于未扰动的初始值，我们发现： (1) 与绝对误差不同，相对误差总是随时间线性增长；(2) 在很长一段时间内，A 的非最右特征值相关的相对误差的贡献消失了。