Let $B$ and $C$ be square complex matrices. The differential equation \begin{equation*} x''(t)+Bx'(t)+Cx(t)=f(t) \end{equation*} is considered. A solvent is a matrix solution $X$ of the equation $X^2+BX+C=\mathbf0$. A pair of solvents $X$ and $Z$ is called complete if the matrix $X-Z$ is invertible. Knowing a complete pair of solvents $X$ and $Z$ allows us to reduce the solution of the initial value problem to the calculation of two matrix exponentials $e^{Xt}$ and $e^{Zt}$. The problem of finding a complete pair $X$ and $Z$, which leads to small rounding errors in solving the differential equation, is discussed.

翻译：令 $B$ 和 $C$ 为复方阵。考虑微分方程 \begin{equation*} x''(t)+Bx'(t)+Cx(t)=f(t) \end{equation*}。溶剂是方程 $X^2+BX+C=\mathbf0$ 的矩阵解 $X$。若矩阵 $X-Z$ 可逆，则溶剂对 $X$ 与 $Z$ 称为完全对。已知完全溶剂对 $X$ 与 $Z$ 可将初值问题的求解简化为计算两个矩阵指数 $e^{Xt}$ 和 $e^{Zt}$。本文讨论如何寻找使微分方程求解中舍入误差较小的完全对 $X$ 与 $Z$。