A complete pair of solvents of a quadratic matrix pencil
<|reference_start|>A complete pair of solvents of a quadratic matrix pencil: 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.<|reference_end|>
@article{kurbatov2024a,
    title={A complete pair of solvents of a quadratic matrix pencil},
    author={V. G. Kurbatov and I. V. Kurbatova},
    journal={arXiv preprint arXiv:2405.07210},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.07210},
    primaryClass={math.NA cs.NA math.DS math.FA math.SP}
}