A complete pair of solvents of a quadratic matrix pencil
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929340879470592 |
|---|---|
| author | Kurbatov, V. G. Kurbatova, I. V. |
| author_facet | Kurbatov, V. G. Kurbatova, I. V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07210 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A complete pair of solvents of a quadratic matrix pencil Kurbatov, V. G. Kurbatova, I. V. Numerical Analysis Dynamical Systems Functional Analysis Spectral Theory 65F60, 15A69, 46B28, 30E10, 97N50 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. |
| title | A complete pair of solvents of a quadratic matrix pencil |
| topic | Numerical Analysis Dynamical Systems Functional Analysis Spectral Theory 65F60, 15A69, 46B28, 30E10, 97N50 |
| url | https://arxiv.org/abs/2405.07210 |